TPTP Problem File: ITP211_4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP211_4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00607_017823
% 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 : 0024_Assertions_00607_017823 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 8415 (2975 unt;1399 typ; 0 def)
% Number of atoms : 13123 (6093 equ)
% Maximal formula atoms : 16 ( 1 avg)
% Number of connectives : 13155 (1788 ~; 212 |;1078 &)
% (1517 <=>;8560 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Maximal term depth : 37 ( 2 avg)
% Number of FOOLs : 511 ( 317 fml; 194 var)
% Number of X terms : 340 ( 0 []; 330 ite; 10 let)
% Number of types : 11 ( 10 usr)
% Number of type conns : 1378 (1005 >; 373 *; 0 +; 0 <<)
% Number of predicates : 261 ( 258 usr; 2 prp; 0-9 aty)
% Number of functors : 1139 (1139 usr; 43 con; 0-8 aty)
% Number of variables : 27366 (24751 !; 475 ?;27366 :)
% (2140 !>; 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 14:51:34.634
%------------------------------------------------------------------------------
% Could-be-implicit typings (22)
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_Typerep_Otyperep,type,
typerep: $tType ).
tff(ty_t_Assertions_Oassn,type,
assn: $tType ).
tff(ty_t_Sum__Type_Osum,type,
sum_sum: ( $tType * $tType ) > $tType ).
tff(ty_t_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
tff(ty_t_String_Ochar,type,
char: $tType ).
tff(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
tff(ty_t_List_Olist,type,
list: $tType > $tType ).
tff(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
tff(ty_t_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 ).
% Explicit typings (1377)
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_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_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_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
tff(sy_cl_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_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_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_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
tff(sy_cl_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_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[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__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__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ab____,type,
aTP_Lamp_ab:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__af____,type,
aTP_Lamp_af:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ag____,type,
aTP_Lamp_ag:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ah____,type,
aTP_Lamp_ah:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aj____,type,
aTP_Lamp_aj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ak____,type,
aTP_Lamp_ak:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__al____,type,
aTP_Lamp_al:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__am____,type,
aTP_Lamp_am:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__an____,type,
aTP_Lamp_an:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ao____,type,
aTP_Lamp_ao:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ap____,type,
aTP_Lamp_ap:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aq____,type,
aTP_Lamp_aq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ar____,type,
aTP_Lamp_ar:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__as____,type,
aTP_Lamp_as:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__at____,type,
aTP_Lamp_at:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__au____,type,
aTP_Lamp_au:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__av____,type,
aTP_Lamp_av:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aw____,type,
aTP_Lamp_aw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ax____,type,
aTP_Lamp_ax:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ay____,type,
aTP_Lamp_ay:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__az____,type,
aTP_Lamp_az:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ba____,type,
aTP_Lamp_ba:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bd____,type,
aTP_Lamp_bd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__be____,type,
aTP_Lamp_be:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bf____,type,
aTP_Lamp_bf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bg____,type,
aTP_Lamp_bg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bh____,type,
aTP_Lamp_bh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bi____,type,
aTP_Lamp_bi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bj____,type,
aTP_Lamp_bj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bk____,type,
aTP_Lamp_bk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bl____,type,
aTP_Lamp_bl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bm____,type,
aTP_Lamp_bm:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bu____,type,
aTP_Lamp_bu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bv____,type,
aTP_Lamp_bv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bw____,type,
aTP_Lamp_bw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cj____,type,
aTP_Lamp_cj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ck____,type,
aTP_Lamp_ck:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cl____,type,
aTP_Lamp_cl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cm____,type,
aTP_Lamp_cm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cn____,type,
aTP_Lamp_cn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__co____,type,
aTP_Lamp_co:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cz____,type,
aTP_Lamp_cz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__da____,type,
aTP_Lamp_da:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__db____,type,
aTP_Lamp_db:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hl____,type,
aTP_Lamp_hl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hm____,type,
aTP_Lamp_hm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ho____,type,
aTP_Lamp_ho:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hp____,type,
aTP_Lamp_hp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hq____,type,
aTP_Lamp_hq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hr____,type,
aTP_Lamp_hr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hs____,type,
aTP_Lamp_hs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ht____,type,
aTP_Lamp_ht:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hu____,type,
aTP_Lamp_hu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hy____,type,
aTP_Lamp_hy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hz____,type,
aTP_Lamp_hz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ia____,type,
aTP_Lamp_ia:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ic____,type,
aTP_Lamp_ic:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__id____,type,
aTP_Lamp_id:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ie____,type,
aTP_Lamp_ie:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__il____,type,
aTP_Lamp_il:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__im____,type,
aTP_Lamp_im:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__in____,type,
aTP_Lamp_in:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__io____,type,
aTP_Lamp_io:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jb____,type,
aTP_Lamp_jb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jc____,type,
aTP_Lamp_jc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jd____,type,
aTP_Lamp_jd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__je____,type,
aTP_Lamp_je:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jf____,type,
aTP_Lamp_jf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jg____,type,
aTP_Lamp_jg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jh____,type,
aTP_Lamp_jh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ji____,type,
aTP_Lamp_ji:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jj____,type,
aTP_Lamp_jj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jk____,type,
aTP_Lamp_jk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jl____,type,
aTP_Lamp_jl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jm____,type,
aTP_Lamp_jm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jn____,type,
aTP_Lamp_jn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jo____,type,
aTP_Lamp_jo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jq____,type,
aTP_Lamp_jq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jr____,type,
aTP_Lamp_jr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__js____,type,
aTP_Lamp_js:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jt____,type,
aTP_Lamp_jt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ju____,type,
aTP_Lamp_ju:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jv____,type,
aTP_Lamp_jv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jw____,type,
aTP_Lamp_jw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jx____,type,
aTP_Lamp_jx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jy____,type,
aTP_Lamp_jy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ka____,type,
aTP_Lamp_ka:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kb____,type,
aTP_Lamp_kb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kc____,type,
aTP_Lamp_kc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kd____,type,
aTP_Lamp_kd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ke____,type,
aTP_Lamp_ke:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kg____,type,
aTP_Lamp_kg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kh____,type,
aTP_Lamp_kh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ki____,type,
aTP_Lamp_ki:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kn____,type,
aTP_Lamp_kn:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kw____,type,
aTP_Lamp_kw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lj____,type,
aTP_Lamp_lj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lk____,type,
aTP_Lamp_lk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ll____,type,
aTP_Lamp_ll:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lm____,type,
aTP_Lamp_lm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lo____,type,
aTP_Lamp_lo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lp____,type,
aTP_Lamp_lp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lq____,type,
aTP_Lamp_lq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lr____,type,
aTP_Lamp_lr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lv____,type,
aTP_Lamp_lv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lw____,type,
aTP_Lamp_lw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lx____,type,
aTP_Lamp_lx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ly____,type,
aTP_Lamp_ly:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lz____,type,
aTP_Lamp_lz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ma____,type,
aTP_Lamp_ma:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mb____,type,
aTP_Lamp_mb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mc____,type,
aTP_Lamp_mc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__md____,type,
aTP_Lamp_md:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__me____,type,
aTP_Lamp_me:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mf____,type,
aTP_Lamp_mf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mw____,type,
aTP_Lamp_mw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mx____,type,
aTP_Lamp_mx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__my____,type,
aTP_Lamp_my:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mz____,type,
aTP_Lamp_mz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__na____,type,
aTP_Lamp_na:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nb____,type,
aTP_Lamp_nb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nc____,type,
aTP_Lamp_nc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nd____,type,
aTP_Lamp_nd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ne____,type,
aTP_Lamp_ne:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nf____,type,
aTP_Lamp_nf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ng____,type,
aTP_Lamp_ng:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nh____,type,
aTP_Lamp_nh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ni____,type,
aTP_Lamp_ni:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nj____,type,
aTP_Lamp_nj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nk____,type,
aTP_Lamp_nk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nl____,type,
aTP_Lamp_nl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nm____,type,
aTP_Lamp_nm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nn____,type,
aTP_Lamp_nn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__no____,type,
aTP_Lamp_no:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__np____,type,
aTP_Lamp_np:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nq____,type,
aTP_Lamp_nq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nr____,type,
aTP_Lamp_nr:
!>[A: $tType,B: $tType] : 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nz____,type,
aTP_Lamp_nz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oa____,type,
aTP_Lamp_oa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ob____,type,
aTP_Lamp_ob:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oc____,type,
aTP_Lamp_oc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__on____,type,
aTP_Lamp_on:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oo____,type,
aTP_Lamp_oo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ot____,type,
aTP_Lamp_ot:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ou____,type,
aTP_Lamp_ou:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ov____,type,
aTP_Lamp_ov:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pt____,type,
aTP_Lamp_pt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pv____,type,
aTP_Lamp_pv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pw____,type,
aTP_Lamp_pw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__px____,type,
aTP_Lamp_px:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__py____,type,
aTP_Lamp_py:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qa____,type,
aTP_Lamp_qa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qb____,type,
aTP_Lamp_qb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qc____,type,
aTP_Lamp_qc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qd____,type,
aTP_Lamp_qd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qe____,type,
aTP_Lamp_qe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qf____,type,
aTP_Lamp_qf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qg____,type,
aTP_Lamp_qg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qh____,type,
aTP_Lamp_qh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qi____,type,
aTP_Lamp_qi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ql____,type,
aTP_Lamp_ql:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qm____,type,
aTP_Lamp_qm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qn____,type,
aTP_Lamp_qn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qo____,type,
aTP_Lamp_qo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qp____,type,
aTP_Lamp_qp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qq____,type,
aTP_Lamp_qq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qr____,type,
aTP_Lamp_qr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qu____,type,
aTP_Lamp_qu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qv____,type,
aTP_Lamp_qv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qw____,type,
aTP_Lamp_qw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qx____,type,
aTP_Lamp_qx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qy____,type,
aTP_Lamp_qy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qz____,type,
aTP_Lamp_qz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ra____,type,
aTP_Lamp_ra:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rb____,type,
aTP_Lamp_rb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rc____,type,
aTP_Lamp_rc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rd____,type,
aTP_Lamp_rd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__re____,type,
aTP_Lamp_re:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rf____,type,
aTP_Lamp_rf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rg____,type,
aTP_Lamp_rg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rh____,type,
aTP_Lamp_rh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ri____,type,
aTP_Lamp_ri:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rj____,type,
aTP_Lamp_rj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rk____,type,
aTP_Lamp_rk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rl____,type,
aTP_Lamp_rl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rm____,type,
aTP_Lamp_rm:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rs____,type,
aTP_Lamp_rs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rt____,type,
aTP_Lamp_rt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ru____,type,
aTP_Lamp_ru:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rv____,type,
aTP_Lamp_rv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rw____,type,
aTP_Lamp_rw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rx____,type,
aTP_Lamp_rx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ry____,type,
aTP_Lamp_ry:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rz____,type,
aTP_Lamp_rz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sa____,type,
aTP_Lamp_sa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sc____,type,
aTP_Lamp_sc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sd____,type,
aTP_Lamp_sd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__se____,type,
aTP_Lamp_se:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sg____,type,
aTP_Lamp_sg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sh____,type,
aTP_Lamp_sh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__si____,type,
aTP_Lamp_si:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sj____,type,
aTP_Lamp_sj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sk____,type,
aTP_Lamp_sk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sl____,type,
aTP_Lamp_sl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sm____,type,
aTP_Lamp_sm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sn____,type,
aTP_Lamp_sn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__so____,type,
aTP_Lamp_so:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sp____,type,
aTP_Lamp_sp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sq____,type,
aTP_Lamp_sq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sr____,type,
aTP_Lamp_sr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ss____,type,
aTP_Lamp_ss:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__st____,type,
aTP_Lamp_st:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__su____,type,
aTP_Lamp_su:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sv____,type,
aTP_Lamp_sv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sw____,type,
aTP_Lamp_sw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sx____,type,
aTP_Lamp_sx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sy____,type,
aTP_Lamp_sy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sz____,type,
aTP_Lamp_sz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ta____,type,
aTP_Lamp_ta:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tb____,type,
aTP_Lamp_tb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tc____,type,
aTP_Lamp_tc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__td____,type,
aTP_Lamp_td:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__te____,type,
aTP_Lamp_te:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tf____,type,
aTP_Lamp_tf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tg____,type,
aTP_Lamp_tg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__th____,type,
aTP_Lamp_th:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ti____,type,
aTP_Lamp_ti:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tj____,type,
aTP_Lamp_tj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tk____,type,
aTP_Lamp_tk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tl____,type,
aTP_Lamp_tl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tm____,type,
aTP_Lamp_tm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tn____,type,
aTP_Lamp_tn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__to____,type,
aTP_Lamp_to:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tp____,type,
aTP_Lamp_tp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tq____,type,
aTP_Lamp_tq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tr____,type,
aTP_Lamp_tr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ts____,type,
aTP_Lamp_ts:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tt____,type,
aTP_Lamp_tt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tu____,type,
aTP_Lamp_tu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tv____,type,
aTP_Lamp_tv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tw____,type,
aTP_Lamp_tw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tx____,type,
aTP_Lamp_tx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ty____,type,
aTP_Lamp_ty:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tz____,type,
aTP_Lamp_tz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ua____,type,
aTP_Lamp_ua:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ub____,type,
aTP_Lamp_ub:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uc____,type,
aTP_Lamp_uc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ud____,type,
aTP_Lamp_ud:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ue____,type,
aTP_Lamp_ue:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uf____,type,
aTP_Lamp_uf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ug____,type,
aTP_Lamp_ug:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uh____,type,
aTP_Lamp_uh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ui____,type,
aTP_Lamp_ui:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uj____,type,
aTP_Lamp_uj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uk____,type,
aTP_Lamp_uk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ul____,type,
aTP_Lamp_ul:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vi____,type,
aTP_Lamp_vi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vj____,type,
aTP_Lamp_vj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vk____,type,
aTP_Lamp_vk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vl____,type,
aTP_Lamp_vl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vm____,type,
aTP_Lamp_vm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vn____,type,
aTP_Lamp_vn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vo____,type,
aTP_Lamp_vo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vp____,type,
aTP_Lamp_vp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vq____,type,
aTP_Lamp_vq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vr____,type,
aTP_Lamp_vr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vs____,type,
aTP_Lamp_vs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vt____,type,
aTP_Lamp_vt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vu____,type,
aTP_Lamp_vu:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vv____,type,
aTP_Lamp_vv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vw____,type,
aTP_Lamp_vw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vx____,type,
aTP_Lamp_vx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vy____,type,
aTP_Lamp_vy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vz____,type,
aTP_Lamp_vz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wa____,type,
aTP_Lamp_wa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wb____,type,
aTP_Lamp_wb:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wt____,type,
aTP_Lamp_wt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wu____,type,
aTP_Lamp_wu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wv____,type,
aTP_Lamp_wv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ww____,type,
aTP_Lamp_ww:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wx____,type,
aTP_Lamp_wx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wy____,type,
aTP_Lamp_wy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wz____,type,
aTP_Lamp_wz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xa____,type,
aTP_Lamp_xa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xb____,type,
aTP_Lamp_xb:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xj____,type,
aTP_Lamp_xj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xk____,type,
aTP_Lamp_xk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xl____,type,
aTP_Lamp_xl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xm____,type,
aTP_Lamp_xm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xn____,type,
aTP_Lamp_xn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xo____,type,
aTP_Lamp_xo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xp____,type,
aTP_Lamp_xp:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).
tff(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn) ).
tff(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( fun(A,assn) > assn ) ).
tff(sy_c_Assertions_Oin__range,type,
in_range: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Ois__pure__assn,type,
is_pure_assn: assn > $o ).
tff(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Oproper,type,
proper: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o) ).
tff(sy_c_Assertions_Opure__assn,type,
pure_assn: $o > assn ).
tff(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( $o > fun(product_prod(A,set(B)),$o) ) ).
tff(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : fun(product_prod($o,product_prod(A,set(B))),fun(product_prod($o,product_prod(A,set(B))),$o)) ).
tff(sy_c_Assertions_OrelH,type,
relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).
tff(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).
tff(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array(A) * list(A) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).
tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref(A) * A ) > assn ) ).
tff(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref(A) * A ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).
tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Owand__assn,type,
wand_assn: ( assn * assn ) > assn ).
tff(sy_c_Assertions_Owand__raw,type,
wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : fun(A,A) ).
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(A,B) * fun(A,C) ) > fun(A,product_prod(B,C)) ) ).
tff(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set(A) * fun(B,C) * fun(D,C) * fun(A,B) * fun(A,D) ) > $o ) ).
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_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,fun(C,$o)) ) > fun(product_prod(A,C),product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,$o)) * fun(A,fun(C,$o)) * B * C ) > A ) ).
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_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(sum_sum(A,B),fun(sum_sum(C,D),$o)) ) ).
tff(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,$o)) * fun(A,fun(B,$o)) ) > fun(product_prod(C,B),product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) * fun(D,fun(E,C)) ) > fun(A,fun(B,C)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_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_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__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(A) * fun(product_prod(A,B),C) * A ) > fun(B,C) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(A,A)),set(A)) ) ).
tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_Basic__BNF__LFPs_Octor__rec,type,
basic_BNF_ctor_rec:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) ) > fun(product_prod(A,B),nat) ) ).
tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) ) > fun(sum_sum(A,B),nat) ) ).
tff(sy_c_Basic__BNF__LFPs_Oxtor,type,
basic_BNF_xtor:
!>[A: $tType] : fun(A,A) ).
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(B,$o) * 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) * 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_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(A)) ).
tff(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(B)) ).
tff(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).
tff(sy_c_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod(A,B) * B ) > $o ) ).
tff(sy_c_Binomial_Obinomial,type,
binomial: nat > fun(nat,nat) ).
tff(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > fun(nat,A) ) ).
tff(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).
tff(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: ( nat * int * int ) > int ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > fun(nat,$o) ) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).
tff(sy_c_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)) ) > $o ) ).
tff(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
tff(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
tff(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: fun(code_integer,int) ).
tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: fun(int,code_integer) ).
tff(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: fun(code_natural,nat) ).
tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: fun(nat,code_natural) ).
tff(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: fun(int,fun(code_integer,$o)) ).
tff(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: fun(nat,fun(code_natural,$o)) ).
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_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( fun(A,A) > set(A) ) ).
tff(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,fun(B,$o)) * fun(A,B) ) > $o ) ).
tff(sy_c_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_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_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Equiv__Relations_Opart__equivp,type,
equiv_part_equivp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,set(A)) ) ).
tff(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).
tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Filter_Oabstract__filter,type,
abstract_filter:
!>[A: $tType] : ( fun(product_unit,filter(A)) > filter(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_Ocofinite,type,
cofinite:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).
tff(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : fun(fun(fun(A,$o),$o),filter(A)) ).
tff(sy_c_Filter_Ofilter_ORep__filter,type,
rep_filter:
!>[A: $tType] : fun(filter(A),fun(fun(A,$o),$o)) ).
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(fun(A,B),fun(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_Ofrequently,type,
frequently:
!>[A: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).
tff(sy_c_Filter_Ois__filter,type,
is_filter:
!>[A: $tType] : fun(fun(fun(A,$o),$o),$o) ).
tff(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( set(A) > fun(fun(A,B),fun(filter(A),filter(B))) ) ).
tff(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : fun(set(A),filter(A)) ).
tff(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : fun(filter(A),fun(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)) > fun(filter(A),fun(filter(B),$o)) ) ).
tff(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( set(B) > nat ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : fun(set(A),$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_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) * B ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).
tff(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(B,C) ) > 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) ) > fun(B,A) ) ).
tff(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[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,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $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] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
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_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
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(fun(C,A),fun(set(C),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
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__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > 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_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
tff(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
tff(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( array(A) > nat ) ).
tff(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ref(A) > nat ) ).
tff(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > list(nat) ) ).
tff(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( heap_ext(Z) > nat ) ).
tff(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > nat ) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,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: $tType] : ( ( set(A) * nat ) > A ) ).
tff(sy_c_Int_OAbs__Integ,type,
abs_Integ: fun(product_prod(nat,nat),int) ).
tff(sy_c_Int_ONeg,type,
neg: num > int ).
tff(sy_c_Int_OPos,type,
pos: num > int ).
tff(sy_c_Int_ORep__Integ,type,
rep_Integ: fun(int,product_prod(nat,nat)) ).
tff(sy_c_Int_Ointrel,type,
intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).
tff(sy_c_Int_Onat,type,
nat2: 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,type,
semilattice_neutr:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
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__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) * B ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
lattic8678736583308907530nf_fin:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * 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__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
lattic4630905495605216202up_fin:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( 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_List_Oappend,type,
append:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( list(A) > set(A) ) ).
tff(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list(A) * 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_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) * 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) * B ) > B ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
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)) * 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_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$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) * B * 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__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( set(A) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) * list(A) ) > B ) ).
tff(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,$o) * list(A) ) > $o ) ).
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))) * list(A) ) > C ) ).
tff(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : fun(list(A),set(A)) ).
tff(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( 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_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).
tff(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( set(A) > set(list(A)) ) ).
tff(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).
tff(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,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_Onull,type,
null:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( fun(A,$o) > fun(list(A),product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_OremoveAll,type,
removeAll:
!>[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_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).
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_Oupt,type,
upt: ( nat * nat ) > list(nat) ).
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__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_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__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,$o)) * list(A) * list(B) ) > $o ) ).
tff(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o)) ).
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__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),$o)) ).
tff(sy_c_Misc_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_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * fun(A,B) ) > $o ) ).
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] : ( ( A * multiset(A) ) > multiset(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__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(multiset(A),A) ) ).
tff(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : fun(multiset(A),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_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__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) * multiset(A) ) > nat ) ).
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] : ( ( nat * A ) > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set(nat) ).
tff(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set(nat) > nat ).
tff(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: sum_sum(nat,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_Onat__of__num,type,
nat_of_num: num > nat ).
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_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
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: num > num ).
tff(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
tff(sy_c_Num_Onum_OOne,type,
one2: num ).
tff(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
tff(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : fun(num,A) ).
tff(sy_c_Num_Opow,type,
pow: ( num * num ) > num ).
tff(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
tff(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_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_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).
tff(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(option(A),fun(option(B),$o)) ) ).
tff(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( option(A) > set(A) ) ).
tff(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : fun(option(A),A) ).
tff(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( set(option(A)) > set(A) ) ).
tff(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),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_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_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $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_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).
tff(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( ( A * set(A) ) > A ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : fun(A,fun(nat,A)) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
tff(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ).
tff(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( ( T * T * $o ) > T ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( ( T * T * $o ) > fun(T,$o) ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( ( T * product_unit ) > fun(T,$o) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( ( T * product_unit ) > T ) ).
tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).
tff(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).
tff(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).
tff(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).
tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: fun($o,product_unit) ).
tff(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: fun(product_unit,$o) ).
tff(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( ( A * product_unit ) > A ) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
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_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_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_Rat_OAbs__Rat,type,
abs_Rat: fun(product_prod(int,int),rat) ).
tff(sy_c_Rat_OFract,type,
fract: ( int * int ) > rat ).
tff(sy_c_Rat_ORep__Rat,type,
rep_Rat: fun(rat,product_prod(int,int)) ).
tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
tff(sy_c_Rat_Onormalize,type,
normalize: product_prod(int,int) > product_prod(int,int) ).
tff(sy_c_Rat_Opcr__rat,type,
pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).
tff(sy_c_Rat_Opositive,type,
positive: fun(rat,$o) ).
tff(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod(int,int) ).
tff(sy_c_Rat_Oratrel,type,
ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).
tff(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > 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_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(A)) ).
tff(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(A,$o) ) ).
tff(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : fun(set(product_prod(A,A)),set(A)) ).
tff(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * set(A) ) > set(B) ) ).
tff(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( fun(A,$o) > fun(set(A),$o) ) ).
tff(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(B)) ).
tff(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(B,$o) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
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_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,$o)) * fun(A,B) * A * 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_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Oreflp,type,
reflp:
!>[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: $tType] : ( ( fun(A,fun(B,$o)) * 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_Osym,type,
sym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Osymp,type,
symp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( fun(A,fun(A,$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] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( 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: $tType] : ( set(A) > fun(fun(A,$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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).
tff(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : fun(set(A),fun(set(A),$o)) ).
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,
insert2:
!>[A: $tType] : fun(A,fun(set(A),set(A))) ).
tff(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( ( A * set(A) ) > set(A) ) ).
tff(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * 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_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr3752724095348155675AtMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr287244882034783167ssThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( 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 * $o ) > char ).
tff(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
tff(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).
tff(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(sum_sum(A,B),sum_sum(C,D)) ) ).
tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) ) > fun(sum_sum(A,B),C) ) ).
tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).
tff(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
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_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).
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_Oacc,type,
acc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,$o) ) ).
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_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $o ) ).
tff(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).
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_Zorn_OChains,type,
chains:
!>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).
tff(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).
tff(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).
tff(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).
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 * set(A) ) > $o ) ).
tff(sy_v_h,type,
h: product_prod(heap_ext(product_unit),set(nat)) ).
% Relevant facts (6379)
tff(fact_0_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X) = aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Y) )
<=> ( X = Y ) ) ).
% Rep_assn_inject
tff(fact_1_Rep__assn__inverse,axiom,
! [X: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X)) = X ).
% Rep_assn_inverse
tff(fact_2_Abs__assn__eqI_I2_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H) )
=> ( Pr = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) ) ) ).
% Abs_assn_eqI(2)
tff(fact_3_Abs__assn__eqI_I1_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H) )
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) = Pr ) ) ).
% Abs_assn_eqI(1)
tff(fact_4_empty__iff,axiom,
! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).
% empty_iff
tff(fact_5_all__not__in__conv,axiom,
! [A: $tType,A3: set(A)] :
( ! [X2: A] : ~ member(A,X2,A3)
<=> ( A3 = bot_bot(set(A)) ) ) ).
% all_not_in_conv
tff(fact_6_Collect__empty__eq,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
<=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).
% Collect_empty_eq
tff(fact_7_empty__Collect__eq,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
<=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).
% empty_Collect_eq
tff(fact_8_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).
% bot_apply
tff(fact_9_emptyE,axiom,
! [A: $tType,A4: A] : ~ member(A,A4,bot_bot(set(A))) ).
% emptyE
tff(fact_10_equals0D,axiom,
! [A: $tType,A3: set(A),A4: A] :
( ( A3 = bot_bot(set(A)) )
=> ~ member(A,A4,A3) ) ).
% equals0D
tff(fact_11_equals0I,axiom,
! [A: $tType,A3: set(A)] :
( ! [Y2: A] : ~ member(A,Y2,A3)
=> ( A3 = bot_bot(set(A)) ) ) ).
% equals0I
tff(fact_12_ex__in__conv,axiom,
! [A: $tType,A3: set(A)] :
( ? [X2: A] : member(A,X2,A3)
<=> ( A3 != bot_bot(set(A)) ) ) ).
% ex_in_conv
tff(fact_13_set__notEmptyE,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ~ ! [X3: A] : ~ member(A,X3,S) ) ).
% set_notEmptyE
tff(fact_14_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set(A)] :
( member(A,X,S)
=> ( S != bot_bot(set(A)) ) ) ).
% memb_imp_not_empty
tff(fact_15_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X4: A] : aa(A,B,bot_bot(fun(A,B)),X4) = bot_bot(B) ) ).
% bot_fun_def
tff(fact_16_bot__set__def,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).
% bot_set_def
tff(fact_17_mod__false,axiom,
! [Ha: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,bot_bot(assn)),Ha) ).
% mod_false
tff(fact_18_Set_Ois__empty__def,axiom,
! [A: $tType,A3: set(A)] :
( is_empty(A,A3)
<=> ( A3 = bot_bot(set(A)) ) ) ).
% Set.is_empty_def
tff(fact_19_Collect__empty__eq__bot,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
<=> ( P = bot_bot(fun(A,$o)) ) ) ).
% Collect_empty_eq_bot
tff(fact_20_bot__empty__eq,axiom,
! [A: $tType,X4: A] :
( aa(A,$o,bot_bot(fun(A,$o)),X4)
<=> member(A,X4,bot_bot(set(A))) ) ).
% bot_empty_eq
tff(fact_21_wand__assn__def,axiom,
! [P: assn,Q: assn] : wand_assn(P,Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,wand_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).
% wand_assn_def
tff(fact_22_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),V: B] : map_mmupd(A,B,M,bot_bot(set(A)),V) = M ).
% map_mmupd_empty
tff(fact_23_one__assn__def,axiom,
one_one(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,one_assn_raw) ).
% one_assn_def
tff(fact_24_Abs__assn__inverse,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y)) = Y ) ) ).
% Abs_assn_inverse
tff(fact_25_one__assn__raw_Osimps,axiom,
! [Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ( As = bot_bot(set(nat)) ) ) ).
% one_assn_raw.simps
tff(fact_26_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ( As2 != bot_bot(set(nat)) ) ) ) ) ).
% one_assn_raw.elims(1)
tff(fact_27_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( As2 != bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(2)
tff(fact_28_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( As2 = bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(3)
tff(fact_29_is__singletonI_H,axiom,
! [A: $tType,A3: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> ( X3 = Y2 ) ) )
=> is_singleton(A,A3) ) ) ).
% is_singletonI'
tff(fact_30_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K2: set(A),M: fun(A,option(B)),V: B] :
( ~ member(A,K,K2)
=> ( aa(A,option(B),map_mmupd(A,B,M,K2,V),K) = aa(A,option(B),M,K) ) ) ).
% mmupd_notin_upd
tff(fact_31_wand__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,wand_raw(P,Q)) ).
% wand_proper
tff(fact_32_one__assn__proper,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,one_assn_raw) ).
% one_assn_proper
tff(fact_33_assn__basic__inequalities_I3_J,axiom,
bot_bot(assn) != one_one(assn) ).
% assn_basic_inequalities(3)
tff(fact_34_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
~ ! [F: fun(A,B),A5: A,B2: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2)) ).
% pairself.cases
tff(fact_35_bex2I,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,S: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S)
=> ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S)
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) )
=> ? [A5: A,B2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2),S)
& aa(B,$o,aa(A,fun(B,$o),P,A5),B2) ) ) ) ).
% bex2I
tff(fact_36_one__assn__raw_Ocases,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
~ ! [H: heap_ext(product_unit),As2: set(nat)] : X != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) ).
% one_assn_raw.cases
tff(fact_37_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A4: A,B3: B,P: fun(B,$o)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),X))
=> aa(B,$o,P,B3) ) ) ).
% sndE
tff(fact_38_Rep__assn,axiom,
! [X: assn] : member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).
% Rep_assn
tff(fact_39_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: fun(A,$o)] :
( member(A,A4,aa(fun(A,$o),set(A),collect(A),P))
<=> aa(A,$o,P,A4) ) ).
% mem_Collect_eq
tff(fact_40_Collect__mem__eq,axiom,
! [A: $tType,A3: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)) = A3 ).
% Collect_mem_eq
tff(fact_41_Collect__cong,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,Q,X3) )
=> ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).
% Collect_cong
tff(fact_42_ext,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
( ! [X3: A] : aa(A,B,F2,X3) = aa(A,B,G,X3)
=> ( F2 = G ) ) ).
% ext
tff(fact_43_Rep__assn__cases,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ~ ! [X3: assn] : Y != aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X3) ) ).
% Rep_assn_cases
tff(fact_44_Rep__assn__induct,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o),P: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( ! [X3: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X3))
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,Y) ) ) ).
% Rep_assn_induct
tff(fact_45_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( ( X = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2) )
=> ~ member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ) ).
% Abs_assn_cases
tff(fact_46_Abs__assn__induct,axiom,
! [P: fun(assn,$o),X: assn] :
( ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> aa(assn,$o,P,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2)) )
=> aa(assn,$o,P,X) ) ).
% Abs_assn_induct
tff(fact_47_Abs__assn__inject,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),X,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_assn_inject
tff(fact_48_mod__h__bot__indep,axiom,
! [P: assn,Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),bot_bot(set(nat))))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ).
% mod_h_bot_indep
tff(fact_49_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A6: A,B4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
<=> ( ( A4 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
tff(fact_50_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_51_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z2: B] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z2 ) ) ).
% sndI
tff(fact_52_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A4: A] :
( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
tff(fact_53_snd__conv,axiom,
! [B: $tType,A: $tType,X1: B,X22: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X22)) = X22 ).
% snd_conv
tff(fact_54_eq__snd__iff,axiom,
! [B: $tType,A: $tType,B3: A,P2: product_prod(B,A)] :
( ( B3 = aa(product_prod(B,A),A,product_snd(B,A),P2) )
<=> ? [A7: B] : P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B3) ) ).
% eq_snd_iff
tff(fact_55_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A4: B,B3: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3)) = aa(C,A,aa(B,fun(C,A),F2,A4),B3) ).
% uncurry_apply
tff(fact_56_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ( ( As2 = bot_bot(set(B)) )
& (X) ) ) ) ).
% pure_assn_raw.elims(3)
tff(fact_57_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ~ ( ( As2 = bot_bot(set(B)) )
& (X) ) ) ) ).
% pure_assn_raw.elims(2)
tff(fact_58_pure__assn__raw_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B)),Y: $o] :
( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
<=> (Y) )
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ( (Y)
<=> ~ ( ( As2 = bot_bot(set(B)) )
& (X) ) ) ) ) ).
% pure_assn_raw.elims(1)
tff(fact_59_pure__assn__raw_Osimps,axiom,
! [B: $tType,A: $tType,B3: $o,Ha: A,As: set(B)] :
( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(B3)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),Ha),As))
<=> ( ( As = bot_bot(set(B)) )
& (B3) ) ) ).
% pure_assn_raw.simps
tff(fact_60_pure__assn__proper,axiom,
! [B3: $o] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,pure_assn_raw(heap_ext(product_unit),nat,(B3))) ).
% pure_assn_proper
tff(fact_61_times__assn__raw_Ocases,axiom,
! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [P3: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),P3),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ).
% times_assn_raw.cases
tff(fact_62_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [R: ref(A),X3: A,H: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ).
% sngr_assn_raw.cases
tff(fact_63_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [R: array(A),X3: list(A),H: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ).
% snga_assn_raw.cases
tff(fact_64_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod($o,product_prod(A,set(B)))] :
~ ! [B2: $o,H: A,As2: set(B)] : X != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(B2)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2)) ).
% pure_assn_raw.cases
tff(fact_65_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A5: A,B2: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) ).
% old.prod.exhaust
tff(fact_66_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
? [X3: A,Y2: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) ).
% surj_pair
tff(fact_67_prod__cases,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
( ! [A5: A,B2: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2))
=> aa(product_prod(A,B),$o,P,P2) ) ).
% prod_cases
tff(fact_68_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A6: A,B4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
=> ~ ( ( A4 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
tff(fact_69_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A5: A,B2: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)) ).
% prod_cases3
tff(fact_70_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A5: A,B2: B,C3: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).
% prod_cases4
tff(fact_71_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A5: A,B2: B,C3: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).
% prod_cases5
tff(fact_72_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
~ ! [A5: A,B2: B,C3: C,D2: D,E2: E,F: F3] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B2),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C3),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D2),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F))))) ).
% prod_cases6
tff(fact_73_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
~ ! [A5: A,B2: B,C3: C,D2: D,E2: E,F: F3,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B2),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C3),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D2),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F),G3)))))) ).
% prod_cases7
tff(fact_74_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),$o),X: product_prod(A,product_prod(B,C))] :
( ! [A5: A,B2: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)))
=> aa(product_prod(A,product_prod(B,C)),$o,P,X) ) ).
% prod_induct3
tff(fact_75_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
( ! [A5: A,B2: B,C3: C,D2: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))))
=> aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,X) ) ).
% prod_induct4
tff(fact_76_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
( ! [A5: A,B2: B,C3: C,D2: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,X) ) ).
% prod_induct5
tff(fact_77_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
( ! [A5: A,B2: B,C3: C,D2: D,E2: E,F: F3] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B2),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C3),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D2),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F))))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,X) ) ).
% prod_induct6
tff(fact_78_prod__induct7,axiom,
! [G2: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
( ! [A5: A,B2: B,C3: C,D2: D,E2: E,F: F3,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B2),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C3),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D2),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F),G3)))))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,X) ) ).
% prod_induct7
tff(fact_79_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
% one_reorient
tff(fact_80_old_Oprod_Orec,axiom,
! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A4: B,B3: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3)) = aa(C,A,aa(B,fun(C,A),F1,A4),B3) ).
% old.prod.rec
tff(fact_81_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A4: B,B3: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3)) = aa(C,A,aa(B,fun(C,A),C2,A4),B3) ).
% internal_case_prod_conv
tff(fact_82_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : uncurry(A,B,C,aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),F2)) = F2 ).
% uncurry_curry_id
tff(fact_83_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C))] : aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),uncurry(A,B,C,F2)) = F2 ).
% curry_uncurry_id
tff(fact_84_snga__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,snga_assn_raw(A,R2,X)) ) ).
% snga_assn_proper
tff(fact_85_sngr__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,sngr_assn_raw(A,R2,X)) ) ).
% sngr_assn_proper
tff(fact_86_type__definition__assn,axiom,
type_definition(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,abs_assn,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).
% type_definition_assn
tff(fact_87_times__assn__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,times_assn_raw(P,Q)) ) ) ).
% times_assn_proper
tff(fact_88_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn(one_one(assn)) ).
% is_pure_assn_basic_simps(2)
tff(fact_89_bijective__def,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
( bijective(A,B,R3)
<=> ( ! [X2: A,Y3: B,Z3: B] :
( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3),R3)
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z3),R3) )
=> ( Y3 = Z3 ) )
& ! [X2: A,Y3: A,Z3: B] :
( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z3),R3)
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z3),R3) )
=> ( X2 = Y3 ) ) ) ) ).
% bijective_def
tff(fact_90_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R3: set(product_prod(A,B)),S3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S2),R3)
=> ( ( S3 = S2 )
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S3),R3) ) ) ).
% ssubst_Pair_rhs
tff(fact_91_is__singletonI,axiom,
! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% is_singletonI
tff(fact_92_curryI,axiom,
! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A4: A,B3: B] :
( aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),product_curry(A,B,$o),F2),A4),B3) ) ).
% curryI
tff(fact_93_insertCI,axiom,
! [A: $tType,A4: A,B5: set(A),B3: A] :
( ( ~ member(A,A4,B5)
=> ( A4 = B3 ) )
=> member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B5)) ) ).
% insertCI
tff(fact_94_insert__iff,axiom,
! [A: $tType,A4: A,B3: A,A3: set(A)] :
( member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),A3))
<=> ( ( A4 = B3 )
| member(A,A4,A3) ) ) ).
% insert_iff
tff(fact_95_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3) ).
% insert_absorb2
tff(fact_96_bijective__Empty,axiom,
! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).
% bijective_Empty
tff(fact_97_singletonI,axiom,
! [A: $tType,A4: A] : member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ).
% singletonI
tff(fact_98_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(product_prod(B,C),A),A4: B,B3: C] : aa(C,A,aa(B,fun(C,A),aa(fun(product_prod(B,C),A),fun(B,fun(C,A)),product_curry(B,C,A),F2),A4),B3) = aa(product_prod(B,C),A,F2,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3)) ).
% curry_conv
tff(fact_99_insertE,axiom,
! [A: $tType,A4: A,B3: A,A3: set(A)] :
( member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),A3))
=> ( ( A4 != B3 )
=> member(A,A4,A3) ) ) ).
% insertE
tff(fact_100_insertI1,axiom,
! [A: $tType,A4: A,B5: set(A)] : member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) ).
% insertI1
tff(fact_101_insertI2,axiom,
! [A: $tType,A4: A,B5: set(A),B3: A] :
( member(A,A4,B5)
=> member(A,A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B5)) ) ).
% insertI2
tff(fact_102_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set(A)] :
( member(A,X,A3)
=> ~ ! [B6: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B6) )
=> member(A,X,B6) ) ) ).
% Set.set_insert
tff(fact_103_insert__ident,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
( ~ member(A,X,A3)
=> ( ~ member(A,X,B5)
=> ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5) )
<=> ( A3 = B5 ) ) ) ) ).
% insert_ident
tff(fact_104_insert__absorb,axiom,
! [A: $tType,A4: A,A3: set(A)] :
( member(A,A4,A3)
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) = A3 ) ) ).
% insert_absorb
tff(fact_105_insert__eq__iff,axiom,
! [A: $tType,A4: A,A3: set(A),B3: A,B5: set(A)] :
( ~ member(A,A4,A3)
=> ( ~ member(A,B3,B5)
=> ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B5) )
<=> $ite(
A4 = B3,
A3 = B5,
? [C4: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),C4) )
& ~ member(A,B3,C4)
& ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),C4) )
& ~ member(A,A4,C4) ) ) ) ) ) ).
% insert_eq_iff
tff(fact_106_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) ).
% insert_commute
tff(fact_107_mk__disjoint__insert,axiom,
! [A: $tType,A4: A,A3: set(A)] :
( member(A,A4,A3)
=> ? [B6: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B6) )
& ~ member(A,A4,B6) ) ) ).
% mk_disjoint_insert
tff(fact_108_singletonD,axiom,
! [A: $tType,B3: A,A4: A] :
( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))
=> ( B3 = A4 ) ) ).
% singletonD
tff(fact_109_singleton__iff,axiom,
! [A: $tType,B3: A,A4: A] :
( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))
<=> ( B3 = A4 ) ) ).
% singleton_iff
tff(fact_110_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B3: A,C2: A,D3: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D3),bot_bot(set(A)))) )
<=> ( ( ( A4 = C2 )
& ( B3 = D3 ) )
| ( ( A4 = D3 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
tff(fact_111_insert__not__empty,axiom,
! [A: $tType,A4: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) != bot_bot(set(A)) ).
% insert_not_empty
tff(fact_112_singleton__inject,axiom,
! [A: $tType,A4: A,B3: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) )
=> ( A4 = B3 ) ) ).
% singleton_inject
tff(fact_113_curryE,axiom,
! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),product_curry(A,B,$o),F2),A4),B3)
=> aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)) ) ).
% curryE
tff(fact_114_curryD,axiom,
! [A: $tType,B: $tType,F2: fun(product_prod(A,B),$o),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),product_curry(A,B,$o),F2),A4),B3)
=> aa(product_prod(A,B),$o,F2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)) ) ).
% curryD
tff(fact_115_is__singleton__def,axiom,
! [A: $tType,A3: set(A)] :
( is_singleton(A,A3)
<=> ? [X2: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).
% is_singleton_def
tff(fact_116_is__singletonE,axiom,
! [A: $tType,A3: set(A)] :
( is_singleton(A,A3)
=> ~ ! [X3: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).
% is_singletonE
tff(fact_117_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn(bot_bot(assn)) ).
% is_pure_assn_basic_simps(1)
tff(fact_118_snga__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),A4: list(A)] : snga_assn(A,R2,A4) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,snga_assn_raw(A,R2,A4)) ) ).
% snga_assn_def
tff(fact_119_sngr__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : sngr_assn(A,R2,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,sngr_assn_raw(A,R2,X)) ) ).
% sngr_assn_def
tff(fact_120_is__singleton__the__elem,axiom,
! [A: $tType,A3: set(A)] :
( is_singleton(A,A3)
<=> ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A3)),bot_bot(set(A))) ) ) ).
% is_singleton_the_elem
tff(fact_121_the__elem__eq,axiom,
! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ).
% the_elem_eq
tff(fact_122_pure__assn__def,axiom,
! [B3: $o] : pure_assn((B3)) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,pure_assn_raw(heap_ext(product_unit),nat,(B3))) ).
% pure_assn_def
tff(fact_123_times__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,times_assn_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).
% times_assn_def
tff(fact_124_type__definition__def,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B)] :
( type_definition(A,B,Rep,Abs,A3)
<=> ( ! [X2: A] : member(B,aa(A,B,Rep,X2),A3)
& ! [X2: A] : aa(B,A,Abs,aa(A,B,Rep,X2)) = X2
& ! [Y3: B] :
( member(B,Y3,A3)
=> ( aa(A,B,Rep,aa(B,A,Abs,Y3)) = Y3 ) ) ) ) ).
% type_definition_def
tff(fact_125_type__definition_ORep__inverse,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),X: A] :
( type_definition(A,B,Rep,Abs,A3)
=> ( aa(B,A,Abs,aa(A,B,Rep,X)) = X ) ) ).
% type_definition.Rep_inverse
tff(fact_126_type__definition_OAbs__inverse,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),Y: B] :
( type_definition(A,B,Rep,Abs,A3)
=> ( member(B,Y,A3)
=> ( aa(A,B,Rep,aa(B,A,Abs,Y)) = Y ) ) ) ).
% type_definition.Abs_inverse
tff(fact_127_type__definition_ORep__inject,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),X: A,Y: A] :
( type_definition(A,B,Rep,Abs,A3)
=> ( ( aa(A,B,Rep,X) = aa(A,B,Rep,Y) )
<=> ( X = Y ) ) ) ).
% type_definition.Rep_inject
tff(fact_128_type__definition_ORep__induct,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),Y: B,P: fun(B,$o)] :
( type_definition(A,B,Rep,Abs,A3)
=> ( member(B,Y,A3)
=> ( ! [X3: A] : aa(B,$o,P,aa(A,B,Rep,X3))
=> aa(B,$o,P,Y) ) ) ) ).
% type_definition.Rep_induct
tff(fact_129_type__definition_OAbs__inject,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),X: B,Y: B] :
( type_definition(A,B,Rep,Abs,A3)
=> ( member(B,X,A3)
=> ( member(B,Y,A3)
=> ( ( aa(B,A,Abs,X) = aa(B,A,Abs,Y) )
<=> ( X = Y ) ) ) ) ) ).
% type_definition.Abs_inject
tff(fact_130_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( pure_assn((P)) = pure_assn((Q)) )
<=> ( (P)
<=> (Q) ) ) ).
% pure_assn_eq_conv
tff(fact_131_mult_Oright__neutral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),one_one(A)) = A4 ) ).
% mult.right_neutral
tff(fact_132_mult__1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A4) = A4 ) ).
% mult_1
tff(fact_133_merge__pure__star,axiom,
! [A4: $o,B3: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((A4))),pure_assn((B3))) = pure_assn(( (A4)
& (B3) )) ).
% merge_pure_star
tff(fact_134_star__false__right,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).
% star_false_right
tff(fact_135_star__false__left,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).
% star_false_left
tff(fact_136_pure__false,axiom,
pure_assn($false) = bot_bot(assn) ).
% pure_false
tff(fact_137_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( pure_assn((P)) = bot_bot(assn) )
<=> ~ (P) ) ).
% pure_assn_eq_false_iff
tff(fact_138_is__pure__assn__starI,axiom,
! [A4: assn,B3: assn] :
( is_pure_assn(A4)
=> ( is_pure_assn(B3)
=> is_pure_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)) ) ) ).
% is_pure_assn_starI
tff(fact_139_is__pure__assn__pure,axiom,
! [P: $o] : is_pure_assn(pure_assn((P))) ).
% is_pure_assn_pure
tff(fact_140_pure__true,axiom,
pure_assn($true) = one_one(assn) ).
% pure_true
tff(fact_141_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( pure_assn((P)) = one_one(assn) )
<=> (P) ) ).
% pure_assn_eq_emp_iff
tff(fact_142_sngr__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P2: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P2,X)),sngr_assn(A,P2,Y)) = bot_bot(assn) ) ).
% sngr_same_false
tff(fact_143_snga__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P2: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P2,X)),snga_assn(A,P2,Y)) = bot_bot(assn) ) ).
% snga_same_false
tff(fact_144_assn__times__comm,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P) ).
% assn_times_comm
tff(fact_145_assn__times__assoc,axiom,
! [P: assn,Q: assn,R3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),R3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),R3)) ).
% assn_times_assoc
tff(fact_146_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [B3: A,A4: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_147_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4) ) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_148_mult_Oassoc,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% mult.assoc
tff(fact_149_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A4) = A4 ) ).
% comm_monoid_mult_class.mult_1
tff(fact_150_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),one_one(A)) = A4 ) ).
% mult.comm_neutral
tff(fact_151_mod__starE,axiom,
! [A4: assn,B3: assn,Ha: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),Ha)
=> ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A4),X_1)
=> ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B3),H_2) ) ) ).
% mod_starE
tff(fact_152_mod__starD,axiom,
! [A3: assn,B5: assn,Ha: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B5)),Ha)
=> ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A3),H1)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B5),H22) ) ) ).
% mod_starD
tff(fact_153_assn__one__left,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P) = P ).
% assn_one_left
tff(fact_154_is__pure__assnE,axiom,
! [A4: assn] :
( is_pure_assn(A4)
=> ~ ! [P3: $o] : A4 != pure_assn((P3)) ) ).
% is_pure_assnE
tff(fact_155_is__pure__assn__def,axiom,
! [A4: assn] :
( is_pure_assn(A4)
<=> ? [P4: $o] : A4 = pure_assn((P4)) ) ).
% is_pure_assn_def
tff(fact_156_type__definition_ORep,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),X: A] :
( type_definition(A,B,Rep,Abs,A3)
=> member(B,aa(A,B,Rep,X),A3) ) ).
% type_definition.Rep
tff(fact_157_type__definition_Ointro,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),A3: set(B),Abs: fun(B,A)] :
( ! [X3: A] : member(B,aa(A,B,Rep,X3),A3)
=> ( ! [X3: A] : aa(B,A,Abs,aa(A,B,Rep,X3)) = X3
=> ( ! [Y2: B] :
( member(B,Y2,A3)
=> ( aa(A,B,Rep,aa(B,A,Abs,Y2)) = Y2 ) )
=> type_definition(A,B,Rep,Abs,A3) ) ) ) ).
% type_definition.intro
tff(fact_158_type__definition_OAbs__cases,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),X: A] :
( type_definition(A,B,Rep,Abs,A3)
=> ~ ! [Y2: B] :
( ( X = aa(B,A,Abs,Y2) )
=> ~ member(B,Y2,A3) ) ) ).
% type_definition.Abs_cases
tff(fact_159_type__definition_ORep__cases,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),Y: B] :
( type_definition(A,B,Rep,Abs,A3)
=> ( member(B,Y,A3)
=> ~ ! [X3: A] : Y != aa(A,B,Rep,X3) ) ) ).
% type_definition.Rep_cases
tff(fact_160_type__definition_OAbs__induct,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),P: fun(A,$o),X: A] :
( type_definition(A,B,Rep,Abs,A3)
=> ( ! [Y2: B] :
( member(B,Y2,A3)
=> aa(A,$o,P,aa(B,A,Abs,Y2)) )
=> aa(A,$o,P,X) ) ) ).
% type_definition.Abs_induct
tff(fact_161_total__on__singleton,axiom,
! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% total_on_singleton
tff(fact_162_refl__on__singleton,axiom,
! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% refl_on_singleton
tff(fact_163_Id__on__empty,axiom,
! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).
% Id_on_empty
tff(fact_164_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% linear_order_on_singleton
tff(fact_165_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod(A,set(B)),Y: $o] :
( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
<=> (Y) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ( ( (Y)
<=> ( ( As2 = bot_bot(set(B)) )
& (X) ) )
=> ~ aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2))) ) ) ) ) ).
% pure_assn_raw.pelims(1)
tff(fact_166_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2)))
=> ~ ( ( As2 = bot_bot(set(B)) )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
tff(fact_167_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H: A,As2: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2)))
=> ( ( As2 = bot_bot(set(B)) )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
tff(fact_168_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A4: B] :
( member(A,B3,image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B)))))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),B3),R2) ) ).
% Image_singleton_iff
tff(fact_169_mult_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% mult.right_assoc
tff(fact_170_mult_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) ) ).
% mult.right_commute
tff(fact_171_in__inv__imagep,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),X: B,Y: B] :
( inv_imagep(A,B,R2,F2,X,Y)
<=> aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,F2,X)),aa(B,A,F2,Y)) ) ).
% in_inv_imagep
tff(fact_172_in__range__empty,axiom,
! [Ha: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),bot_bot(set(nat)))) ).
% in_range_empty
tff(fact_173_ImageI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set(product_prod(A,B)),A3: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> ( member(A,A4,A3)
=> member(B,B3,image(A,B,R2,A3)) ) ) ).
% ImageI
tff(fact_174_Image__empty2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(B,A))] : image(B,A,R3,bot_bot(set(B))) = bot_bot(set(A)) ).
% Image_empty2
tff(fact_175_Id__onI,axiom,
! [A: $tType,A4: A,A3: set(A)] :
( member(A,A4,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),id_on(A,A3)) ) ).
% Id_onI
tff(fact_176_bool__assn__proper_I1_J,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,in_range) ).
% bool_assn_proper(1)
tff(fact_177_Image__empty1,axiom,
! [B: $tType,A: $tType,X5: set(B)] : image(B,A,bot_bot(set(product_prod(B,A))),X5) = bot_bot(set(A)) ).
% Image_empty1
tff(fact_178_refl__on__Id__on,axiom,
! [A: $tType,A3: set(A)] : refl_on(A,A3,id_on(A,A3)) ).
% refl_on_Id_on
tff(fact_179_refl__on__domain,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A,B3: A] :
( refl_on(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(A,A4,A3)
& member(A,B3,A3) ) ) ) ).
% refl_on_domain
tff(fact_180_rev__ImageI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set(A),B3: B,R2: set(product_prod(A,B))] :
( member(A,A4,A3)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> member(B,B3,image(A,B,R2,A3)) ) ) ).
% rev_ImageI
tff(fact_181_Image__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A3: set(B)] :
( member(A,B3,image(B,A,R2,A3))
<=> ? [X2: B] :
( member(B,X2,A3)
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),B3),R2) ) ) ).
% Image_iff
tff(fact_182_ImageE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A3: set(B)] :
( member(A,B3,image(B,A,R2,A3))
=> ~ ! [X3: B] :
( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B3),R2)
=> ~ member(B,X3,A3) ) ) ).
% ImageE
tff(fact_183_refl__onD2,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( refl_on(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,Y,A3) ) ) ).
% refl_onD2
tff(fact_184_refl__onD1,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( refl_on(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,X,A3) ) ) ).
% refl_onD1
tff(fact_185_refl__onD,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A] :
( refl_on(A,A3,R2)
=> ( member(A,A4,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2) ) ) ).
% refl_onD
tff(fact_186_total__on__def,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( total_on(A,A3,R2)
<=> ! [X2: A] :
( member(A,X2,A3)
=> ! [Xa2: A] :
( member(A,Xa2,A3)
=> ( ( X2 != Xa2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),R2) ) ) ) ) ) ).
% total_on_def
tff(fact_187_total__onI,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> ( ( X3 != Y2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),R2) ) ) ) )
=> total_on(A,A3,R2) ) ).
% total_onI
tff(fact_188_total__on__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).
% total_on_empty
tff(fact_189_models__in__range,axiom,
! [P: assn,Ha: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),Ha)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Ha) ) ).
% models_in_range
tff(fact_190_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),id_on(A,A3))
<=> ( ( X = Y )
& member(A,X,A3) ) ) ).
% Id_on_iff
tff(fact_191_Id__on__eqI,axiom,
! [A: $tType,A4: A,B3: A,A3: set(A)] :
( ( A4 = B3 )
=> ( member(A,A4,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),id_on(A,A3)) ) ) ).
% Id_on_eqI
tff(fact_192_Id__onE,axiom,
! [A: $tType,C2: product_prod(A,A),A3: set(A)] :
( member(product_prod(A,A),C2,id_on(A,A3))
=> ~ ! [X3: A] :
( member(A,X3,A3)
=> ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).
% Id_onE
tff(fact_193_lnear__order__on__empty,axiom,
! [A: $tType] : order_679001287576687338der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% lnear_order_on_empty
tff(fact_194_refl__on__empty,axiom,
! [A: $tType] : refl_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% refl_on_empty
tff(fact_195_properD1,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Ha: heap_ext(product_unit),As: set(nat)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As)) ) ) ).
% properD1
tff(fact_196_proper__iff,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( relH(As,Ha,H2)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ) ) ).
% proper_iff
tff(fact_197_proper__def,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
<=> ! [H3: heap_ext(product_unit),H4: heap_ext(product_unit),As3: set(nat)] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)) )
& ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& relH(As3,H3,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3)) ) ) ) ).
% proper_def
tff(fact_198_properD2,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Ha: heap_ext(product_unit),As: set(nat),H2: heap_ext(product_unit)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> ( relH(As,Ha,H2)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ) ) ).
% properD2
tff(fact_199_properI,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( ! [As2: set(nat),H: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) )
=> ( ! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ( relH(As2,H,H5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2)) ) ) )
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P) ) ) ).
% properI
tff(fact_200_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E3: set(product_prod(B,A))] : vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U),E3) = image(B,A,E3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),U),bot_bot(set(B)))) ).
% pair_vimage_is_Image
tff(fact_201_Range__insert,axiom,
! [A: $tType,B: $tType,A4: B,B3: A,R2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ).
% Range_insert
tff(fact_202_Domain__insert,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ).
% Domain_insert
tff(fact_203_preorder__on__empty,axiom,
! [A: $tType] : order_preorder_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% preorder_on_empty
tff(fact_204_trancl__single,axiom,
! [A: $tType,A4: A,B3: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),bot_bot(set(product_prod(A,A)))) ).
% trancl_single
tff(fact_205_trans__singleton,axiom,
! [A: $tType,A4: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),bot_bot(set(product_prod(A,A))))) ).
% trans_singleton
tff(fact_206_mult_Osafe__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [X: A,Y: A,A4: A,B3: A] :
( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),A4)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4) ) ) ) ).
% mult.safe_commute
tff(fact_207_vimage__eq,axiom,
! [A: $tType,B: $tType,A4: A,F2: fun(A,B),B5: set(B)] :
( member(A,A4,vimage(A,B,F2,B5))
<=> member(B,aa(A,B,F2,A4),B5) ) ).
% vimage_eq
tff(fact_208_vimageI,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A4: B,B3: A,B5: set(A)] :
( ( aa(B,A,F2,A4) = B3 )
=> ( member(A,B3,B5)
=> member(B,A4,vimage(B,A,F2,B5)) ) ) ).
% vimageI
tff(fact_209_vimage__empty,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,bot_bot(set(B))) = bot_bot(set(A)) ).
% vimage_empty
tff(fact_210_Domain__Id__on,axiom,
! [A: $tType,A3: set(A)] : aa(set(product_prod(A,A)),set(A),domain(A,A),id_on(A,A3)) = A3 ).
% Domain_Id_on
tff(fact_211_Range__Id__on,axiom,
! [A: $tType,A3: set(A)] : aa(set(product_prod(A,A)),set(A),range2(A,A),id_on(A,A3)) = A3 ).
% Range_Id_on
tff(fact_212_Domain__empty,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),domain(A,B),bot_bot(set(product_prod(A,B)))) = bot_bot(set(A)) ).
% Domain_empty
tff(fact_213_Range__empty,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),range2(B,A),bot_bot(set(product_prod(B,A)))) = bot_bot(set(A)) ).
% Range_empty
tff(fact_214_relH__sym,axiom,
! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( relH(As,Ha,H2)
=> relH(As,H2,Ha) ) ).
% relH_sym
tff(fact_215_relH__trans,axiom,
! [As: set(nat),H12: heap_ext(product_unit),H23: heap_ext(product_unit),H32: heap_ext(product_unit)] :
( relH(As,H12,H23)
=> ( relH(As,H23,H32)
=> relH(As,H12,H32) ) ) ).
% relH_trans
tff(fact_216_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: fun(B,$o),F2: fun(A,B),Q: fun(A,$o)] :
( ! [X3: A] :
( aa(B,$o,P,aa(A,B,F2,X3))
<=> aa(A,$o,Q,X3) )
=> ( vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),P)) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).
% vimage_Collect
tff(fact_217_vimageI2,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A4: B,A3: set(A)] :
( member(A,aa(B,A,F2,A4),A3)
=> member(B,A4,vimage(B,A,F2,A3)) ) ).
% vimageI2
tff(fact_218_vimageE,axiom,
! [A: $tType,B: $tType,A4: A,F2: fun(A,B),B5: set(B)] :
( member(A,A4,vimage(A,B,F2,B5))
=> member(B,aa(A,B,F2,A4),B5) ) ).
% vimageE
tff(fact_219_vimageD,axiom,
! [A: $tType,B: $tType,A4: A,F2: fun(A,B),A3: set(B)] :
( member(A,A4,vimage(A,B,F2,A3))
=> member(B,aa(A,B,F2,A4),A3) ) ).
% vimageD
tff(fact_220_transD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ).
% transD
tff(fact_221_transE,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ).
% transE
tff(fact_222_transI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z4),R2) ) )
=> trans(A,R2) ) ).
% transI
tff(fact_223_trans__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
<=> ! [X2: A,Y3: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3),R2) ) ) ) ).
% trans_def
tff(fact_224_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A4: A,R2: set(product_prod(A,B))] :
( member(A,A4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
=> ~ ! [B2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2),R2) ) ).
% Domain.cases
tff(fact_225_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,R2: set(product_prod(A,B))] :
( member(A,A4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
<=> ? [A7: A,B7: B] :
( ( A4 = A7 )
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7),R2) ) ) ).
% Domain.simps
tff(fact_226_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> member(A,A4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).
% Domain.DomainI
tff(fact_227_DomainE,axiom,
! [B: $tType,A: $tType,A4: A,R2: set(product_prod(A,B))] :
( member(A,A4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
=> ~ ! [B2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2),R2) ) ).
% DomainE
tff(fact_228_Domain__iff,axiom,
! [B: $tType,A: $tType,A4: A,R2: set(product_prod(A,B))] :
( member(A,A4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
<=> ? [Y3: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),Y3),R2) ) ).
% Domain_iff
tff(fact_229_Range_Ocases,axiom,
! [A: $tType,B: $tType,A4: A,R2: set(product_prod(B,A))] :
( member(A,A4,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
=> ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),A4),R2) ) ).
% Range.cases
tff(fact_230_Range_Osimps,axiom,
! [A: $tType,B: $tType,A4: A,R2: set(product_prod(B,A))] :
( member(A,A4,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
<=> ? [A7: B,B7: A] :
( ( A4 = B7 )
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B7),R2) ) ) ).
% Range.simps
tff(fact_231_Range_Ointros,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> member(B,B3,aa(set(product_prod(A,B)),set(B),range2(A,B),R2)) ) ).
% Range.intros
tff(fact_232_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A))] :
( member(A,B3,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
=> ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B3),R2) ) ).
% RangeE
tff(fact_233_Range__iff,axiom,
! [A: $tType,B: $tType,A4: A,R2: set(product_prod(B,A))] :
( member(A,A4,aa(set(product_prod(B,A)),set(A),range2(B,A),R2))
<=> ? [Y3: B] : member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y3),A4),R2) ) ).
% Range_iff
tff(fact_234_trans__empty,axiom,
! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).
% trans_empty
tff(fact_235_trans__Id__on,axiom,
! [A: $tType,A3: set(A)] : trans(A,id_on(A,A3)) ).
% trans_Id_on
tff(fact_236_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A4: A,F2: fun(A,B),B3: B] :
( member(A,A4,vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B)))))
<=> ( aa(A,B,F2,A4) = B3 ) ) ).
% vimage_singleton_eq
tff(fact_237_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( ( aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = bot_bot(set(A)) )
<=> ( R2 = bot_bot(set(product_prod(A,B))) ) ) ).
% Domain_empty_iff
tff(fact_238_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
( ( aa(set(product_prod(B,A)),set(A),range2(B,A),R2) = bot_bot(set(A)) )
<=> ( R2 = bot_bot(set(product_prod(B,A))) ) ) ).
% Range_empty_iff
tff(fact_239_relH__in__rangeI_I2_J,axiom,
! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( relH(As,Ha,H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ).
% relH_in_rangeI(2)
tff(fact_240_relH__in__rangeI_I1_J,axiom,
! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( relH(As,Ha,H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As)) ) ).
% relH_in_rangeI(1)
tff(fact_241_relH__refl,axiom,
! [Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> relH(As,Ha,Ha) ) ).
% relH_refl
tff(fact_242_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R3: set(product_prod(A,A)),V: A] :
( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_trancl(A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).
% Image_empty_trancl_Image_empty
tff(fact_243_mod__relH,axiom,
! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit),P: assn] :
( relH(As,Ha,H2)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).
% mod_relH
tff(fact_244_trancl__empty,axiom,
! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).
% trancl_empty
tff(fact_245_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R3: set(product_prod(A,B))] :
( member(A,X,aa(set(product_prod(A,B)),set(A),domain(A,B),R3))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),fun_of_rel(A,B,R3,X)),R3) ) ).
% for_in_RI
tff(fact_246_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),R2)
=> ~ ! [B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B2),transitive_trancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A22),R2) ) ) ) ).
% trancl.cases
tff(fact_247_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
<=> ( ? [A7: A,B7: A] :
( ( A1 = A7 )
& ( A22 = B7 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),R2) )
| ? [A7: A,B7: A,C5: A] :
( ( A1 = A7 )
& ( A22 = C5 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_trancl(A,R2))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5),R2) ) ) ) ).
% trancl.simps
tff(fact_248_trancl_Or__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2)) ) ).
% trancl.r_into_trancl
tff(fact_249_tranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ~ ! [C3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C3),transitive_trancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C3),B3),R2) ) ) ) ).
% tranclE
tff(fact_250_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2)) ) ) ).
% trancl_trans
tff(fact_251_trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),Y2),R2)
=> aa(A,$o,P,Y2) )
=> ( ! [Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),Y2),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( aa(A,$o,P,Y2)
=> aa(A,$o,P,Z4) ) ) )
=> aa(A,$o,P,B3) ) ) ) ).
% trancl_induct
tff(fact_252_r__r__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R3: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R3)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R3)) ) ) ).
% r_r_into_trancl
tff(fact_253_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2)
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),transitive_trancl(A,R2)) ) ) ) ).
% converse_tranclE
tff(fact_254_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R2))
=> ( ! [A5: A,B2: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),R2)
=> aa(B,$o,aa(A,fun(B,$o),P,A5),B2) )
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),transitive_trancl(product_prod(A,B),R2))
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
=> ( aa(B,$o,aa(A,fun(B,$o),P,A5),B2)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% trancl_induct2
tff(fact_255_converse__trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B3),R2)
=> aa(A,$o,P,Y2) )
=> ( ! [Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B3),transitive_trancl(A,R2))
=> ( aa(A,$o,P,Z4)
=> aa(A,$o,P,Y2) ) ) )
=> aa(A,$o,P,A4) ) ) ) ).
% converse_trancl_induct
tff(fact_256_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),P: fun(A,fun(A,$o))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R2))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> aa(A,$o,aa(A,fun(A,$o),P,X3),Y2) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),transitive_trancl(A,R2))
=> ( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),transitive_trancl(A,R2))
=> ( aa(A,$o,aa(A,fun(A,$o),P,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),P,X3),Z4) ) ) ) )
=> aa(A,$o,aa(A,fun(A,$o),P,X),Y) ) ) ) ).
% trancl_trans_induct
tff(fact_257_trancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R2)) ) ) ).
% trancl_into_trancl2
tff(fact_258_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R2)) ) ) ).
% Transitive_Closure.trancl_into_trancl
tff(fact_259_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
tff(fact_260_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))),image(A,A,transitive_trancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A))))) ) ).
% trancl_Image_advance_ss
tff(fact_261_wand__assnI,axiom,
! [Ha: heap_ext(product_unit),As: set(nat),Q: assn,R3: assn] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> ( ! [H5: heap_ext(product_unit),As4: set(nat)] :
( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As4) = bot_bot(set(nat)) )
=> ( relH(As,Ha,H5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,R3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As4))) ) ) ) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,wand_assn(Q,R3)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As)) ) ) ).
% wand_assnI
tff(fact_262_trancl__Image__in__Range,axiom,
! [A: $tType,R3: set(product_prod(A,A)),V2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,R3),V2)),aa(set(product_prod(A,A)),set(A),range2(A,A),R3)) ).
% trancl_Image_in_Range
tff(fact_263_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A3: set(B),F2: fun(B,set(A))] :
vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,A3,F2)) = $ite(member(B,X,A3),aa(B,set(A),F2,X),bot_bot(set(A))) ).
% Pair_vimage_Sigma
tff(fact_264_proj__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B] : aa(B,set(A),equiv_proj(B,A,R2),X) = image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))) ).
% proj_def
tff(fact_265_relH__def,axiom,
! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( relH(As,Ha,H2)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [T2: typerep,X2: nat] :
( member(nat,X2,As)
=> ( ( refs(product_unit,Ha,T2,X2) = refs(product_unit,H2,T2,X2) )
& ( arrays(product_unit,Ha,T2,X2) = arrays(product_unit,H2,T2,X2) ) ) ) ) ) ).
% relH_def
tff(fact_266_vimage__insert,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A4: B,B5: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B))))),vimage(A,B,F2,B5)) ).
% vimage_insert
tff(fact_267_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V1),V22)),E3)))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W),transitive_trancl(A,E3))
=> ~ ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1),transitive_rtrancl(A,E3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W),transitive_rtrancl(A,E3)) ) ) ) ).
% trancl_over_edgeE
tff(fact_268_partial__order__on__empty,axiom,
! [A: $tType] : order_7125193373082350890der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% partial_order_on_empty
tff(fact_269_antisym__singleton,axiom,
! [A: $tType,X: product_prod(A,A)] : antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),X),bot_bot(set(product_prod(A,A))))) ).
% antisym_singleton
tff(fact_270_prod__set__simps_I2_J,axiom,
! [B: $tType,A: $tType,X: B,Y: A] : basic_snds(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))) ).
% prod_set_simps(2)
tff(fact_271_dual__order_Orefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),A4) ) ).
% dual_order.refl
tff(fact_272_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X) ) ).
% order_refl
tff(fact_273_subset__antisym,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( A3 = B5 ) ) ) ).
% subset_antisym
tff(fact_274_subsetI,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ! [X3: A] :
( member(A,X3,A3)
=> member(A,X3,B5) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% subsetI
tff(fact_275_Int__iff,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
<=> ( member(A,C2,A3)
& member(A,C2,B5) ) ) ).
% Int_iff
tff(fact_276_IntI,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,A3)
=> ( member(A,C2,B5)
=> member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ) ) ).
% IntI
tff(fact_277_Un__iff,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
<=> ( member(A,C2,A3)
| member(A,C2,B5) ) ) ).
% Un_iff
tff(fact_278_UnCI,axiom,
! [A: $tType,C2: A,B5: set(A),A3: set(A)] :
( ( ~ member(A,C2,B5)
=> member(A,C2,A3) )
=> member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ) ).
% UnCI
tff(fact_279_empty__subsetI,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A3) ).
% empty_subsetI
tff(fact_280_subset__empty,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A)))
<=> ( A3 = bot_bot(set(A)) ) ) ).
% subset_empty
tff(fact_281_insert__subset,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),B5)
<=> ( member(A,X,B5)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ) ).
% insert_subset
tff(fact_282_SigmaI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set(A),B3: B,B5: fun(A,set(B))] :
( member(A,A4,A3)
=> ( member(B,B3,aa(A,set(B),B5,A4))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),product_Sigma(A,B,A3,B5)) ) ) ).
% SigmaI
tff(fact_283_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set(A),B5: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),product_Sigma(A,B,A3,B5))
<=> ( member(A,A4,A3)
& member(B,B3,aa(A,set(B),B5,A4)) ) ) ).
% mem_Sigma_iff
tff(fact_284_Un__empty,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = bot_bot(set(A)) )
<=> ( ( A3 = bot_bot(set(A)) )
& ( B5 = bot_bot(set(A)) ) ) ) ).
% Un_empty
tff(fact_285_Int__subset__iff,axiom,
! [A: $tType,C6: set(A),A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),A3)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),B5) ) ) ).
% Int_subset_iff
tff(fact_286_Int__insert__right__if1,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] :
( member(A,A4,A3)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ) ) ).
% Int_insert_right_if1
tff(fact_287_Int__insert__right__if0,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] :
( ~ member(A,A4,A3)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ) ) ).
% Int_insert_right_if0
tff(fact_288_insert__inter__insert,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ).
% insert_inter_insert
tff(fact_289_Int__insert__left__if1,axiom,
! [A: $tType,A4: A,C6: set(A),B5: set(A)] :
( member(A,A4,C6)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)),C6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) ) ) ).
% Int_insert_left_if1
tff(fact_290_Int__insert__left__if0,axiom,
! [A: $tType,A4: A,C6: set(A),B5: set(A)] :
( ~ member(A,A4,C6)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6) ) ) ).
% Int_insert_left_if0
tff(fact_291_Un__subset__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),C6)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6) ) ) ).
% Un_subset_iff
tff(fact_292_Un__insert__right,axiom,
! [A: $tType,A3: set(A),A4: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ).
% Un_insert_right
tff(fact_293_Un__insert__left,axiom,
! [A: $tType,A4: A,B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)),C6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) ).
% Un_insert_left
tff(fact_294_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)),S) = S ).
% Un_Int_eq(1)
tff(fact_295_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)),T3) = T3 ).
% Un_Int_eq(2)
tff(fact_296_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = S ).
% Un_Int_eq(3)
tff(fact_297_Un__Int__eq_I4_J,axiom,
! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = T3 ).
% Un_Int_eq(4)
tff(fact_298_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)),S) = S ).
% Int_Un_eq(1)
tff(fact_299_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)),T3) = T3 ).
% Int_Un_eq(2)
tff(fact_300_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)) = S ).
% Int_Un_eq(3)
tff(fact_301_Int__Un__eq_I4_J,axiom,
! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)) = T3 ).
% Int_Un_eq(4)
tff(fact_302_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B5: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B5)) ).
% vimage_Int
tff(fact_303_vimage__Un,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B5: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B5)) ).
% vimage_Un
tff(fact_304_relH__dist__union,axiom,
! [As: set(nat),As5: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit)] :
( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5),Ha,H2)
<=> ( relH(As,Ha,H2)
& relH(As5,Ha,H2) ) ) ).
% relH_dist_union
tff(fact_305_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A4: A,A3: set(A)] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) )
<=> ( ( A4 = B3 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ).
% singleton_insert_inj_eq
tff(fact_306_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A4: A,A3: set(A),B3: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) )
<=> ( ( A4 = B3 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ).
% singleton_insert_inj_eq'
tff(fact_307_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set(A),B3: A,B5: set(A)] :
( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B5)) )
<=> ( ~ member(A,B3,A3)
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ) ) ) ).
% disjoint_insert(2)
tff(fact_308_disjoint__insert_I1_J,axiom,
! [A: $tType,B5: set(A),A4: A,A3: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = bot_bot(set(A)) )
<=> ( ~ member(A,A4,B5)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A3) = bot_bot(set(A)) ) ) ) ).
% disjoint_insert(1)
tff(fact_309_insert__disjoint_I2_J,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] :
( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)),B5) )
<=> ( ~ member(A,A4,B5)
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ) ) ) ).
% insert_disjoint(2)
tff(fact_310_insert__disjoint_I1_J,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)),B5) = bot_bot(set(A)) )
<=> ( ~ member(A,A4,B5)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) ) ) ) ).
% insert_disjoint(1)
tff(fact_311_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B5: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B5) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty1
tff(fact_312_Image__Id__on,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : image(A,A,id_on(A,A3),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ).
% Image_Id_on
tff(fact_313_in__range__dist__union,axiom,
! [Ha: heap_ext(product_unit),As: set(nat),As5: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5)))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As5)) ) ) ).
% in_range_dist_union
tff(fact_314_reachable__mono,axiom,
! [A: $tType,R3: set(product_prod(A,A)),R4: set(product_prod(A,A)),X5: set(A),X6: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),R4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),X6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,R3),X5)),image(A,A,transitive_rtrancl(A,R4),X6)) ) ) ).
% reachable_mono
tff(fact_315_antisym__subset,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
=> ( antisym(A,S2)
=> antisym(A,R2) ) ) ).
% antisym_subset
tff(fact_316_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% order_antisym_conv
tff(fact_317_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_le_cases
tff(fact_318_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( ( aa(A,B,F2,B3) = C2 )
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% ord_le_eq_subst
tff(fact_319_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( ( A4 = aa(B,A,F2,B3) )
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% ord_eq_le_subst
tff(fact_320_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_linear
tff(fact_321_subset__Collect__conv,axiom,
! [A: $tType,S: set(A),P: fun(A,$o)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(fun(A,$o),set(A),collect(A),P))
<=> ! [X2: A] :
( member(A,X2,S)
=> aa(A,$o,P,X2) ) ) ).
% subset_Collect_conv
tff(fact_322_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),S4)
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S4)) ) ).
% rtrancl_mono_rightI
tff(fact_323_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( ( X = Y )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% order_eq_refl
tff(fact_324_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( A4 = B3 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> ( ( C2 = D3 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),D3) ) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_325_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% order_subst2
tff(fact_326_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(B,A,F2,B3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% order_subst1
tff(fact_327_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ).
% Orderings.order_eq_iff
tff(fact_328_inter__eq__subsetI,axiom,
! [A: $tType,S: set(A),S4: set(A),A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),S4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S4) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S) ) ) ) ).
% inter_eq_subsetI
tff(fact_329_Int__left__commute,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6)) ).
% Int_left_commute
tff(fact_330_Int__Collect__mono,axiom,
! [A: $tType,A3: set(A),B5: set(A),P: fun(A,$o),Q: fun(A,$o)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) ) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(fun(A,$o),set(A),collect(A),Q))) ) ) ).
% Int_Collect_mono
tff(fact_331_Collect__mono__iff,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q))
<=> ! [X2: A] :
( aa(A,$o,P,X2)
=> aa(A,$o,Q,X2) ) ) ).
% Collect_mono_iff
tff(fact_332_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F2: fun(A,B),G: fun(A,B)] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
<=> ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) ) ) ).
% le_fun_def
tff(fact_333_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set(product_prod(A,A)),V2: set(product_prod(A,A)),X: product_prod(A,A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V2)
=> ( member(product_prod(A,A),X,transitive_rtrancl(A,U2))
=> member(product_prod(A,A),X,transitive_rtrancl(A,V2)) ) ) ).
% rtrancl_mono_mp
tff(fact_334_Un__left__commute,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C6)) ).
% Un_left_commute
tff(fact_335_Un__Int__distrib2,axiom,
! [A: $tType,B5: set(A),C6: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C6),A3)) ).
% Un_Int_distrib2
tff(fact_336_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),A3) ) ).
% Un_Int_assoc_eq
tff(fact_337_Int__left__absorb,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ).
% Int_left_absorb
tff(fact_338_Int__Un__distrib2,axiom,
! [A: $tType,B5: set(A),C6: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),A3)) ).
% Int_Un_distrib2
tff(fact_339_Un__left__absorb,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) ).
% Un_left_absorb
tff(fact_340_Un__Int__distrib,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C6)) ).
% Un_Int_distrib
tff(fact_341_Int__Un__distrib,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6)) ).
% Int_Un_distrib
tff(fact_342_set__eq__subset,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( A3 = B5 )
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).
% set_eq_subset
tff(fact_343_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F2: fun(A,B),G: fun(A,B)] :
( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
=> aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).
% le_funI
tff(fact_344_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F2: fun(A,B),G: fun(A,B),X: A] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).
% le_funE
tff(fact_345_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F2: fun(A,B),G: fun(A,B),X: A] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).
% le_funD
tff(fact_346_antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( A4 = B3 ) ) ) ) ).
% antisym
tff(fact_347_r__le__rtrancl,axiom,
! [A: $tType,S: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S)) ).
% r_le_rtrancl
tff(fact_348_subset__trans,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6) ) ) ).
% subset_trans
tff(fact_349_subset__Un__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = B5 ) ) ).
% subset_Un_eq
tff(fact_350_Un__Int__crazy,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C6),A3)) ).
% Un_Int_crazy
tff(fact_351_Int__greatest,axiom,
! [A: $tType,C6: set(A),A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ) ) ).
% Int_greatest
tff(fact_352_Collect__mono,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ) ).
% Collect_mono
tff(fact_353_subset__refl,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A3) ).
% subset_refl
tff(fact_354_Int__commute,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A3) ).
% Int_commute
tff(fact_355_Int__absorb2,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = A3 ) ) ).
% Int_absorb2
tff(fact_356_Int__absorb1,axiom,
! [A: $tType,B5: set(A),A3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = B5 ) ) ).
% Int_absorb1
tff(fact_357_subset__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
<=> ! [T2: A] :
( member(A,T2,A3)
=> member(A,T2,B5) ) ) ).
% subset_iff
tff(fact_358_subset__UnE,axiom,
! [A: $tType,C6: set(A),A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
=> ~ ! [A8: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),A3)
=> ! [B8: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),B5)
=> ( C6 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B8) ) ) ) ) ).
% subset_UnE
tff(fact_359_equalityD2,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( A3 = B5 )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ).
% equalityD2
tff(fact_360_equalityD1,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( A3 = B5 )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% equalityD1
tff(fact_361_Un__commute,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A3) ).
% Un_commute
tff(fact_362_Un__absorb2,axiom,
! [A: $tType,B5: set(A),A3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = A3 ) ) ).
% Un_absorb2
tff(fact_363_Un__absorb1,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = B5 ) ) ).
% Un_absorb1
tff(fact_364_Int__lower2,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),B5) ).
% Int_lower2
tff(fact_365_Int__lower1,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),A3) ).
% Int_lower1
tff(fact_366_Int__absorb,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),A3) = A3 ).
% Int_absorb
tff(fact_367_subset__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
<=> ! [X2: A] :
( member(A,X2,A3)
=> member(A,X2,B5) ) ) ).
% subset_eq
tff(fact_368_equalityE,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( A3 = B5 )
=> ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).
% equalityE
tff(fact_369_Un__upper2,axiom,
! [A: $tType,B5: set(A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ).
% Un_upper2
tff(fact_370_Un__upper1,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ).
% Un_upper1
tff(fact_371_Un__absorb,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),A3) = A3 ).
% Un_absorb
tff(fact_372_Int__assoc,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) ).
% Int_assoc
tff(fact_373_Un__least,axiom,
! [A: $tType,A3: set(A),C6: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),C6) ) ) ).
% Un_least
tff(fact_374_Un__assoc,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) ).
% Un_assoc
tff(fact_375_Int__mono,axiom,
! [A: $tType,A3: set(A),C6: set(A),B5: set(A),D4: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),D4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),D4)) ) ) ).
% Int_mono
tff(fact_376_subsetD,axiom,
! [A: $tType,A3: set(A),B5: set(A),C2: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( member(A,C2,A3)
=> member(A,C2,B5) ) ) ).
% subsetD
tff(fact_377_in__mono,axiom,
! [A: $tType,A3: set(A),B5: set(A),X: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( member(A,X,A3)
=> member(A,X,B5) ) ) ).
% in_mono
tff(fact_378_ball__Un,axiom,
! [A: $tType,A3: set(A),B5: set(A),P: fun(A,$o)] :
( ! [X2: A] :
( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
=> aa(A,$o,P,X2) )
<=> ( ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,P,X2) )
& ! [X2: A] :
( member(A,X2,B5)
=> aa(A,$o,P,X2) ) ) ) ).
% ball_Un
tff(fact_379_Un__mono,axiom,
! [A: $tType,A3: set(A),C6: set(A),B5: set(A),D4: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),D4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C6),D4)) ) ) ).
% Un_mono
tff(fact_380_bex__Un,axiom,
! [A: $tType,A3: set(A),B5: set(A),P: fun(A,$o)] :
( ? [X2: A] :
( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
& aa(A,$o,P,X2) )
<=> ( ? [X2: A] :
( member(A,X2,A3)
& aa(A,$o,P,X2) )
| ? [X2: A] :
( member(A,X2,B5)
& aa(A,$o,P,X2) ) ) ) ).
% bex_Un
tff(fact_381_IntD2,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> member(A,C2,B5) ) ).
% IntD2
tff(fact_382_IntD1,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> member(A,C2,A3) ) ).
% IntD1
tff(fact_383_UnI2,axiom,
! [A: $tType,C2: A,B5: set(A),A3: set(A)] :
( member(A,C2,B5)
=> member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ) ).
% UnI2
tff(fact_384_UnI1,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,A3)
=> member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ) ).
% UnI1
tff(fact_385_IntE,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> ~ ( member(A,C2,A3)
=> ~ member(A,C2,B5) ) ) ).
% IntE
tff(fact_386_UnE,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
=> ( ~ member(A,C2,A3)
=> member(A,C2,B5) ) ) ).
% UnE
tff(fact_387_dual__order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A4) ) ) ) ).
% dual_order.trans
tff(fact_388_dual__order_Oantisym,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
tff(fact_389_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% dual_order.eq_iff
tff(fact_390_linorder__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,$o)),A4: A,B3: A] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B2)
=> aa(A,$o,aa(A,fun(A,$o),P,A5),B2) )
=> ( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,B2),A5)
=> aa(A,$o,aa(A,fun(A,$o),P,A5),B2) )
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) ) ) ) ).
% linorder_wlog
tff(fact_391_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).
% order_trans
tff(fact_392_order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),C2) ) ) ) ).
% order.trans
tff(fact_393_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( X = Y ) ) ) ) ).
% order_antisym
tff(fact_394_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( ( B3 = C2 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),C2) ) ) ) ).
% ord_le_eq_trans
tff(fact_395_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),C2) ) ) ) ).
% ord_eq_le_trans
tff(fact_396_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ( X = Y )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).
% order_class.order_eq_iff
tff(fact_397_le__cases3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X) )
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ) ) ) ).
% le_cases3
tff(fact_398_nle__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: A,B3: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
& ( B3 != A4 ) ) ) ) ).
% nle_le
tff(fact_399_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I: set(A),J: set(A),C6: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I),J),C6) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I,C6)),product_Sigma(A,B,J,C6)) ).
% Sigma_Int_distrib1
tff(fact_400_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I: set(A),J: set(A),C6: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),J),C6) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I,C6)),product_Sigma(A,B,J,C6)) ).
% Sigma_Un_distrib1
tff(fact_401_Sigma__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),C6: set(A),B5: fun(A,set(B)),D4: fun(A,set(B))] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B5,X3)),aa(A,set(B),D4,X3)) )
=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,B5)),product_Sigma(A,B,C6,D4)) ) ) ).
% Sigma_mono
tff(fact_402_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(B,A)),B5: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),A3),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),A3)),aa(set(product_prod(B,A)),set(A),range2(B,A),B5))) ).
% Range_Int_subset
tff(fact_403_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B)),B5: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),A3),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A3)),aa(set(product_prod(A,B)),set(A),domain(A,B),B5))) ).
% Domain_Int_subset
tff(fact_404_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A4: A,B3: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B3),transitive_rtrancl(A,P))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),Q)
=> ( Y2 = X3 ) ) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,P)) ) ) ).
% rtrancl_Un_separator_converseE
tff(fact_405_rtrancl__Un__separatorE,axiom,
! [A: $tType,A4: A,B3: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),X3),transitive_rtrancl(A,P))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),Q)
=> ( X3 = Y2 ) ) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,P)) ) ) ).
% rtrancl_Un_separatorE
tff(fact_406_disjoint__mono,axiom,
! [A: $tType,A4: set(A),A6: set(A),B3: set(A),B4: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),A6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),B4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B4) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ) ) ).
% disjoint_mono
tff(fact_407_rtrancl__image__unfold__right,axiom,
! [A: $tType,E3: set(product_prod(A,A)),V2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E3,image(A,A,transitive_rtrancl(A,E3),V2))),image(A,A,transitive_rtrancl(A,E3),V2)) ).
% rtrancl_image_unfold_right
tff(fact_408_rtrancl__reachable__induct,axiom,
! [A: $tType,I: set(A),INV: set(A),E3: set(product_prod(A,A))] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),I),INV)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E3,INV)),INV)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,E3),I)),INV) ) ) ).
% rtrancl_reachable_induct
tff(fact_409_Un__Image,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S: set(product_prod(B,A)),A3: set(B)] : image(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R3),S),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(B,A,R3,A3)),image(B,A,S,A3)) ).
% Un_Image
tff(fact_410_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R3,A3)),image(B,A,R3,B5))) ).
% Image_Int_subset
tff(fact_411_Image__mono,axiom,
! [B: $tType,A: $tType,R5: set(product_prod(A,B)),R2: set(product_prod(A,B)),A9: set(A),A3: set(A)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R5),R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A9),A3)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R5,A9)),image(A,B,R2,A3)) ) ) ).
% Image_mono
tff(fact_412_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B)),B5: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A3)),aa(set(product_prod(A,B)),set(A),domain(A,B),B5)) ).
% Domain_Un_eq
tff(fact_413_refl__on__Un,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),B5: set(A),S2: set(product_prod(A,A))] :
( refl_on(A,A3,R2)
=> ( refl_on(A,B5,S2)
=> refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ).
% refl_on_Un
tff(fact_414_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: product_prod(A,A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),X),R3))) ).
% rtrancl_sub_insert_rtrancl
tff(fact_415_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(B,A)),B5: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(B,A)),set(A),range2(B,A),A3)),aa(set(product_prod(B,A)),set(A),range2(B,A),B5)) ).
% Range_Un_eq
tff(fact_416_refl__on__Int,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),B5: set(A),S2: set(product_prod(A,A))] :
( refl_on(A,A3,R2)
=> ( refl_on(A,B5,S2)
=> refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).
% refl_on_Int
tff(fact_417_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)),aa(set(product_prod(A,B)),set(A),domain(A,B),S2)) ) ).
% Domain_mono
tff(fact_418_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(A,B)),set(B),range2(A,B),R2)),aa(set(product_prod(A,B)),set(B),range2(A,B),S2)) ) ).
% Range_mono
tff(fact_419_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E3: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E3),U2)))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,E3))
=> ? [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),X3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E3),U2)))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),U2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),W),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E3),U2))) ) ) ) ).
% trancl_union_outside
tff(fact_420_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod(A,B),A3: set(A),B5: fun(A,set(B))] :
( member(product_prod(A,B),C2,product_Sigma(A,B,A3,B5))
=> ~ ! [X3: A] :
( member(A,X3,A3)
=> ! [Y2: B] :
( member(B,Y2,aa(A,set(B),B5,X3))
=> ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) ) ) ) ) ).
% SigmaE
tff(fact_421_SigmaD1,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set(A),B5: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),product_Sigma(A,B,A3,B5))
=> member(A,A4,A3) ) ).
% SigmaD1
tff(fact_422_SigmaD2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set(A),B5: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),product_Sigma(A,B,A3,B5))
=> member(B,B3,aa(A,set(B),B5,A4)) ) ).
% SigmaD2
tff(fact_423_SigmaE2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set(A),B5: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),product_Sigma(A,B,A3,B5))
=> ~ ( member(A,A4,A3)
=> ~ member(B,B3,aa(A,set(B),B5,A4)) ) ) ).
% SigmaE2
tff(fact_424_trancl__image__by__rtrancl,axiom,
! [A: $tType,E3: set(product_prod(A,A)),Vi: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(A,A,transitive_trancl(A,E3),Vi)),Vi) = image(A,A,transitive_rtrancl(A,E3),Vi) ).
% trancl_image_by_rtrancl
tff(fact_425_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,R3))
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V),transitive_rtrancl(A,R3)) ) ) ) ) ).
% converse_rtranclE'
tff(fact_426_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
=> ( ( A22 != A1 )
=> ~ ! [B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B2),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A22),R2) ) ) ) ).
% rtrancl.cases
tff(fact_427_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
<=> ( ? [A7: A] :
( ( A1 = A7 )
& ( A22 = A7 ) )
| ? [A7: A,B7: A,C5: A] :
( ( A1 = A7 )
& ( A22 = C5 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_rtrancl(A,R2))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5),R2) ) ) ) ).
% rtrancl.simps
tff(fact_428_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),transitive_rtrancl(A,R2)) ).
% rtrancl.rtrancl_refl
tff(fact_429_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_rtrancl(A,R2)) ) ) ).
% rtrancl.rtrancl_into_rtrancl
tff(fact_430_rtranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( ( A4 != B3 )
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),Y2),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B3),R2) ) ) ) ).
% rtranclE
tff(fact_431_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2)) ) ) ).
% rtrancl_trans
tff(fact_432_rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,A4)
=> ( ! [Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),Y2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( aa(A,$o,P,Y2)
=> aa(A,$o,P,Z4) ) ) )
=> aa(A,$o,P,B3) ) ) ) ).
% rtrancl_induct
tff(fact_433_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2))
=> ( ( X != Z2 )
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),transitive_rtrancl(A,R2)) ) ) ) ).
% converse_rtranclE
tff(fact_434_converse__rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,B3)
=> ( ! [Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B3),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,Z4)
=> aa(A,$o,P,Y2) ) ) )
=> aa(A,$o,P,A4) ) ) ) ).
% converse_rtrancl_induct
tff(fact_435_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_rtrancl(A,R2)) ) ) ).
% converse_rtrancl_into_rtrancl
tff(fact_436_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),transitive_rtrancl(product_prod(A,B),R2))
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
=> ( aa(B,$o,aa(A,fun(B,$o),P,A5),B2)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% rtrancl_induct2
tff(fact_437_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R2))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
=> ~ ! [A5: A,B2: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),R2)
=> ~ member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R2)) ) ) ) ).
% converse_rtranclE2
tff(fact_438_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
=> aa(B,$o,aa(A,fun(B,$o),P,A5),B2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).
% converse_rtrancl_induct2
tff(fact_439_Un__empty__right,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).
% Un_empty_right
tff(fact_440_Un__empty__left,axiom,
! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B5) = B5 ).
% Un_empty_left
tff(fact_441_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
<=> ! [X2: A] :
( member(A,X2,A3)
=> ! [Xa2: A] :
( member(A,Xa2,B5)
=> ( X2 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_442_Int__empty__right,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).
% Int_empty_right
tff(fact_443_Int__empty__left,axiom,
! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B5) = bot_bot(set(A)) ).
% Int_empty_left
tff(fact_444_disjoint__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
<=> ! [X2: A] :
( member(A,X2,A3)
=> ~ member(A,X2,B5) ) ) ).
% disjoint_iff
tff(fact_445_Int__emptyI,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ! [X3: A] :
( member(A,X3,A3)
=> ~ member(A,X3,B5) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) ) ) ).
% Int_emptyI
tff(fact_446_disjointI,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ! [X3: A] :
( member(A,X3,A4)
=> ~ member(A,X3,B3) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).
% disjointI
tff(fact_447_Int__insert__right,axiom,
! [A: $tType,A3: set(A),A4: A,B5: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = $ite(member(A,A4,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ).
% Int_insert_right
tff(fact_448_Int__insert__left,axiom,
! [A: $tType,A4: A,B5: set(A),C6: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)),C6) = $ite(member(A,A4,C6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) ).
% Int_insert_left
tff(fact_449_subrelI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( ! [X3: A,Y2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),S2) )
=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).
% subrelI
tff(fact_450_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),bot_bot(A))
=> ( A4 = bot_bot(A) ) ) ) ).
% bot.extremum_uniqueI
tff(fact_451_bot_Oextremum__unique,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),bot_bot(A))
<=> ( A4 = bot_bot(A) ) ) ) ).
% bot.extremum_unique
tff(fact_452_bot_Oextremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A4) ) ).
% bot.extremum
tff(fact_453_subset__emptyI,axiom,
! [A: $tType,A3: set(A)] :
( ! [X3: A] : ~ member(A,X3,A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A))) ) ).
% subset_emptyI
tff(fact_454_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set(A),X5: set(A)] :
( member(A,X,A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),X5)),A3) ) ) ).
% insert_subsetI
tff(fact_455_subset__insertI2,axiom,
! [A: $tType,A3: set(A),B5: set(A),B3: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B5)) ) ).
% subset_insertI2
tff(fact_456_subset__insertI,axiom,
! [A: $tType,B5: set(A),A4: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) ).
% subset_insertI
tff(fact_457_subset__insert,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
( ~ member(A,X,A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5))
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ) ).
% subset_insert
tff(fact_458_insert__mono,axiom,
! [A: $tType,C6: set(A),D4: set(A),A4: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),D4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),C6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),D4)) ) ).
% insert_mono
tff(fact_459_Image__Un,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: set(B),B5: set(B)] : image(B,A,R3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(B,A,R3,A3)),image(B,A,R3,B5)) ).
% Image_Un
tff(fact_460_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod(A,A),R3: set(product_prod(A,A)),R2: product_prod(A,A)] :
( member(product_prod(A,A),X,transitive_rtrancl(A,R3))
=> member(product_prod(A,A),X,transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),R2),R3))) ) ).
% in_rtrancl_insert
tff(fact_461_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,B),G: fun(A,B),Y: set(B)] :
( ! [W2: A] :
( member(A,W2,S)
=> ( aa(A,B,F2,W2) = aa(A,B,G,W2) ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Y)),S) ) ) ).
% vimage_inter_cong
tff(fact_462_trancl__mono__mp,axiom,
! [A: $tType,U2: set(product_prod(A,A)),V2: set(product_prod(A,A)),X: product_prod(A,A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V2)
=> ( member(product_prod(A,A),X,transitive_trancl(A,U2))
=> member(product_prod(A,A),X,transitive_trancl(A,V2)) ) ) ).
% trancl_mono_mp
tff(fact_463_trancl__sub,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),transitive_trancl(A,R3)) ).
% trancl_sub
tff(fact_464_subset__vimage__iff,axiom,
! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B5: set(B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),vimage(A,B,F2,B5))
<=> ! [X2: A] :
( member(A,X2,A3)
=> member(B,aa(A,B,F2,X2),B5) ) ) ).
% subset_vimage_iff
tff(fact_465_vimage__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(A),F2: fun(B,A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,A3)),vimage(B,A,F2,B5)) ) ).
% vimage_mono
tff(fact_466_relH__subset,axiom,
! [Bs: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit),As: set(nat)] :
( relH(Bs,Ha,H2)
=> ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),Bs)
=> relH(As,Ha,H2) ) ) ).
% relH_subset
tff(fact_467_rtrancl__apply__insert,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: A,S: set(A)] : image(A,A,transitive_rtrancl(A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),image(A,A,transitive_rtrancl(A,R3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ).
% rtrancl_apply_insert
tff(fact_468_antisymD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( antisym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> ( A4 = B3 ) ) ) ) ).
% antisymD
tff(fact_469_antisymI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),R2)
=> ( X3 = Y2 ) ) )
=> antisym(A,R2) ) ).
% antisymI
tff(fact_470_antisym__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( antisym(A,R2)
<=> ! [X2: A,Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R2)
=> ( X2 = Y3 ) ) ) ) ).
% antisym_def
tff(fact_471_Image__absorb__rtrancl,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(A),C6: set(A)] :
( trans(A,A3)
=> ( refl_on(A,B5,A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),B5)
=> ( image(A,A,transitive_rtrancl(A,A3),C6) = image(A,A,A3,C6) ) ) ) ) ).
% Image_absorb_rtrancl
tff(fact_472_antisym__empty,axiom,
! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).
% antisym_empty
tff(fact_473_times__assn__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As22)) ) ) ).
% times_assn_raw.simps
tff(fact_474_times__assn__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ~ ? [As1: set(nat),As22: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) ) ) ) ).
% times_assn_raw.elims(1)
tff(fact_475_times__assn__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ~ ? [As12: set(nat),As23: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ).
% times_assn_raw.elims(2)
tff(fact_476_times__assn__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ? [As13: set(nat),As24: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As24)) ) ) ) ).
% times_assn_raw.elims(3)
tff(fact_477_antisym__Id__on,axiom,
! [A: $tType,A3: set(A)] : antisym(A,id_on(A,A3)) ).
% antisym_Id_on
tff(fact_478_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))),image(A,A,transitive_rtrancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A))))) ) ).
% rtrancl_Image_advance_ss
tff(fact_479_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q3: A,R3: set(product_prod(A,A)),Q0: set(A),X: A] :
( member(A,Q3,image(A,A,transitive_rtrancl(A,R3),Q0))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X),transitive_rtrancl(A,R3))
=> member(A,X,image(A,A,transitive_rtrancl(A,R3),Q0)) ) ) ).
% rtrancl_image_advance_rtrancl
tff(fact_480_rtrancl__image__advance,axiom,
! [A: $tType,Q3: A,R3: set(product_prod(A,A)),Q0: set(A),X: A] :
( member(A,Q3,image(A,A,transitive_rtrancl(A,R3),Q0))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X),R3)
=> member(A,X,image(A,A,transitive_rtrancl(A,R3),Q0)) ) ) ).
% rtrancl_image_advance
tff(fact_481_trancl__rtrancl__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R2)) ) ) ).
% trancl_rtrancl_trancl
tff(fact_482_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_trancl_trancl
tff(fact_483_rtrancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_into_trancl2
tff(fact_484_rtrancl__into__trancl1,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),C2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_into_trancl1
tff(fact_485_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R3))
<=> ( ( X = Y )
| ( ( X != Y )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R3)) ) ) ) ).
% rtrancl_eq_or_trancl
tff(fact_486_trancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2)) ) ).
% trancl_into_rtrancl
tff(fact_487_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R3))
=> ? [Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4),transitive_rtrancl(A,R3))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y),R3) ) ) ).
% tranclD2
tff(fact_488_rtranclD,axiom,
! [A: $tType,A4: A,B3: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R3))
=> ( ( A4 = B3 )
| ( ( A4 != B3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R3)) ) ) ) ).
% rtranclD
tff(fact_489_tranclD,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R3))
=> ? [Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4),R3)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y),transitive_rtrancl(A,R3)) ) ) ).
% tranclD
tff(fact_490_insert__is__Un,axiom,
! [A: $tType,A4: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))),A3) ).
% insert_is_Un
tff(fact_491_Un__singleton__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A),X: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
<=> ( ( ( A3 = bot_bot(set(A)) )
& ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
| ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B5 = bot_bot(set(A)) ) )
| ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).
% Un_singleton_iff
tff(fact_492_singleton__Un__iff,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) )
<=> ( ( ( A3 = bot_bot(set(A)) )
& ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
| ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B5 = bot_bot(set(A)) ) )
| ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).
% singleton_Un_iff
tff(fact_493_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R3: set(product_prod(A,A)),Y: A] :
( ~ member(A,X,aa(set(product_prod(A,A)),set(A),domain(A,A),R3))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R3))
<=> ( X = Y ) ) ) ).
% Not_Domain_rtrancl
tff(fact_494_trancl__Image__unfold__right,axiom,
! [A: $tType,E3: set(product_prod(A,A)),S: set(A)] : image(A,A,transitive_trancl(A,E3),S) = image(A,A,E3,image(A,A,transitive_rtrancl(A,E3),S)) ).
% trancl_Image_unfold_right
tff(fact_495_trancl__Image__unfold__left,axiom,
! [A: $tType,E3: set(product_prod(A,A)),S: set(A)] : image(A,A,transitive_trancl(A,E3),S) = image(A,A,transitive_rtrancl(A,E3),image(A,A,E3,S)) ).
% trancl_Image_unfold_left
tff(fact_496_subset__singletonD,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
=> ( ( A3 = bot_bot(set(A)) )
| ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ).
% subset_singletonD
tff(fact_497_subset__singleton__iff,axiom,
! [A: $tType,X5: set(A),A4: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))
<=> ( ( X5 = bot_bot(set(A)) )
| ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) ) ) ) ).
% subset_singleton_iff
tff(fact_498_mod__star__conv,axiom,
! [A3: assn,B5: assn,Ha: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B5)),Ha)
<=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
( ( Ha = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As22)) ) ) ).
% mod_star_conv
tff(fact_499_star__assnI,axiom,
! [P: assn,Ha: heap_ext(product_unit),As: set(nat),Q: assn,As5: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As5))
=> ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))) ) ) ) ).
% star_assnI
tff(fact_500_in__range__subset,axiom,
! [As: set(nat),As5: set(nat),Ha: heap_ext(product_unit)] :
( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),As5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As5))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As)) ) ) ).
% in_range_subset
tff(fact_501_trancl__sub__insert__trancl,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: product_prod(A,A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),X),R3))) ).
% trancl_sub_insert_trancl
tff(fact_502_wand__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
& ! [H4: heap_ext(product_unit),As6: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As6) = bot_bot(set(nat)) )
& relH(As,Ha,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6))) ) ) ) ).
% wand_raw.simps
tff(fact_503_wand__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H4: heap_ext(product_unit),As6: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As6) = bot_bot(set(nat)) )
& relH(As2,H,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6))) ) ) ) ) ) ).
% wand_raw.elims(1)
tff(fact_504_wand__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H6: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As7) = bot_bot(set(nat)) )
& relH(As2,H,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As7)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7))) ) ) ) ) ).
% wand_raw.elims(2)
tff(fact_505_wand__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H5: heap_ext(product_unit),As4: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As4) = bot_bot(set(nat)) )
& relH(As2,H,H5)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_506_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R3: set(product_prod(A,A)),V: A] :
( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_rtrancl(A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))) ) ) ).
% Image_empty_rtrancl_Image_id
tff(fact_507_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),bot_bot(A)) = A4 ) ).
% sup_bot.right_neutral
tff(fact_508_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A4: A,B3: A] :
( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3) )
<=> ( ( A4 = bot_bot(A) )
& ( B3 = bot_bot(A) ) ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_509_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A4) = A4 ) ).
% sup_bot.left_neutral
tff(fact_510_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3) = bot_bot(A) )
<=> ( ( A4 = bot_bot(A) )
& ( B3 = bot_bot(A) ) ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_511_sup__eq__bot__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
<=> ( ( X = bot_bot(A) )
& ( Y = bot_bot(A) ) ) ) ) ).
% sup_eq_bot_iff
tff(fact_512_bot__eq__sup__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A,Y: A] :
( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
<=> ( ( X = bot_bot(A) )
& ( Y = bot_bot(A) ) ) ) ) ).
% bot_eq_sup_iff
tff(fact_513_sup__bot__right,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).
% sup_bot_right
tff(fact_514_sup__bot__left,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).
% sup_bot_left
tff(fact_515_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).
% boolean_algebra.conj_zero_right
tff(fact_516_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).
% boolean_algebra.conj_zero_left
tff(fact_517_inf__bot__right,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).
% inf_bot_right
tff(fact_518_merge__pure__or,axiom,
! [A4: $o,B3: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn((A4))),pure_assn((B3))) = pure_assn(( (A4)
| (B3) )) ).
% merge_pure_or
tff(fact_519_merge__pure__and,axiom,
! [A4: $o,B3: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((A4))),pure_assn((B3))) = pure_assn(( (A4)
& (B3) )) ).
% merge_pure_and
tff(fact_520_inf__bot__left,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).
% inf_bot_left
tff(fact_521_trans__Int,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( trans(A,R2)
=> ( trans(A,S2)
=> trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).
% trans_Int
tff(fact_522_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).
% boolean_algebra.disj_zero_right
tff(fact_523_prod__set__defs_I2_J,axiom,
! [A: $tType,B: $tType,X4: product_prod(A,B)] : basic_snds(A,B,X4) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(product_prod(A,B),B,product_snd(A,B),X4)),bot_bot(set(B))) ).
% prod_set_defs(2)
tff(fact_524_snds_Ocases,axiom,
! [A: $tType,B: $tType,A4: A,P2: product_prod(B,A)] :
( member(A,A4,basic_snds(B,A,P2))
=> ( A4 = aa(product_prod(B,A),A,product_snd(B,A),P2) ) ) ).
% snds.cases
tff(fact_525_snds_Osimps,axiom,
! [A: $tType,B: $tType,A4: A,P2: product_prod(B,A)] :
( member(A,A4,basic_snds(B,A,P2))
<=> ( A4 = aa(product_prod(B,A),A,product_snd(B,A),P2) ) ) ).
% snds.simps
tff(fact_526_snds_Ointros,axiom,
! [A: $tType,B: $tType,P2: product_prod(B,A)] : member(A,aa(product_prod(B,A),A,product_snd(B,A),P2),basic_snds(B,A,P2)) ).
% snds.intros
tff(fact_527_less__by__empty,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( ( A3 = bot_bot(set(product_prod(A,A))) )
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A3),B5) ) ).
% less_by_empty
tff(fact_528_wand__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H4: heap_ext(product_unit),As6: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As6) = bot_bot(set(nat)) )
& relH(As2,H,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6))) ) ) )
=> ~ aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))) ) ) ) ) ).
% wand_raw.pelims(1)
tff(fact_529_wand__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H6: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As7) = bot_bot(set(nat)) )
& relH(As2,H,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As7)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7))) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
tff(fact_530_wand__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H5: heap_ext(product_unit),As4: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As4) = bot_bot(set(nat)) )
& relH(As2,H,H5)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As4)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_531_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).
% le_numeral_extra(4)
tff(fact_532_less__eq__assn__def,axiom,
! [A4: assn,B3: assn] :
( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A4),B3)
<=> ( A4 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),B3) ) ) ).
% less_eq_assn_def
tff(fact_533_times__assn__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ? [As13: set(nat),As24: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As24)) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
tff(fact_534_times__assn__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ~ ? [As12: set(nat),As23: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
tff(fact_535_times__assn__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) )
=> ~ aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))) ) ) ) ) ).
% times_assn_raw.pelims(1)
tff(fact_536_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2) ) ) ) ) ).
% subset_Image1_Image1_iff
tff(fact_537_Field__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ).
% Field_insert
tff(fact_538_dom__ran__disj__comp,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R3)),aa(set(product_prod(A,A)),set(A),range2(A,A),R3)) = bot_bot(set(A)) )
=> ( relcomp(A,A,A,R3,R3) = bot_bot(set(product_prod(A,A))) ) ) ).
% dom_ran_disj_comp
tff(fact_539_proj__iff,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A3)
=> ( ( aa(A,set(A),equiv_proj(A,A,R2),X) = aa(A,set(A),equiv_proj(A,A,R2),Y) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% proj_iff
tff(fact_540_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( antisym(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
<=> ( A4 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
tff(fact_541_equiv__class__nondisjoint,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,A4: A,B3: A] :
( equiv_equiv(A,A3,R2)
=> ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ).
% equiv_class_nondisjoint
tff(fact_542_finite__reachable__advance,axiom,
! [A: $tType,E3: set(product_prod(A,A)),V0: A,V: A] :
( aa(set(A),$o,finite_finite2(A),image(A,A,transitive_rtrancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V0),bot_bot(set(A)))))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V),transitive_rtrancl(A,E3))
=> aa(set(A),$o,finite_finite2(A),image(A,A,transitive_rtrancl(A,E3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))) ) ) ).
% finite_reachable_advance
tff(fact_543_finite__Field__eq__finite,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R3))
<=> aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R3) ) ).
% finite_Field_eq_finite
tff(fact_544_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(A,C))] : relcomp(A,C,B,R3,bot_bot(set(product_prod(C,B)))) = bot_bot(set(product_prod(A,B))) ).
% relcomp_empty2
tff(fact_545_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(C,B))] : relcomp(A,C,B,bot_bot(set(product_prod(A,C))),R3) = bot_bot(set(product_prod(A,B))) ).
% relcomp_empty1
tff(fact_546_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set(product_prod(A,C)),T3: set(product_prod(A,C)),R3: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C))),sup_sup(set(product_prod(A,C))),S),T3),R3) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,S,R3)),relcomp(A,C,B,T3,R3)) ).
% relcomp_distrib2
tff(fact_547_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set(product_prod(A,C)),S: set(product_prod(C,B)),T3: set(product_prod(C,B))] : relcomp(A,C,B,R3,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),S),T3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,R3,S)),relcomp(A,C,B,R3,T3)) ).
% relcomp_distrib
tff(fact_548_Field__empty,axiom,
! [A: $tType] : aa(set(product_prod(A,A)),set(A),field2(A),bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).
% Field_empty
tff(fact_549_Field__Un,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S2)) ).
% Field_Un
tff(fact_550_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R3: set(product_prod(A,D)),S: set(product_prod(D,C)),T3: set(product_prod(C,B))] : relcomp(A,C,B,relcomp(A,D,C,R3,S),T3) = relcomp(A,D,B,R3,relcomp(D,C,B,S,T3)) ).
% O_assoc
tff(fact_551_finite__Field,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
=> aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).
% finite_Field
tff(fact_552_finite__Image,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),A3: set(A)] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R3)
=> aa(set(B),$o,finite_finite2(B),image(A,B,R3,A3)) ) ).
% finite_Image
tff(fact_553_finite__Domain,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
=> aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).
% finite_Domain
tff(fact_554_finite__Range,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
=> aa(set(B),$o,finite_finite2(B),aa(set(product_prod(A,B)),set(B),range2(A,B),R2)) ) ).
% finite_Range
tff(fact_555_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A4: A,C2: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),C2),relcomp(A,C,B,R2,S2))
=> ~ ! [B2: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),B2),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B2),C2),S2) ) ) ).
% relcompEpair
tff(fact_556_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,S2))
=> ~ ! [X3: A,Y2: C,Z4: B] :
( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z4) )
=> ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y2),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y2),Z4),S2) ) ) ) ).
% relcompE
tff(fact_557_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A4: A,B3: B,R2: set(product_prod(A,B)),C2: C,S2: set(product_prod(B,C))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C2),S2)
=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),C2),relcomp(A,B,C,R2,S2)) ) ) ).
% relcomp.relcompI
tff(fact_558_relcomp_Osimps,axiom,
! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S2))
<=> ? [A7: A,B7: C,C5: B] :
( ( A1 = A7 )
& ( A22 = C5 )
& member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),B7),R2)
& member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B7),C5),S2) ) ) ).
% relcomp.simps
tff(fact_559_relcomp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S2))
=> ~ ! [B2: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B2),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B2),A22),S2) ) ) ).
% relcomp.cases
tff(fact_560_FieldI2,axiom,
! [A: $tType,I2: A,J2: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2),R3)
=> member(A,J2,aa(set(product_prod(A,A)),set(A),field2(A),R3)) ) ).
% FieldI2
tff(fact_561_FieldI1,axiom,
! [A: $tType,I2: A,J2: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2),R3)
=> member(A,I2,aa(set(product_prod(A,A)),set(A),field2(A),R3)) ) ).
% FieldI1
tff(fact_562_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: set(product_prod(A,B)),R2: set(product_prod(A,B)),S3: set(product_prod(B,C)),S2: set(product_prod(B,C))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R5),R2)
=> ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S3),S2)
=> aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R5,S3)),relcomp(A,B,C,R2,S2)) ) ) ).
% relcomp_mono
tff(fact_563_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X5: set(product_prod(B,C)),Y4: set(product_prod(C,A)),Z5: set(B)] : image(B,A,relcomp(B,C,A,X5,Y4),Z5) = image(C,A,Y4,image(B,C,X5,Z5)) ).
% relcomp_Image
tff(fact_564_trans__O__subset,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,R2)),R2) ) ).
% trans_O_subset
tff(fact_565_mono__Field,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S2)) ) ).
% mono_Field
tff(fact_566_union__comp__emptyL,axiom,
! [A: $tType,A3: set(product_prod(A,A)),C6: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( ( relcomp(A,A,A,A3,C6) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,B5,C6) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A3),B5),C6) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyL
tff(fact_567_union__comp__emptyR,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(product_prod(A,A)),C6: set(product_prod(A,A))] :
( ( relcomp(A,A,A,A3,B5) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,A3,C6) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,A,A,A3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),B5),C6)) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyR
tff(fact_568_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(B,A)),B5: set(B),C6: set(product_prod(B,A))] :
( aa(set(A),$o,finite_finite2(A),image(B,A,A3,B5))
=> ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),C6),A3)
=> aa(set(A),$o,finite_finite2(A),image(B,A,C6,B5)) ) ) ).
% finite_Image_subset
tff(fact_569_equiv__class__self,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A] :
( equiv_equiv(A,A3,R2)
=> ( member(A,A4,A3)
=> member(A,A4,image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) ) ) ).
% equiv_class_self
tff(fact_570_Field__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),R2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range2(A,A),R2)) ).
% Field_def
tff(fact_571_equiv__class__eq__iff,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
<=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
& member(A,X,A3)
& member(A,Y,A3) ) ) ) ).
% equiv_class_eq_iff
tff(fact_572_eq__equiv__class__iff,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(A,X,A3)
=> ( member(A,Y,A3)
=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ).
% eq_equiv_class_iff
tff(fact_573_equiv__class__eq,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A,B3: A] :
( equiv_equiv(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ) ).
% equiv_class_eq
tff(fact_574_eq__equiv__class,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A,A3: set(A)] :
( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
=> ( equiv_equiv(A,A3,R2)
=> ( member(A,B3,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ) ).
% eq_equiv_class
tff(fact_575_rtrancl__Image__in__Field,axiom,
! [A: $tType,R3: set(product_prod(A,A)),V2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,R3),V2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R3)),V2)) ).
% rtrancl_Image_in_Field
tff(fact_576_refines__equiv__class__eq2,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A)),A3: set(A),A4: A] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),S)
=> ( equiv_equiv(A,A3,R3)
=> ( equiv_equiv(A,A3,S)
=> ( image(A,A,S,image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq2
tff(fact_577_refines__equiv__class__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A)),A3: set(A),A4: A] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),S)
=> ( equiv_equiv(A,A3,R3)
=> ( equiv_equiv(A,A3,S)
=> ( image(A,A,R3,image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq
tff(fact_578_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
<=> ( A4 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
tff(fact_579_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set(A),I: set(A),E3: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite2(A),Q)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),I),Q)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),range2(A,A),E3)),Q)
=> aa(set(A),$o,finite_finite2(A),image(A,A,transitive_rtrancl(A,E3),I)) ) ) ) ).
% finite_reachable_restrictedI
tff(fact_580_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B5: set(A)] :
( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,A3)),image(A,A,R2,B5))
<=> ! [X2: A] :
( member(A,X2,A3)
=> ? [Xa2: A] :
( member(A,Xa2,B5)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),R2) ) ) ) ) ) ) ).
% subset_Image_Image_iff
tff(fact_581_equiv__class__subset,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A,B3: A] :
( equiv_equiv(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ).
% equiv_class_subset
tff(fact_582_subset__equiv__class,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),B3: A,A4: A] :
( equiv_equiv(A,A3,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))
=> ( member(A,B3,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ) ).
% subset_equiv_class
tff(fact_583_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F4: set(A),Ha: fun(B,A),A3: set(B)] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( ! [Y2: A] :
( member(A,Y2,F4)
=> aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,Ha,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A3)) )
=> aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,Ha,F4)),A3)) ) ) ).
% finite_finite_vimage_IntI
tff(fact_584_finite__insert,axiom,
! [A: $tType,A4: A,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))
<=> aa(set(A),$o,finite_finite2(A),A3) ) ).
% finite_insert
tff(fact_585_finite__subset__induct_H,axiom,
! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A5: A,F5: set(A)] :
( aa(set(A),$o,finite_finite2(A),F5)
=> ( member(A,A5,A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
=> ( ~ member(A,A5,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F5)) ) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_subset_induct'
tff(fact_586_finite__subset__induct,axiom,
! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A5: A,F5: set(A)] :
( aa(set(A),$o,finite_finite2(A),F5)
=> ( member(A,A5,A3)
=> ( ~ member(A,A5,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F5)) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_subset_induct
tff(fact_587_finite__ranking__induct,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,S5: set(A)] :
( aa(set(A),$o,finite_finite2(A),S5)
=> ( ! [Y5: A] :
( member(A,Y5,S5)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y5)),aa(A,B,F2,X3)) )
=> ( aa(set(A),$o,P,S5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),S5)) ) ) )
=> aa(set(A),$o,P,S) ) ) ) ) ).
% finite_ranking_induct
tff(fact_588_infinite__finite__induct,axiom,
! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
( ! [A10: set(A)] :
( ~ aa(set(A),$o,finite_finite2(A),A10)
=> aa(set(A),$o,P,A10) )
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite2(A),F5)
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) )
=> aa(set(A),$o,P,A3) ) ) ) ).
% infinite_finite_induct
tff(fact_589_finite__ne__induct,axiom,
! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( ( F4 != bot_bot(set(A)) )
=> ( ! [X3: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite2(A),F5)
=> ( ( F5 != bot_bot(set(A)) )
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_ne_induct
tff(fact_590_finite__induct,axiom,
! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite2(A),F5)
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ).
% finite_induct
tff(fact_591_finite_Osimps,axiom,
! [A: $tType,A4: set(A)] :
( aa(set(A),$o,finite_finite2(A),A4)
<=> ( ( A4 = bot_bot(set(A)) )
| ? [A11: set(A),A7: A] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),A11) )
& aa(set(A),$o,finite_finite2(A),A11) ) ) ) ).
% finite.simps
tff(fact_592_infinite__imp__nonempty,axiom,
! [A: $tType,S: set(A)] :
( ~ aa(set(A),$o,finite_finite2(A),S)
=> ( S != bot_bot(set(A)) ) ) ).
% infinite_imp_nonempty
tff(fact_593_finite_OemptyI,axiom,
! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).
% finite.emptyI
tff(fact_594_finite_OinsertI,axiom,
! [A: $tType,A3: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) ) ).
% finite.insertI
tff(fact_595_finite__has__minimal,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,A3)
& ! [Xa3: A] :
( member(A,Xa3,A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa3),X3)
=> ( X3 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_596_finite__has__maximal,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,A3)
& ! [Xa3: A] :
( member(A,Xa3,A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3)
=> ( X3 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_597_finite_Ocases,axiom,
! [A: $tType,A4: set(A)] :
( aa(set(A),$o,finite_finite2(A),A4)
=> ( ( A4 != bot_bot(set(A)) )
=> ~ ! [A10: set(A)] :
( ? [A5: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),A10)
=> ~ aa(set(A),$o,finite_finite2(A),A10) ) ) ) ).
% finite.cases
tff(fact_598_min__ext__compat,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S)),R3)
=> aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R3),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R3)) ) ).
% min_ext_compat
tff(fact_599_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S: set(A),Y: A,F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( member(A,Y,S)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Y)) ) ) ) ) ).
% arg_min_least
tff(fact_600_Total__subset__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
=> ( ( R2 = bot_bot(set(product_prod(A,A))) )
| ? [A5: A] : R2 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).
% Total_subset_Id
tff(fact_601_finite__transitivity__chain,axiom,
! [A: $tType,A3: set(A),R3: fun(A,fun(A,$o))] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),R3,X3),X3)
=> ( ! [X3: A,Y2: A,Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R3,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),R3,X3),Z4) ) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> ? [Y5: A] :
( member(A,Y5,A3)
& aa(A,$o,aa(A,fun(A,$o),R3,X3),Y5) ) )
=> ( A3 = bot_bot(set(A)) ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_602_disjnt__equiv__class,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A,B3: A] :
( equiv_equiv(A,A3,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ).
% disjnt_equiv_class
tff(fact_603_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,A3)) ) ) ) ) ).
% Sup_fin.insert
tff(fact_604_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ).
% Inf_fin.insert
tff(fact_605_in__quotient__imp__in__rel,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),X5)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% in_quotient_imp_in_rel
tff(fact_606_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] : basic_fsts(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).
% prod_set_simps(1)
tff(fact_607_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,E3))
=> ( ~ member(A,X,R3)
=> ( ~ member(A,Y,R3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E3,R3)),R3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,rel_restrict(A,E3,R3))) ) ) ) ) ).
% rel_restrict_tranclI
tff(fact_608_pair__in__Id__conv,axiom,
! [A: $tType,A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),id2(A))
<=> ( A4 = B3 ) ) ).
% pair_in_Id_conv
tff(fact_609_IdI,axiom,
! [A: $tType,A4: A] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),id2(A)) ).
% IdI
tff(fact_610_quotient__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R2) = bot_bot(set(set(A))) ).
% quotient_empty
tff(fact_611_quotient__is__empty,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( ( equiv_quotient(A,A3,R2) = bot_bot(set(set(A))) )
<=> ( A3 = bot_bot(set(A)) ) ) ).
% quotient_is_empty
tff(fact_612_quotient__is__empty2,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( ( bot_bot(set(set(A))) = equiv_quotient(A,A3,R2) )
<=> ( A3 = bot_bot(set(A)) ) ) ).
% quotient_is_empty2
tff(fact_613_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),S),S)
<=> ( S = bot_bot(set(A)) ) ) ).
% disjnt_self_iff_empty
tff(fact_614_disjnt__insert1,axiom,
! [A: $tType,A4: A,X5: set(A),Y4: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5)),Y4)
<=> ( ~ member(A,A4,Y4)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X5),Y4) ) ) ).
% disjnt_insert1
tff(fact_615_disjnt__insert2,axiom,
! [A: $tType,Y4: set(A),A4: A,X5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5))
<=> ( ~ member(A,A4,Y4)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y4),X5) ) ) ).
% disjnt_insert2
tff(fact_616_rel__restrict__empty,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : rel_restrict(A,R3,bot_bot(set(A))) = R3 ).
% rel_restrict_empty
tff(fact_617_disjnt__Un1,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)),C6)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),C6)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B5),C6) ) ) ).
% disjnt_Un1
tff(fact_618_disjnt__Un2,axiom,
! [A: $tType,C6: set(A),A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C6),A3)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C6),B5) ) ) ).
% disjnt_Un2
tff(fact_619_Image__Id,axiom,
! [A: $tType,A3: set(A)] : image(A,A,id2(A),A3) = A3 ).
% Image_Id
tff(fact_620_Id__O__R,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : relcomp(A,A,B,id2(A),R3) = R3 ).
% Id_O_R
tff(fact_621_R__O__Id,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : relcomp(A,B,B,R3,id2(B)) = R3 ).
% R_O_Id
tff(fact_622_bijective__Id,axiom,
! [A: $tType] : bijective(A,A,id2(A)) ).
% bijective_Id
tff(fact_623_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% Sup_fin.singleton
tff(fact_624_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% Inf_fin.singleton
tff(fact_625_rtrancl__empty,axiom,
! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).
% rtrancl_empty
tff(fact_626_disjnt__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),B5)
<=> ! [X2: A] :
~ ( member(A,X2,A3)
& member(A,X2,B5) ) ) ).
% disjnt_iff
tff(fact_627_disjnt__sym,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B5),A3) ) ).
% disjnt_sym
tff(fact_628_IdE,axiom,
! [A: $tType,P2: product_prod(A,A)] :
( member(product_prod(A,A),P2,id2(A))
=> ~ ! [X3: A] : P2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).
% IdE
tff(fact_629_disjnt__empty1,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),bot_bot(set(A))),A3) ).
% disjnt_empty1
tff(fact_630_disjnt__empty2,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),bot_bot(set(A))) ).
% disjnt_empty2
tff(fact_631_disjnt__subset1,axiom,
! [A: $tType,X5: set(A),Y4: set(A),Z5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X5),Y4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),X5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Z5),Y4) ) ) ).
% disjnt_subset1
tff(fact_632_disjnt__subset2,axiom,
! [A: $tType,X5: set(A),Y4: set(A),Z5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X5),Y4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),Y4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X5),Z5) ) ) ).
% disjnt_subset2
tff(fact_633_disjnt__insert,axiom,
! [A: $tType,X: A,N: set(A),M2: set(A)] :
( ~ member(A,X,N)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),M2),N)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),M2)),N) ) ) ).
% disjnt_insert
tff(fact_634_trans__Id,axiom,
! [A: $tType] : trans(A,id2(A)) ).
% trans_Id
tff(fact_635_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,A3,R3))
=> ~ member(A,Y,R3) ) ).
% rel_restrict_notR(2)
tff(fact_636_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,A3,R3))
=> ~ member(A,X,R3) ) ).
% rel_restrict_notR(1)
tff(fact_637_rel__restrictI,axiom,
! [A: $tType,X: A,R3: set(A),Y: A,E3: set(product_prod(A,A))] :
( ~ member(A,X,R3)
=> ( ~ member(A,Y,R3)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,E3,R3)) ) ) ) ).
% rel_restrictI
tff(fact_638_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,E3,R3))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E3) ) ).
% rel_restrict_lift
tff(fact_639_rel__restrict__union,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A),B5: set(A)] : rel_restrict(A,R3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = rel_restrict(A,rel_restrict(A,R3,A3),B5) ).
% rel_restrict_union
tff(fact_640_antisym__Id,axiom,
! [A: $tType] : antisym(A,id2(A)) ).
% antisym_Id
tff(fact_641_rel__restrict__mono,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(product_prod(A,A)),R3: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A3),B5)
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A3,R3)),rel_restrict(A,B5,R3)) ) ).
% rel_restrict_mono
tff(fact_642_rel__restrict__sub,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R3,A3)),R3) ).
% rel_restrict_sub
tff(fact_643_finite__rel__restrict,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] :
( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R3)
=> aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),rel_restrict(A,R3,A3)) ) ).
% finite_rel_restrict
tff(fact_644_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A3)),lattic5882676163264333800up_fin(A,A3)) ) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_645_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: set(A),C6: fun(A,set(B)),B5: set(A)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A3,C6)),product_Sigma(A,B,B5,C6))
<=> ( ! [X2: A] :
( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> ( aa(A,set(B),C6,X2) = bot_bot(set(B)) ) )
| aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),B5) ) ) ).
% disjnt_Sigma_iff
tff(fact_646_disjnt__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),B5)
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) ) ) ).
% disjnt_def
tff(fact_647_quotient__eqI,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( member(set(A),Y4,equiv_quotient(A,A3,R2))
=> ( member(A,X,X5)
=> ( member(A,Y,Y4)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( X5 = Y4 ) ) ) ) ) ) ) ).
% quotient_eqI
tff(fact_648_quotient__eq__iff,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( member(set(A),Y4,equiv_quotient(A,A3,R2))
=> ( member(A,X,X5)
=> ( member(A,Y,Y4)
=> ( ( X5 = Y4 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ) ) ).
% quotient_eq_iff
tff(fact_649_in__quotient__imp__closed,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( member(A,X,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,Y,X5) ) ) ) ) ).
% in_quotient_imp_closed
tff(fact_650_in__quotient__imp__non__empty,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( X5 != bot_bot(set(A)) ) ) ) ).
% in_quotient_imp_non_empty
tff(fact_651_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,rel_restrict(A,E3,R3)))
=> ~ member(A,W,R3) ) ).
% rel_restrict_trancl_notR(2)
tff(fact_652_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,rel_restrict(A,E3,R3)))
=> ~ member(A,V,R3) ) ).
% rel_restrict_trancl_notR(1)
tff(fact_653_rel__restrict__trancl__mem,axiom,
! [A: $tType,A4: A,B3: A,A3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,rel_restrict(A,A3,R3)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),rel_restrict(A,transitive_trancl(A,A3),R3)) ) ).
% rel_restrict_trancl_mem
tff(fact_654_rel__restrict__mono2,axiom,
! [A: $tType,R3: set(A),S: set(A),A3: set(product_prod(A,A))] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R3),S)
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A3,S)),rel_restrict(A,A3,R3)) ) ).
% rel_restrict_mono2
tff(fact_655_rel__restrict__trancl__sub,axiom,
! [A: $tType,A3: set(product_prod(A,A)),R3: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,A3,R3))),rel_restrict(A,transitive_trancl(A,A3),R3)) ).
% rel_restrict_trancl_sub
tff(fact_656_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),bot_bot(set(set(A)))) ).
% singleton_quotient
tff(fact_657_quotientI,axiom,
! [A: $tType,X: A,A3: set(A),R2: set(product_prod(A,A))] :
( member(A,X,A3)
=> member(set(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),equiv_quotient(A,A3,R2)) ) ).
% quotientI
tff(fact_658_quotientE,axiom,
! [A: $tType,X5: set(A),A3: set(A),R2: set(product_prod(A,A))] :
( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ~ ! [X3: A] :
( ( X5 = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))) )
=> ~ member(A,X3,A3) ) ) ).
% quotientE
tff(fact_659_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3))
=> ! [A12: A] :
( member(A,A12,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A12) ) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_660_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A5) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ).
% Inf_fin.boundedI
tff(fact_661_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X)
=> ! [A12: A] :
( member(A,A12,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A12),X) ) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_662_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X) ) ) ) ) ).
% Sup_fin.boundedI
tff(fact_663_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A3))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_664_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),X)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_665_quotient__disj,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),Y4: set(A)] :
( equiv_equiv(A,A3,R2)
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( member(set(A),Y4,equiv_quotient(A,A3,R2))
=> ( ( X5 = Y4 )
| ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),Y4) = bot_bot(set(A)) ) ) ) ) ) ).
% quotient_disj
tff(fact_666_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( trans(A,A3)
=> ( transitive_rtrancl(A,A3) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A3),id2(A)) ) ) ).
% trans_rtrancl_eq_reflcl
tff(fact_667_rel__restrict__Int__empty,axiom,
! [A: $tType,A3: set(A),R3: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(product_prod(A,A)),set(A),field2(A),R3)) = bot_bot(set(A)) )
=> ( rel_restrict(A,R3,A3) = R3 ) ) ).
% rel_restrict_Int_empty
tff(fact_668_eq__equiv__class__iff2,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A3,R2)
=> ( member(A,X,A3)
=> ( member(A,Y,A3)
=> ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))),R2) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ).
% eq_equiv_class_iff2
tff(fact_669_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B5)),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_670_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B5)) ) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_671_Inf__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,B5)),lattic7752659483105999362nf_fin(A,A3)) = lattic7752659483105999362nf_fin(A,A3) ) ) ) ) ) ).
% Inf_fin.subset
tff(fact_672_Sup__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,B5)),lattic5882676163264333800up_fin(A,A3)) = lattic5882676163264333800up_fin(A,A3) ) ) ) ) ) ).
% Sup_fin.subset
tff(fact_673_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic7752659483105999362nf_fin(A,A3),A3) ) ) ) ) ).
% Inf_fin.closed
tff(fact_674_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_675_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic5882676163264333800up_fin(A,A3),A3) ) ) ) ) ).
% Sup_fin.closed
tff(fact_676_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,A3)) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_677_Inf__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,A3)),lattic7752659483105999362nf_fin(A,B5)) ) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_678_Sup__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B5)) ) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_679_refl__on__reflcl__Image,axiom,
! [A: $tType,B5: set(A),A3: set(product_prod(A,A)),C6: set(A)] :
( refl_on(A,B5,A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),B5)
=> ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A3),id2(A)),C6) = image(A,A,A3,C6) ) ) ) ).
% refl_on_reflcl_Image
tff(fact_680_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> member(A,lattic7623131987881927897min_on(A,B,F2,S),S) ) ) ) ).
% arg_min_if_finite(1)
tff(fact_681_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,E3))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E3,R3)),R3)
=> ( ~ member(A,V,R3)
=> ~ ( ~ member(A,U,R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E3,R3))) ) ) ) ) ).
% E_closed_restr_reach_cases
tff(fact_682_max__ext__compat,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S)),R3)
=> aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R3),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R3)) ) ).
% max_ext_compat
tff(fact_683_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),id2(A))
=> ( A4 = B3 ) ) ).
% BNF_Greatest_Fixpoint.IdD
tff(fact_684_Sup__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic5882676163264333800up_fin(A,A3) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).
% Sup_fin.remove
tff(fact_685_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).
% Sup_fin.insert_remove
tff(fact_686_Inf__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic7752659483105999362nf_fin(A,A3) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).
% Inf_fin.remove
tff(fact_687_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).
% Inf_fin.insert_remove
tff(fact_688_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
<=> ( transitive_acyclic(A,R2)
& ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2)) ) ) ).
% acyclic_insert
tff(fact_689_irrefl__tranclI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R2)),transitive_rtrancl(A,R2)) = bot_bot(set(product_prod(A,A))) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R2)) ) ).
% irrefl_tranclI
tff(fact_690_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
=> ( ~ aa(set(B),$o,finite_finite2(B),A3)
=> ? [X3: A] :
( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
& ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))),A3)) ) ) ) ).
% inf_img_fin_dom'
tff(fact_691_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
=> ( ~ aa(set(B),$o,finite_finite2(B),A3)
=> ~ ! [Y2: A] :
( member(A,Y2,aa(set(B),set(A),image2(B,A,F2),A3))
=> aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A3)) ) ) ) ).
% inf_img_fin_domE'
tff(fact_692_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: fun(B,A),X: B,A3: set(B)] :
( ( B3 = aa(B,A,F2,X) )
=> ( member(B,X,A3)
=> member(A,B3,aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).
% image_eqI
tff(fact_693_Diff__idemp,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : minus_minus(set(A),minus_minus(set(A),A3,B5),B5) = minus_minus(set(A),A3,B5) ).
% Diff_idemp
tff(fact_694_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,minus_minus(set(A),A3,B5))
<=> ( member(A,C2,A3)
& ~ member(A,C2,B5) ) ) ).
% Diff_iff
tff(fact_695_DiffI,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,A3)
=> ( ~ member(A,C2,B5)
=> member(A,C2,minus_minus(set(A),A3,B5)) ) ) ).
% DiffI
tff(fact_696_converse__converse,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : converse(B,A,converse(A,B,R2)) = R2 ).
% converse_converse
tff(fact_697_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
( ( converse(B,A,R2) = converse(B,A,S2) )
<=> ( R2 = S2 ) ) ).
% converse_inject
tff(fact_698_image__is__empty,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
( ( aa(set(B),set(A),image2(B,A,F2),A3) = bot_bot(set(A)) )
<=> ( A3 = bot_bot(set(B)) ) ) ).
% image_is_empty
tff(fact_699_empty__is__image,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F2),A3) )
<=> ( A3 = bot_bot(set(B)) ) ) ).
% empty_is_image
tff(fact_700_image__empty,axiom,
! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).
% image_empty
tff(fact_701_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
( member(A,X,A3)
=> ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,X)),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),set(B),image2(A,B,F2),A3) ) ) ).
% insert_image
tff(fact_702_image__insert,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: B,B5: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F2,A4)),aa(set(B),set(A),image2(B,A,F2),B5)) ).
% image_insert
tff(fact_703_img__snd,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S)
=> member(B,B3,aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S)) ) ).
% img_snd
tff(fact_704_Diff__cancel,axiom,
! [A: $tType,A3: set(A)] : minus_minus(set(A),A3,A3) = bot_bot(set(A)) ).
% Diff_cancel
tff(fact_705_empty__Diff,axiom,
! [A: $tType,A3: set(A)] : minus_minus(set(A),bot_bot(set(A)),A3) = bot_bot(set(A)) ).
% empty_Diff
tff(fact_706_Diff__empty,axiom,
! [A: $tType,A3: set(A)] : minus_minus(set(A),A3,bot_bot(set(A))) = A3 ).
% Diff_empty
tff(fact_707_insert__Diff1,axiom,
! [A: $tType,X: A,B5: set(A),A3: set(A)] :
( member(A,X,B5)
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),B5) = minus_minus(set(A),A3,B5) ) ) ).
% insert_Diff1
tff(fact_708_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
( ~ member(A,X,A3)
=> ( minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5)) = minus_minus(set(A),A3,B5) ) ) ).
% Diff_insert0
tff(fact_709_Un__Diff__cancel2,axiom,
! [A: $tType,B5: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),B5,A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A3) ).
% Un_Diff_cancel2
tff(fact_710_Un__Diff__cancel,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),minus_minus(set(A),B5,A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) ).
% Un_Diff_cancel
tff(fact_711_converse__iff,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),converse(B,A,R2))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A4),R2) ) ).
% converse_iff
tff(fact_712_Field__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),converse(A,A,R2)) = aa(set(product_prod(A,A)),set(A),field2(A),R2) ).
% Field_converse
tff(fact_713_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S2))
<=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),R2),S2) ) ).
% converse_mono
tff(fact_714_converse__empty,axiom,
! [B: $tType,A: $tType] : converse(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(product_prod(A,B))) ).
% converse_empty
tff(fact_715_trans__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,converse(A,A,R2))
<=> trans(A,R2) ) ).
% trans_converse
tff(fact_716_converse__Id,axiom,
! [A: $tType] : converse(A,A,id2(A)) = id2(A) ).
% converse_Id
tff(fact_717_finite__converse,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),converse(B,A,R2))
<=> aa(set(product_prod(B,A)),$o,finite_finite2(product_prod(B,A)),R2) ) ).
% finite_converse
tff(fact_718_refl__on__converse,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,A3,converse(A,A,R2))
<=> refl_on(A,A3,R2) ) ).
% refl_on_converse
tff(fact_719_total__on__converse,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( total_on(A,A3,converse(A,A,R2))
<=> total_on(A,A3,R2) ) ).
% total_on_converse
tff(fact_720_acyclic__empty,axiom,
! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).
% acyclic_empty
tff(fact_721_total__on__diff__Id,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( total_on(A,A3,minus_minus(set(product_prod(A,A)),R2,id2(A)))
<=> total_on(A,A3,R2) ) ).
% total_on_diff_Id
tff(fact_722_antisym__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( antisym(A,converse(A,A,R2))
<=> antisym(A,R2) ) ).
% antisym_converse
tff(fact_723_converse__Id__on,axiom,
! [A: $tType,A3: set(A)] : converse(A,A,id_on(A,A3)) = id_on(A,A3) ).
% converse_Id_on
tff(fact_724_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( minus_minus(set(A),A3,B5) = bot_bot(set(A)) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% Diff_eq_empty_iff
tff(fact_725_insert__Diff__single,axiom,
! [A: $tType,A4: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3) ).
% insert_Diff_single
tff(fact_726_finite__Diff__insert,axiom,
! [A: $tType,A3: set(A),A4: A,B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)))
<=> aa(set(A),$o,finite_finite2(A),minus_minus(set(A),A3,B5)) ) ).
% finite_Diff_insert
tff(fact_727_Diff__disjoint,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),minus_minus(set(A),B5,A3)) = bot_bot(set(A)) ).
% Diff_disjoint
tff(fact_728_below__Id__inv,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),converse(A,A,R3)),id2(A))
<=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),id2(A)) ) ).
% below_Id_inv
tff(fact_729_Domain__converse,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : aa(set(product_prod(A,B)),set(A),domain(A,B),converse(B,A,R2)) = aa(set(product_prod(B,A)),set(A),range2(B,A),R2) ).
% Domain_converse
tff(fact_730_Range__converse,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(B,A)),set(A),range2(B,A),converse(A,B,R2)) = aa(set(product_prod(A,B)),set(A),domain(A,B),R2) ).
% Range_converse
tff(fact_731_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),aa(set(B),set(A),image2(B,A,F2),A3),aa(set(B),set(A),image2(B,A,F2),B5))),aa(set(B),set(A),image2(B,A,F2),minus_minus(set(B),A3,B5))) ).
% image_diff_subset
tff(fact_732_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set(A),B3: B,F2: fun(A,B)] :
( member(A,X,A3)
=> ( ( B3 = aa(A,B,F2,X) )
=> member(B,B3,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).
% rev_image_eqI
tff(fact_733_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
( ! [X3: A] :
( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
=> aa(A,$o,P,X3) )
=> ! [X4: B] :
( member(B,X4,A3)
=> aa(A,$o,P,aa(B,A,F2,X4)) ) ) ).
% ball_imageD
tff(fact_734_image__cong,axiom,
! [B: $tType,A: $tType,M2: set(A),N: set(A),F2: fun(A,B),G: fun(A,B)] :
( ( M2 = N )
=> ( ! [X3: A] :
( member(A,X3,N)
=> ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
=> ( aa(set(A),set(B),image2(A,B,F2),M2) = aa(set(A),set(B),image2(A,B,G),N) ) ) ) ).
% image_cong
tff(fact_735_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
( ? [X4: A] :
( member(A,X4,aa(set(B),set(A),image2(B,A,F2),A3))
& aa(A,$o,P,X4) )
=> ? [X3: B] :
( member(B,X3,A3)
& aa(A,$o,P,aa(B,A,F2,X3)) ) ) ).
% bex_imageD
tff(fact_736_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F2: fun(B,A),A3: set(B)] :
( member(A,Z2,aa(set(B),set(A),image2(B,A,F2),A3))
<=> ? [X2: B] :
( member(B,X2,A3)
& ( Z2 = aa(B,A,F2,X2) ) ) ) ).
% image_iff
tff(fact_737_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
( member(A,X,A3)
=> member(B,aa(A,B,F2,X),aa(set(A),set(B),image2(A,B,F2),A3)) ) ).
% imageI
tff(fact_738_DiffD2,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,minus_minus(set(A),A3,B5))
=> ~ member(A,C2,B5) ) ).
% DiffD2
tff(fact_739_DiffD1,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,minus_minus(set(A),A3,B5))
=> member(A,C2,A3) ) ).
% DiffD1
tff(fact_740_DiffE,axiom,
! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
( member(A,C2,minus_minus(set(A),A3,B5))
=> ~ ( member(A,C2,A3)
=> member(A,C2,B5) ) ) ).
% DiffE
tff(fact_741_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A4: A,C2: A,B3: A] : minus_minus(A,minus_minus(A,A4,C2),B3) = minus_minus(A,minus_minus(A,A4,B3),C2) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_742_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( minus_minus(A,A4,B3) = minus_minus(A,C2,D3) )
=> ( ( A4 = B3 )
<=> ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_743_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I: set(A),J: set(A),C6: fun(A,set(B))] : product_Sigma(A,B,minus_minus(set(A),I,J),C6) = minus_minus(set(product_prod(A,B)),product_Sigma(A,B,I,C6),product_Sigma(A,B,J,C6)) ).
% Sigma_Diff_distrib1
tff(fact_744_in__image__insert__iff,axiom,
! [A: $tType,B5: set(set(A)),X: A,A3: set(A)] :
( ! [C7: set(A)] :
( member(set(A),C7,B5)
=> ~ member(A,X,C7) )
=> ( member(set(A),A3,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),B5))
<=> ( member(A,X,A3)
& member(set(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),B5) ) ) ) ).
% in_image_insert_iff
tff(fact_745_converse_Ocases,axiom,
! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A22),A1),R2) ) ).
% converse.cases
tff(fact_746_converse_Osimps,axiom,
! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
<=> ? [A7: B,B7: A] :
( ( A1 = B7 )
& ( A22 = A7 )
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B7),R2) ) ) ).
% converse.simps
tff(fact_747_converseD,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),converse(B,A,R2))
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A4),R2) ) ).
% converseD
tff(fact_748_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod(A,B),R2: set(product_prod(B,A))] :
( member(product_prod(A,B),Yx,converse(B,A,R2))
=> ~ ! [X3: B,Y2: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X3) )
=> ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y2),R2) ) ) ).
% converseE
tff(fact_749_converseI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R2)
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B3),A4),converse(A,B,R2)) ) ).
% converseI
tff(fact_750_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B)),B5: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),aa(set(product_prod(A,B)),set(A),domain(A,B),A3),aa(set(product_prod(A,B)),set(A),domain(A,B),B5))),aa(set(product_prod(A,B)),set(A),domain(A,B),minus_minus(set(product_prod(A,B)),A3,B5))) ).
% Domain_Diff_subset
tff(fact_751_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(B,A)),B5: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),aa(set(product_prod(B,A)),set(A),range2(B,A),A3),aa(set(product_prod(B,A)),set(A),range2(B,A),B5))),aa(set(product_prod(B,A)),set(A),range2(B,A),minus_minus(set(product_prod(B,A)),A3,B5))) ).
% Range_Diff_subset
tff(fact_752_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( minus_minus(A,A4,B3) = minus_minus(A,C2,D3) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_753_diff__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,A4,C2)),minus_minus(A,B3,C2)) ) ) ).
% diff_right_mono
tff(fact_754_diff__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,C2,A4)),minus_minus(A,C2,B3)) ) ) ).
% diff_left_mono
tff(fact_755_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,D3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,A4,C2)),minus_minus(A,B3,D3)) ) ) ) ).
% diff_mono
tff(fact_756_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set(A),F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F2),A3))
<=> ? [AA: set(B)] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
& ( B5 = aa(set(B),set(A),image2(B,A,F2),AA) ) ) ) ).
% subset_image_iff
tff(fact_757_image__subset__iff,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B5)
<=> ! [X2: B] :
( member(B,X2,A3)
=> member(A,aa(B,A,F2,X2),B5) ) ) ).
% image_subset_iff
tff(fact_758_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set(A),F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F2),A3))
=> ~ ! [C7: set(B)] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C7),A3)
=> ( B5 != aa(set(B),set(A),image2(B,A,F2),C7) ) ) ) ).
% subset_imageE
tff(fact_759_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B5: set(B)] :
( ! [X3: A] :
( member(A,X3,A3)
=> member(B,aa(A,B,F2,X3),B5) )
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B5) ) ).
% image_subsetI
tff(fact_760_image__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(A),F2: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B5)) ) ).
% image_mono
tff(fact_761_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(B,A))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),converse(B,A,S2))
<=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),converse(A,B,R2)),S2) ) ).
% converse_subset_swap
tff(fact_762_image__Un,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B5: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B5)) ).
% image_Un
tff(fact_763_double__diff,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> ( minus_minus(set(A),B5,minus_minus(set(A),C6,A3)) = A3 ) ) ) ).
% double_diff
tff(fact_764_Diff__subset,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),A3,B5)),A3) ).
% Diff_subset
tff(fact_765_Diff__mono,axiom,
! [A: $tType,A3: set(A),C6: set(A),D4: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),A3,B5)),minus_minus(set(A),C6,D4)) ) ) ).
% Diff_mono
tff(fact_766_insert__Diff__if,axiom,
! [A: $tType,X: A,A3: set(A),B5: set(A)] :
minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),B5) = $ite(member(A,X,B5),minus_minus(set(A),A3,B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),minus_minus(set(A),A3,B5))) ).
% insert_Diff_if
tff(fact_767_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(B,C)),S2: set(product_prod(C,A))] : converse(B,A,relcomp(B,C,A,R2,S2)) = relcomp(A,C,B,converse(C,A,S2),converse(B,C,R2)) ).
% converse_relcomp
tff(fact_768_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),R2),S2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S2)) ).
% converse_Int
tff(fact_769_Diff__Int__distrib2,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),minus_minus(set(A),A3,B5)),C6) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) ).
% Diff_Int_distrib2
tff(fact_770_Diff__Int__distrib,axiom,
! [A: $tType,C6: set(A),A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),minus_minus(set(A),A3,B5)) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C6),B5)) ).
% Diff_Int_distrib
tff(fact_771_Diff__Diff__Int,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : minus_minus(set(A),A3,minus_minus(set(A),A3,B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ).
% Diff_Diff_Int
tff(fact_772_Diff__Int2,axiom,
! [A: $tType,A3: set(A),C6: set(A),B5: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6),B5) ).
% Diff_Int2
tff(fact_773_Int__Diff,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),minus_minus(set(A),B5,C6)) ).
% Int_Diff
tff(fact_774_converse__Un,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R2),S2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S2)) ).
% converse_Un
tff(fact_775_set__diff__diff__left,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : minus_minus(set(A),minus_minus(set(A),A3,B5),C6) = minus_minus(set(A),A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) ).
% set_diff_diff_left
tff(fact_776_Un__Diff,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5),C6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A3,C6)),minus_minus(set(A),B5,C6)) ).
% Un_Diff
tff(fact_777_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B5: set(B)] : vimage(A,B,F2,minus_minus(set(B),A3,B5)) = minus_minus(set(A),vimage(A,B,F2,A3),vimage(A,B,F2,B5)) ).
% vimage_Diff
tff(fact_778_cyclic__subset,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,R3)
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),S)
=> ~ transitive_acyclic(A,S) ) ) ).
% cyclic_subset
tff(fact_779_acyclic__union_I2_J,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A3),B5))
=> transitive_acyclic(A,B5) ) ).
% acyclic_union(2)
tff(fact_780_acyclic__union_I1_J,axiom,
! [A: $tType,A3: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A3),B5))
=> transitive_acyclic(A,A3) ) ).
% acyclic_union(1)
tff(fact_781_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,converse(A,A,R2))) ) ).
% rtrancl_converseI
tff(fact_782_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,converse(A,A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,R2)) ) ).
% rtrancl_converseD
tff(fact_783_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),converse(A,A,transitive_trancl(A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,converse(A,A,R2))) ) ).
% trancl_converseI
tff(fact_784_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,converse(A,A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),converse(A,A,transitive_trancl(A,R2))) ) ).
% trancl_converseD
tff(fact_785_diff__shunt__var,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( minus_minus(A,X,Y) = bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% diff_shunt_var
tff(fact_786_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
( member(A,Y,aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))
=> ~ ! [X3: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y),S) ) ).
% in_snd_imageE
tff(fact_787_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B5))) ).
% image_Int_subset
tff(fact_788_subset__minus__empty,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( minus_minus(set(A),A3,B5) = bot_bot(set(A)) ) ) ).
% subset_minus_empty
tff(fact_789_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B5)
<=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),vimage(B,A,F2,B5)) ) ).
% image_subset_iff_subset_vimage
tff(fact_790_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3))),A3) ).
% image_vimage_subset
tff(fact_791_Diff__insert,axiom,
! [A: $tType,A3: set(A),A4: A,B5: set(A)] : minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = minus_minus(set(A),minus_minus(set(A),A3,B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ).
% Diff_insert
tff(fact_792_insert__Diff,axiom,
! [A: $tType,A4: A,A3: set(A)] :
( member(A,A4,A3)
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = A3 ) ) ).
% insert_Diff
tff(fact_793_Diff__insert2,axiom,
! [A: $tType,A3: set(A),A4: A,B5: set(A)] : minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5)) = minus_minus(set(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))),B5) ).
% Diff_insert2
tff(fact_794_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A3: set(A)] :
( ( X != Y )
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))) ) ) ).
% insert_minus_eq
tff(fact_795_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set(A)] :
( ~ member(A,X,A3)
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = A3 ) ) ).
% Diff_insert_absorb
tff(fact_796_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X5: set(A)] :
( ~ member(A,X,X5)
=> ( minus_minus(set(A),X5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_797_subset__Diff__insert,axiom,
! [A: $tType,A3: set(A),B5: set(A),X: A,C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),minus_minus(set(A),B5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),C6)))
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),minus_minus(set(A),B5,C6))
& ~ member(A,X,A3) ) ) ).
% subset_Diff_insert
tff(fact_798_Diff__triv,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( minus_minus(set(A),A3,B5) = A3 ) ) ).
% Diff_triv
tff(fact_799_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),minus_minus(set(A),A3,B5)) = bot_bot(set(A)) ).
% Int_Diff_disjoint
tff(fact_800_disjoint__alt__simp1,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( minus_minus(set(A),A3,B5) = A3 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) ) ) ).
% disjoint_alt_simp1
tff(fact_801_disjoint__alt__simp2,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( minus_minus(set(A),A3,B5) != A3 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp2
tff(fact_802_Diff__subset__conv,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),A3,B5)),C6)
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) ) ).
% Diff_subset_conv
tff(fact_803_Diff__partition,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),minus_minus(set(A),B5,A3)) = B5 ) ) ).
% Diff_partition
tff(fact_804_Un__Diff__Int,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A3,B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) = A3 ).
% Un_Diff_Int
tff(fact_805_Int__Diff__Un,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)),minus_minus(set(A),A3,B5)) = A3 ).
% Int_Diff_Un
tff(fact_806_Diff__Int,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : minus_minus(set(A),A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A3,B5)),minus_minus(set(A),A3,C6)) ).
% Diff_Int
tff(fact_807_Diff__Un,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] : minus_minus(set(A),A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),minus_minus(set(A),A3,B5)),minus_minus(set(A),A3,C6)) ).
% Diff_Un
tff(fact_808_Range__snd,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),R2) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R2) ).
% Range_snd
tff(fact_809_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R3) = aa(set(product_prod(B,A)),set(A),range2(B,A),R3) ).
% snd_eq_Range
tff(fact_810_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X: A] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [Y2: A] :
( member(A,Y2,A3)
=> ( aa(A,B,F2,Y2) = aa(A,B,F2,X) ) )
=> ( the_elem(B,aa(set(A),set(B),image2(A,B,F2),A3)) = aa(A,B,F2,X) ) ) ) ).
% the_elem_image_unique
tff(fact_811_cyclicE,axiom,
! [A: $tType,G: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,G)
=> ~ ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,G)) ) ).
% cyclicE
tff(fact_812_acyclic__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( transitive_acyclic(A,R2)
<=> ! [X2: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),transitive_trancl(A,R2)) ) ).
% acyclic_def
tff(fact_813_acyclicI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R2))
=> transitive_acyclic(A,R2) ) ).
% acyclicI
tff(fact_814_snd__image__mp,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(B,A)),B5: set(A),X: B,Y: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A3)),B5)
=> ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),A3)
=> member(A,Y,B5) ) ) ).
% snd_image_mp
tff(fact_815_snd__in__Field,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R3)),aa(set(product_prod(A,A)),set(A),field2(A),R3)) ).
% snd_in_Field
tff(fact_816_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(B,A)),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,A3,B5)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A3)) ).
% Image_subset_snd_image
tff(fact_817_infinite__remove,axiom,
! [A: $tType,S: set(A),A4: A] :
( ~ aa(set(A),$o,finite_finite2(A),S)
=> ~ aa(set(A),$o,finite_finite2(A),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) ) ).
% infinite_remove
tff(fact_818_infinite__coinduct,axiom,
! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
( aa(set(A),$o,X5,A3)
=> ( ! [A10: set(A)] :
( aa(set(A),$o,X5,A10)
=> ? [X4: A] :
( member(A,X4,A10)
& ( aa(set(A),$o,X5,minus_minus(set(A),A10,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A)))))
| ~ aa(set(A),$o,finite_finite2(A),minus_minus(set(A),A10,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))))) ) ) )
=> ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).
% infinite_coinduct
tff(fact_819_finite__empty__induct,axiom,
! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,P,A3)
=> ( ! [A5: A,A10: set(A)] :
( aa(set(A),$o,finite_finite2(A),A10)
=> ( member(A,A5,A10)
=> ( aa(set(A),$o,P,A10)
=> aa(set(A),$o,P,minus_minus(set(A),A10,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),bot_bot(set(A))))) ) ) )
=> aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).
% finite_empty_induct
tff(fact_820_subset__insert__iff,axiom,
! [A: $tType,A3: set(A),X: A,B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5))
<=> $ite(member(A,X,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)) ) ).
% subset_insert_iff
tff(fact_821_Diff__single__insert,axiom,
! [A: $tType,A3: set(A),X: A,B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5)) ) ).
% Diff_single_insert
tff(fact_822_Field__rel__restrict,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),rel_restrict(A,R3,A3))),minus_minus(set(A),aa(set(product_prod(A,A)),set(A),field2(A),R3),A3)) ).
% Field_rel_restrict
tff(fact_823_Domain__rel__restrict,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),rel_restrict(A,R3,A3))),minus_minus(set(A),aa(set(product_prod(A,A)),set(A),domain(A,A),R3),A3)) ).
% Domain_rel_restrict
tff(fact_824_Range__rel__restrict,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),range2(A,A),rel_restrict(A,R3,A3))),minus_minus(set(A),aa(set(product_prod(A,A)),set(A),range2(A,A),R3),A3)) ).
% Range_rel_restrict
tff(fact_825_trans__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> ( antisym(A,R2)
=> trans(A,minus_minus(set(product_prod(A,A)),R2,id2(A))) ) ) ).
% trans_diff_Id
tff(fact_826_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
=> ( ~ aa(set(B),$o,finite_finite2(B),A3)
=> ~ ! [Y2: A] :
( member(A,Y2,aa(set(B),set(A),image2(B,A,F2),A3))
=> aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_domE
tff(fact_827_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
=> ( ~ aa(set(B),$o,finite_finite2(B),A3)
=> ? [X3: A] :
( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
& ~ aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_dom
tff(fact_828_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set(product_prod(A,A)),X: A,Y: A] :
( transitive_acyclic(A,G)
=> ( ~ transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),G))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,G)) ) ) ).
% acyclic_insert_cyclic
tff(fact_829_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Ha: fun(A,A),N: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,Ha,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,Ha,X3)),aa(A,A,Ha,Y2))
=> ( aa(set(A),$o,finite_finite2(A),N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,Ha,lattic7752659483105999362nf_fin(A,N)) = lattic7752659483105999362nf_fin(A,aa(set(A),set(A),image2(A,A,Ha),N)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_830_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Ha: fun(A,A),N: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,Ha,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,Ha,X3)),aa(A,A,Ha,Y2))
=> ( aa(set(A),$o,finite_finite2(A),N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,Ha,lattic5882676163264333800up_fin(A,N)) = lattic5882676163264333800up_fin(A,aa(set(A),set(A),image2(A,A,Ha),N)) ) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_831_finite__remove__induct,axiom,
! [A: $tType,B5: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),B5)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A10: set(A)] :
( aa(set(A),$o,finite_finite2(A),A10)
=> ( ( A10 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),B5)
=> ( ! [X4: A] :
( member(A,X4,A10)
=> aa(set(A),$o,P,minus_minus(set(A),A10,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))))) )
=> aa(set(A),$o,P,A10) ) ) ) )
=> aa(set(A),$o,P,B5) ) ) ) ).
% finite_remove_induct
tff(fact_832_remove__induct,axiom,
! [A: $tType,P: fun(set(A),$o),B5: set(A)] :
( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ( ~ aa(set(A),$o,finite_finite2(A),B5)
=> aa(set(A),$o,P,B5) )
=> ( ! [A10: set(A)] :
( aa(set(A),$o,finite_finite2(A),A10)
=> ( ( A10 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),B5)
=> ( ! [X4: A] :
( member(A,X4,A10)
=> aa(set(A),$o,P,minus_minus(set(A),A10,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))))) )
=> aa(set(A),$o,P,A10) ) ) ) )
=> aa(set(A),$o,P,B5) ) ) ) ).
% remove_induct
tff(fact_833_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),minus_minus(set(product_prod(A,A)),R2,id2(A))) ) ) ) ) ).
% Linear_order_in_diff_Id
tff(fact_834_max__ext_Omax__extI,axiom,
! [A: $tType,X5: set(A),Y4: set(A),R3: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( aa(set(A),$o,finite_finite2(A),Y4)
=> ( ( Y4 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,Y4)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R3) ) )
=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Y4),max_ext(A,R3)) ) ) ) ) ).
% max_ext.max_extI
tff(fact_835_max__ext_Osimps,axiom,
! [A: $tType,A1: set(A),A22: set(A),R3: set(product_prod(A,A))] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R3))
<=> ( aa(set(A),$o,finite_finite2(A),A1)
& aa(set(A),$o,finite_finite2(A),A22)
& ( A22 != bot_bot(set(A)) )
& ! [X2: A] :
( member(A,X2,A1)
=> ? [Xa2: A] :
( member(A,Xa2,A22)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R3) ) ) ) ) ).
% max_ext.simps
tff(fact_836_max__ext_Ocases,axiom,
! [A: $tType,A1: set(A),A22: set(A),R3: set(product_prod(A,A))] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R3))
=> ~ ( aa(set(A),$o,finite_finite2(A),A1)
=> ( aa(set(A),$o,finite_finite2(A),A22)
=> ( ( A22 != bot_bot(set(A)) )
=> ~ ! [X4: A] :
( member(A,X4,A1)
=> ? [Xa4: A] :
( member(A,Xa4,A22)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),R3) ) ) ) ) ) ) ).
% max_ext.cases
tff(fact_837_inf__period_I2_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
( ! [X3: A,K3: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,P,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) )
=> ( ! [X3: A,K3: A] :
( aa(A,$o,Q,X3)
<=> aa(A,$o,Q,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) )
=> ! [X4: A,K4: A] :
( ( aa(A,$o,P,X4)
| aa(A,$o,Q,X4) )
<=> ( aa(A,$o,P,minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))
| aa(A,$o,Q,minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))) ) ) ) ) ) ).
% inf_period(2)
tff(fact_838_inf__period_I1_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
( ! [X3: A,K3: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,P,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) )
=> ( ! [X3: A,K3: A] :
( aa(A,$o,Q,X3)
<=> aa(A,$o,Q,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) )
=> ! [X4: A,K4: A] :
( ( aa(A,$o,P,X4)
& aa(A,$o,Q,X4) )
<=> ( aa(A,$o,P,minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))
& aa(A,$o,Q,minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))) ) ) ) ) ) ).
% inf_period(1)
tff(fact_839_left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,B3)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% left_diff_distrib
tff(fact_840_right__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),minus_minus(A,B3,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ).
% right_diff_distrib
tff(fact_841_left__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [B3: A,C2: A,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B3,C2)),A4) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)) ) ).
% left_diff_distrib'
tff(fact_842_right__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),minus_minus(A,B3,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ).
% right_diff_distrib'
tff(fact_843_in__lex__prod,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A6: A,B4: B,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4)),lex_prod(A,B,R2,S2))
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A6),R2)
| ( ( A4 = A6 )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B3),B4),S2) ) ) ) ).
% in_lex_prod
tff(fact_844_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( wf(A,minus_minus(set(product_prod(A,A)),R2,id2(A)))
<=> ! [A11: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( A11 != bot_bot(set(A)) )
=> ? [X2: A] :
( member(A,X2,A11)
& ! [Xa2: A] :
( member(A,Xa2,A11)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R2) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
tff(fact_845_remove__def,axiom,
! [A: $tType,X: A,A3: set(A)] : remove(A,X,A3) = minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% remove_def
tff(fact_846_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A,Phi: fun(A,fun(A,$o))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),minus_minus(set(product_prod(A,A)),R2,id2(A)))
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),minus_minus(set(product_prod(A,A)),R2,id2(A))) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) )
=> ( ( ( A4 = B3 )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) ) ) ) ) ).
% wo_rel.cases_Total3
tff(fact_847_wf__max,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,converse(A,A,R3))
=> ( ( R3 != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M3: A] : ~ member(A,M3,minus_minus(set(A),aa(set(product_prod(A,A)),set(A),range2(A,A),R3),aa(set(product_prod(A,A)),set(A),domain(A,A),R3))) ) ) ).
% wf_max
tff(fact_848_wf__Un,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( wf(A,R2)
=> ( wf(A,S2)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range2(A,A),S2)) = bot_bot(set(A)) )
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ) ).
% wf_Un
tff(fact_849_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
<=> ! [A11: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( A11 != bot_bot(set(A)) )
=> ? [X2: A] :
( member(A,X2,A11)
& ! [Xa2: A] :
( member(A,Xa2,A11)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R2) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
tff(fact_850_underS__incr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( trans(A,R2)
=> ( antisym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A4)),order_underS(A,R2,B3)) ) ) ) ).
% underS_incr
tff(fact_851_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A13: set(A),R1: set(product_prod(A,A)),A23: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C))] :
( equiv_equiv(A,A13,R1)
=> ( equiv_equiv(B,A23,R22)
=> ( ! [Y2: A,Z4: A,W2: B] :
( member(B,W2,A23)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R1)
=> ( aa(B,C,aa(A,fun(B,C),F2,Y2),W2) = aa(B,C,aa(A,fun(B,C),F2,Z4),W2) ) ) )
=> ( ! [Y2: B,Z4: B,W2: A] :
( member(A,W2,A13)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z4),R22)
=> ( aa(B,C,aa(A,fun(B,C),F2,W2),Y2) = aa(B,C,aa(A,fun(B,C),F2,W2),Z4) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F2) ) ) ) ) ).
% congruent2I
tff(fact_852_member__remove,axiom,
! [A: $tType,X: A,Y: A,A3: set(A)] :
( member(A,X,remove(A,Y,A3))
<=> ( member(A,X,A3)
& ( X != Y ) ) ) ).
% member_remove
tff(fact_853_wf__empty,axiom,
! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).
% wf_empty
tff(fact_854_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
<=> ( wf(A,R2)
& ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2)) ) ) ).
% wf_insert
tff(fact_855_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A4: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2) )
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A4) ) ) ).
% wo_rel.well_order_induct
tff(fact_856_wf__induct__rule,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A4: A] :
( wf(A,R2)
=> ( ! [X3: A] :
( ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2)
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A4) ) ) ).
% wf_induct_rule
tff(fact_857_wf__eq__minimal,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [Q4: set(A)] :
( ? [X2: A] : member(A,X2,Q4)
=> ? [X2: A] :
( member(A,X2,Q4)
& ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R2)
=> ~ member(A,Y3,Q4) ) ) ) ) ).
% wf_eq_minimal
tff(fact_858_wf__not__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( wf(A,R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2) ) ).
% wf_not_refl
tff(fact_859_wf__not__sym,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,X: A] :
( wf(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),X),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A4),R2) ) ) ).
% wf_not_sym
tff(fact_860_wf__irrefl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( wf(A,R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2) ) ).
% wf_irrefl
tff(fact_861_wf__induct,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A4: A] :
( wf(A,R2)
=> ( ! [X3: A] :
( ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2)
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A4) ) ) ).
% wf_induct
tff(fact_862_wf__asym,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,X: A] :
( wf(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),X),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A4),R2) ) ) ).
% wf_asym
tff(fact_863_wfUNIVI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [P3: fun(A,$o),X3: A] :
( ! [Xa3: A] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R2)
=> aa(A,$o,P3,Y2) )
=> aa(A,$o,P3,Xa3) )
=> aa(A,$o,P3,X3) )
=> wf(A,R2) ) ).
% wfUNIVI
tff(fact_864_wfI__min,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [X3: A,Q2: set(A)] :
( member(A,X3,Q2)
=> ? [Xa3: A] :
( member(A,Xa3,Q2)
& ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R3)
=> ~ member(A,Y2,Q2) ) ) )
=> wf(A,R3) ) ).
% wfI_min
tff(fact_865_wfE__min,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: A,Q: set(A)] :
( wf(A,R3)
=> ( member(A,X,Q)
=> ~ ! [Z4: A] :
( member(A,Z4,Q)
=> ~ ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z4),R3)
=> ~ member(A,Y5,Q) ) ) ) ) ).
% wfE_min
tff(fact_866_wf__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [P4: fun(A,$o)] :
( ! [X2: A] :
( ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R2)
=> aa(A,$o,P4,Y3) )
=> aa(A,$o,P4,X2) )
=> ! [X_12: A] : aa(A,$o,P4,X_12) ) ) ).
% wf_def
tff(fact_867_well__order__on__domain,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A4: A,B3: A] :
( order_well_order_on(A,A3,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> ( member(A,A4,A3)
& member(A,B3,A3) ) ) ) ).
% well_order_on_domain
tff(fact_868_underS__E,axiom,
! [A: $tType,I2: A,R3: set(product_prod(A,A)),J2: A] :
( member(A,I2,order_underS(A,R3,J2))
=> ( ( I2 != J2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2),R3) ) ) ).
% underS_E
tff(fact_869_underS__I,axiom,
! [A: $tType,I2: A,J2: A,R3: set(product_prod(A,A))] :
( ( I2 != J2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2),R3)
=> member(A,I2,order_underS(A,R3,J2)) ) ) ).
% underS_I
tff(fact_870_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I2: A,R3: set(product_prod(A,A)),J2: A] :
( member(A,I2,order_underS(A,R3,J2))
=> member(A,I2,aa(set(product_prod(A,A)),set(A),field2(A),R3)) ) ).
% BNF_Least_Fixpoint.underS_Field
tff(fact_871_wo__rel_OTOTALS,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ! [X4: A] :
( member(A,X4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ! [Xa3: A] :
( member(A,Xa3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4),R2) ) ) ) ) ).
% wo_rel.TOTALS
tff(fact_872_wfE__min_H,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Q: set(A)] :
( wf(A,R3)
=> ( ( Q != bot_bot(set(A)) )
=> ~ ! [Z4: A] :
( member(A,Z4,Q)
=> ~ ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z4),R3)
=> ~ member(A,Y5,Q) ) ) ) ) ).
% wfE_min'
tff(fact_873_well__order__on__empty,axiom,
! [A: $tType] : order_well_order_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% well_order_on_empty
tff(fact_874_wf__no__loop,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ( relcomp(A,A,A,R3,R3) = bot_bot(set(product_prod(A,A))) )
=> wf(A,R3) ) ).
% wf_no_loop
tff(fact_875_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R3)
=> ( wf(A,converse(A,A,R3))
<=> wf(A,R3) ) ) ).
% finite_wf_eq_wf_converse
tff(fact_876_well__order__induct__imp,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A4: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2) )
=> ( member(A,Y5,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> aa(A,$o,P,Y5) ) )
=> ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> aa(A,$o,P,X3) ) )
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> aa(A,$o,P,A4) ) ) ) ).
% well_order_induct_imp
tff(fact_877_underS__empty,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A))] :
( ~ member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( order_underS(A,R2,A4) = bot_bot(set(A)) ) ) ).
% underS_empty
tff(fact_878_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C)),Y1: A,Z1: A,Y22: B,Z22: B] :
( equiv_congruent2(A,B,C,R1,R22,F2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1),R1)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z22),R22)
=> ( aa(B,C,aa(A,fun(B,C),F2,Y1),Y22) = aa(B,C,aa(A,fun(B,C),F2,Z1),Z22) ) ) ) ) ).
% congruent2D
tff(fact_879_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C))] :
( ! [Y12: A,Z12: A,Y23: B,Z23: B] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z12),R1)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z23),R22)
=> ( aa(B,C,aa(A,fun(B,C),F2,Y12),Y23) = aa(B,C,aa(A,fun(B,C),F2,Z12),Z23) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F2) ) ).
% congruent2I'
tff(fact_880_wfI__pf,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [A10: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),image(A,A,R3,A10))
=> ( A10 = bot_bot(set(A)) ) )
=> wf(A,R3) ) ).
% wfI_pf
tff(fact_881_wfE__pf,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] :
( wf(A,R3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),image(A,A,R3,A3))
=> ( A3 = bot_bot(set(A)) ) ) ) ).
% wfE_pf
tff(fact_882_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,$o),K: B,M: fun(B,A)] :
( wf(A,R2)
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),transitive_trancl(A,R2))
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),transitive_rtrancl(A,R2)) )
=> ( aa(B,$o,P,K)
=> ? [X3: B] :
( aa(B,$o,P,X3)
& ! [Y5: B] :
( aa(B,$o,P,Y5)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X3)),aa(B,A,M,Y5)),transitive_rtrancl(A,R2)) ) ) ) ) ) ).
% wf_linord_ex_has_least
tff(fact_883_wf__eq__minimal2,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [A11: set(A)] :
( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ( A11 != bot_bot(set(A)) ) )
=> ? [X2: A] :
( member(A,X2,A11)
& ! [Xa2: A] :
( member(A,Xa2,A11)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),R2) ) ) ) ) ).
% wf_eq_minimal2
tff(fact_884_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A,Phi: fun(A,fun(A,$o))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) )
=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A4),B3) ) ) ) ) ).
% wo_rel.cases_Total
tff(fact_885_wf__no__path,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R3)),aa(set(product_prod(A,A)),set(A),range2(A,A),R3)) = bot_bot(set(A)) )
=> wf(A,R3) ) ).
% wf_no_path
tff(fact_886_wf__min,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,R3)
=> ( ( R3 != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M3: A] : ~ member(A,M3,minus_minus(set(A),aa(set(product_prod(A,A)),set(A),domain(A,A),R3),aa(set(product_prod(A,A)),set(A),range2(A,A),R3))) ) ) ).
% wf_min
tff(fact_887_underS__incl__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A4)),order_underS(A,R2,B3))
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ) ) ).
% underS_incl_iff
tff(fact_888_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,fun(A,B))] :
( equiv_equiv(A,A3,R2)
=> ( ! [Y2: A,Z4: A] :
( member(A,Y2,A3)
=> ( member(A,Z4,A3)
=> ( aa(A,B,aa(A,fun(A,B),F2,Y2),Z4) = aa(A,B,aa(A,fun(A,B),F2,Z4),Y2) ) ) )
=> ( ! [Y2: A,Z4: A,W2: A] :
( member(A,W2,A3)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( aa(A,B,aa(A,fun(A,B),F2,W2),Y2) = aa(A,B,aa(A,fun(A,B),F2,W2),Z4) ) ) )
=> equiv_congruent2(A,A,B,R2,R2,F2) ) ) ) ).
% congruent2_commuteI
tff(fact_889_brk__rel__wf,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,R3)
=> wf(product_prod($o,A),brk_rel(A,A,R3)) ) ).
% brk_rel_wf
tff(fact_890_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( B5 != bot_bot(set(A)) )
=> ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R2,B5,X_1) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
tff(fact_891_Refl__under__underS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( order_under(A,R2,A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R2,A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ) ) ) ).
% Refl_under_underS
tff(fact_892_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( B5 != bot_bot(set(A)) )
=> member(A,bNF_We6954850376910717587_minim(A,R2,B5),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).
% wo_rel.minim_inField
tff(fact_893_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( B5 != bot_bot(set(A)) )
=> member(A,bNF_We6954850376910717587_minim(A,R2,B5),B5) ) ) ) ).
% wo_rel.minim_in
tff(fact_894_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A),A4: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,A4,B5)
=> ( ! [B2: A] :
( member(A,B2,B5)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B2),R2) )
=> ( A4 = bNF_We6954850376910717587_minim(A,R2,B5) ) ) ) ) ) ).
% wo_rel.equals_minim
tff(fact_895_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A),B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,B5)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R2,B5)),B3),R2) ) ) ) ).
% wo_rel.minim_least
tff(fact_896_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),bNF_We1388413361240627857o_max2(A,R2,A4,B3)),R2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_We1388413361240627857o_max2(A,R2,A4,B3)),R2)
& member(A,bNF_We1388413361240627857o_max2(A,R2,A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ) ) ).
% wo_rel.max2_greater_among
tff(fact_897_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,$o)))] :
( wf(A,R3)
=> ( ! [F: fun(A,B),G3: fun(A,B),X3: A,R: B] :
( ! [Z6: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X3),R3)
=> ( aa(A,B,F,Z6) = aa(A,B,G3,Z6) ) )
=> ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F),X3),R)
<=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,G3),X3),R) ) )
=> ( ! [X3: A,F: fun(A,B)] :
( ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R3)
=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F),Y5),aa(A,B,F,Y5)) )
=> ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F),X3),X_13) )
=> ? [F: fun(A,B)] :
! [X4: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F),X4),aa(A,B,F,X4)) ) ) ) ).
% dependent_wf_choice
tff(fact_898_wo__rel_Omax2__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( bNF_We1388413361240627857o_max2(A,R2,A4,B3) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2),B3,A4) ) ) ).
% wo_rel.max2_def
tff(fact_899_under__incr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A4)),order_under(A,R2,B3)) ) ) ).
% under_incr
tff(fact_900_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( B5 != bot_bot(set(A)) )
=> bNF_We4791949203932849705sMinim(A,R2,B5,bNF_We6954850376910717587_minim(A,R2,B5)) ) ) ) ).
% wo_rel.minim_isMinim
tff(fact_901_wo__rel_OisMinim__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( bNF_We4791949203932849705sMinim(A,R2,A3,B3)
<=> ( member(A,B3,A3)
& ! [X2: A] :
( member(A,X2,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),X2),R2) ) ) ) ) ).
% wo_rel.isMinim_def
tff(fact_902_wo__rel_Omax2__greater,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),bNF_We1388413361240627857o_max2(A,R2,A4,B3)),R2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_We1388413361240627857o_max2(A,R2,A4,B3)),R2) ) ) ) ) ).
% wo_rel.max2_greater
tff(fact_903_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( bNF_We1388413361240627857o_max2(A,R2,A4,B3) = B3 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ) ) ).
% wo_rel.max2_equals2
tff(fact_904_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( bNF_We1388413361240627857o_max2(A,R2,A4,B3) = A4 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2) ) ) ) ) ).
% wo_rel.max2_equals1
tff(fact_905_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> member(A,bNF_We1388413361240627857o_max2(A,R2,A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ) ).
% wo_rel.max2_among
tff(fact_906_chains__extend,axiom,
! [A: $tType,C2: set(set(A)),S: set(set(A)),Z2: set(A)] :
( member(set(set(A)),C2,chains2(A,S))
=> ( member(set(A),Z2,S)
=> ( ! [X3: set(A)] :
( member(set(A),X3,C2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Z2) )
=> member(set(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z2),bot_bot(set(set(A))))),C2),chains2(A,S)) ) ) ) ).
% chains_extend
tff(fact_907_Zorns__po__lemma,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ! [C7: set(A)] :
( member(set(A),C7,chains(A,R2))
=> ? [X4: A] :
( member(A,X4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ! [Xa4: A] :
( member(A,Xa4,C7)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X4),R2) ) ) )
=> ? [X3: A] :
( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ! [Xa3: A] :
( member(A,Xa3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R2)
=> ( Xa3 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
tff(fact_908_bsqr__max2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22)),bNF_Wellorder_bsqr(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R2,A1,A22)),bNF_We1388413361240627857o_max2(A,R2,B1,B22)),R2) ) ) ).
% bsqr_max2
tff(fact_909_wf__Union,axiom,
! [A: $tType,R3: set(set(product_prod(A,A)))] :
( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
=> wf(A,X3) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
=> ! [Xa4: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa4,R3)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),X3)),aa(set(product_prod(A,A)),set(A),range2(A,A),Xa4)) = bot_bot(set(A)) ) ) ) )
=> wf(A,aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R3)) ) ) ).
% wf_Union
tff(fact_910_Pow__singleton__iff,axiom,
! [A: $tType,X5: set(A),Y4: set(A)] :
( ( pow2(A,X5) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Y4),bot_bot(set(set(A)))) )
<=> ( ( X5 = bot_bot(set(A)) )
& ( Y4 = bot_bot(set(A)) ) ) ) ).
% Pow_singleton_iff
tff(fact_911_Pow__empty,axiom,
! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_empty
tff(fact_912_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,E3: set(product_prod(A,A)),F2: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,E3))
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,A4)),aa(A,B,F2,B3)),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F2)),E3))) ) ).
% rtrancl_mapI
tff(fact_913_disjoint__image__subset,axiom,
! [A: $tType,A14: set(set(A)),F2: fun(set(A),set(A))] :
( pairwise(set(A),disjnt(A),A14)
=> ( ! [X7: set(A)] :
( member(set(A),X7,A14)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),F2,X7)),X7) )
=> pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F2),A14)) ) ) ).
% disjoint_image_subset
tff(fact_914_pairself_Opelims,axiom,
! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
=> ( aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),Xa))
=> ~ ! [A5: B,B2: B] :
( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B2) )
=> ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A5)),aa(B,A,X,B2)) )
=> ~ aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B2))) ) ) ) ) ).
% pairself.pelims
tff(fact_915_Pow__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( member(set(A),A3,pow2(A,B5))
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% Pow_iff
tff(fact_916_PowI,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> member(set(A),A3,pow2(A,B5)) ) ).
% PowI
tff(fact_917_Pow__Int__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A3)),pow2(A,B5)) ).
% Pow_Int_eq
tff(fact_918_Field__Union,axiom,
! [A: $tType,R3: set(set(product_prod(A,A)))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,A))),set(set(A)),image2(set(product_prod(A,A)),set(A),field2(A)),R3)) ).
% Field_Union
tff(fact_919_pairwise__def,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),S: set(A)] :
( pairwise(A,R3,S)
<=> ! [X2: A] :
( member(A,X2,S)
=> ! [Xa2: A] :
( member(A,Xa2,S)
=> ( ( X2 != Xa2 )
=> aa(A,$o,aa(A,fun(A,$o),R3,X2),Xa2) ) ) ) ) ).
% pairwise_def
tff(fact_920_pairwiseI,axiom,
! [A: $tType,S: set(A),R3: fun(A,fun(A,$o))] :
( ! [X3: A,Y2: A] :
( member(A,X3,S)
=> ( member(A,Y2,S)
=> ( ( X3 != Y2 )
=> aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2) ) ) )
=> pairwise(A,R3,S) ) ).
% pairwiseI
tff(fact_921_pairwiseD,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),S: set(A),X: A,Y: A] :
( pairwise(A,R3,S)
=> ( member(A,X,S)
=> ( member(A,Y,S)
=> ( ( X != Y )
=> aa(A,$o,aa(A,fun(A,$o),R3,X),Y) ) ) ) ) ).
% pairwiseD
tff(fact_922_Pow__top,axiom,
! [A: $tType,A3: set(A)] : member(set(A),A3,pow2(A,A3)) ).
% Pow_top
tff(fact_923_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S: set(A),A3: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
( ! [X3: A] :
( member(A,X3,S)
=> refl_on(B,aa(A,set(B),A3,X3),aa(A,set(product_prod(B,B)),R2,X3)) )
=> refl_on(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),S)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).
% refl_on_UNION
tff(fact_924_Pow__bottom,axiom,
! [A: $tType,B5: set(A)] : member(set(A),bot_bot(set(A)),pow2(A,B5)) ).
% Pow_bottom
tff(fact_925_Pow__mono,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow2(A,A3)),pow2(A,B5)) ) ).
% Pow_mono
tff(fact_926_PowD,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( member(set(A),A3,pow2(A,B5))
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% PowD
tff(fact_927_Un__Pow__subset,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),pow2(A,B5))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) ).
% Un_Pow_subset
tff(fact_928_Pow__not__empty,axiom,
! [A: $tType,A3: set(A)] : pow2(A,A3) != bot_bot(set(set(A))) ).
% Pow_not_empty
tff(fact_929_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),P: fun(B,fun(B,$o))] :
( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> ( ( X3 != Y2 )
=> ( ( aa(A,B,F2,X3) != aa(A,B,F2,Y2) )
=> aa(B,$o,aa(B,fun(B,$o),P,aa(A,B,F2,X3)),aa(A,B,F2,Y2)) ) ) ) )
=> pairwise(B,P,aa(set(A),set(B),image2(A,B,F2),A3)) ) ).
% pairwise_imageI
tff(fact_930_pairwise__empty,axiom,
! [A: $tType,P: fun(A,fun(A,$o))] : pairwise(A,P,bot_bot(set(A))) ).
% pairwise_empty
tff(fact_931_pairwise__subset,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),S: set(A),T3: set(A)] :
( pairwise(A,P,S)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S)
=> pairwise(A,P,T3) ) ) ).
% pairwise_subset
tff(fact_932_pairwise__mono,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),A3: set(A),Q: fun(A,fun(A,$o)),B5: set(A)] :
( pairwise(A,P,A3)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),Q,X3),Y2) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> pairwise(A,Q,B5) ) ) ) ).
% pairwise_mono
tff(fact_933_pairwise__insert,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A,S2: set(A)] :
( pairwise(A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S2))
<=> ( ! [Y3: A] :
( ( member(A,Y3,S2)
& ( Y3 != X ) )
=> ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y3)
& aa(A,$o,aa(A,fun(A,$o),R2,Y3),X) ) )
& pairwise(A,R2,S2) ) ) ).
% pairwise_insert
tff(fact_934_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A3: set(B),A4: B] :
( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3)))
=> ( member(B,A4,A3)
=> aa(set(A),$o,finite_finite2(A),aa(B,set(A),B5,A4)) ) ) ).
% finite_UNION_then_finite
tff(fact_935_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B5: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F2),A3) = B5 )
=> ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A3)) = pow2(A,B5) ) ) ).
% image_Pow_surj
tff(fact_936_Pow__insert,axiom,
! [A: $tType,A4: A,A3: set(A)] : pow2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4)),pow2(A,A3))) ).
% Pow_insert
tff(fact_937_pairself_Osimps,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: B,B3: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F2),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F2,A4)),aa(B,A,F2,B3)) ).
% pairself.simps
tff(fact_938_pairself_Oelims,axiom,
! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
=> ~ ! [A5: B,B2: B] :
( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B2) )
=> ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A5)),aa(B,A,X,B2)) ) ) ) ).
% pairself.elims
tff(fact_939_pairwise__singleton,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),A3: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).
% pairwise_singleton
tff(fact_940_finite__Sup__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2),A3) ) )
=> member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ) ).
% finite_Sup_in
tff(fact_941_Sup__fin__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).
% Sup_fin_Sup
tff(fact_942_insert__partition,axiom,
! [A: $tType,X: set(A),F4: set(set(A))] :
( ~ member(set(A),X,F4)
=> ( ! [X3: set(A)] :
( member(set(A),X3,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4))
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4))
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).
% insert_partition
tff(fact_943_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B5)
=> aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A3))),pow2(A,B5)) ) ).
% image_Pow_mono
tff(fact_944_pairwise__alt,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),S: set(A)] :
( pairwise(A,R3,S)
<=> ! [X2: A] :
( member(A,X2,S)
=> ! [Xa2: A] :
( member(A,Xa2,minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))
=> aa(A,$o,aa(A,fun(A,$o),R3,X2),Xa2) ) ) ) ).
% pairwise_alt
tff(fact_945_Sup__insert,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ).
% Sup_insert
tff(fact_946_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% ccpo_Sup_singleton
tff(fact_947_cSup__singleton,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% cSup_singleton
tff(fact_948_Sup__empty,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).
% Sup_empty
tff(fact_949_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( ( aa(set(A),A,complete_Sup_Sup(A),A3) = bot_bot(A) )
<=> ! [X2: A] :
( member(A,X2,A3)
=> ( X2 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(1)
tff(fact_950_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A3) )
<=> ! [X2: A] :
( member(A,X2,A3)
=> ( X2 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(2)
tff(fact_951_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = lattic5882676163264333800up_fin(A,X5) ) ) ) ) ).
% cSup_eq_Sup_fin
tff(fact_952_Union__image__insert,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),A4: B,B5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),B5))) ).
% Union_image_insert
tff(fact_953_Union__image__empty,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),bot_bot(set(B))))) = A3 ).
% Union_image_empty
tff(fact_954_Sup__SUP__eq,axiom,
! [A: $tType,S: set(fun(A,$o)),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),S),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).
% Sup_SUP_eq
tff(fact_955_empty__Union__conv,axiom,
! [A: $tType,A3: set(set(A))] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) )
<=> ! [X2: set(A)] :
( member(set(A),X2,A3)
=> ( X2 = bot_bot(set(A)) ) ) ) ).
% empty_Union_conv
tff(fact_956_Union__empty__conv,axiom,
! [A: $tType,A3: set(set(A))] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = bot_bot(set(A)) )
<=> ! [X2: set(A)] :
( member(set(A),X2,A3)
=> ( X2 = bot_bot(set(A)) ) ) ) ).
% Union_empty_conv
tff(fact_957_less__eq__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A),U: A] :
( ! [V3: A] :
( member(A,V3,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
=> ( ( A3 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).
% less_eq_Sup
tff(fact_958_cSup__least,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z2) ) ) ) ).
% cSup_least
tff(fact_959_cSup__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A4) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( member(A,X4,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Y2) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = A4 ) ) ) ) ) ).
% cSup_eq_non_empty
tff(fact_960_SUP__eq__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),F2: fun(A,B),X: B] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> ( aa(A,B,F2,I3) = X ) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I)) = X ) ) ) ) ).
% SUP_eq_const
tff(fact_961_Union__disjoint,axiom,
! [A: $tType,C6: set(set(A)),A3: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)),A3) = bot_bot(set(A)) )
<=> ! [X2: set(A)] :
( member(set(A),X2,C6)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),A3) = bot_bot(set(A)) ) ) ) ).
% Union_disjoint
tff(fact_962_Union__empty,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).
% Union_empty
tff(fact_963_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),C2: B,F2: fun(A,B)] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F2,I3)) )
=> ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I)) = C2 )
<=> ! [X2: A] :
( member(A,X2,I)
=> ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).
% SUP_eq_iff
tff(fact_964_cSUP__least,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),M2: B] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),M2) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),M2) ) ) ) ).
% cSUP_least
tff(fact_965_Sup__finite__insert,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ).
% Sup_finite_insert
tff(fact_966_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B5: set(A),A4: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A4) = bot_bot(A) )
<=> ! [X2: A] :
( member(A,X2,B5)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),A4) = bot_bot(A) ) ) ) ) ).
% Sup_inf_eq_bot_iff
tff(fact_967_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A3: fun(B,set(A)),I2: B,B5: set(A),J: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A3,I2,B5)),J)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),minus_minus(set(B),J,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I2),bot_bot(set(B))))))),
$ite(member(B,I2,J),B5,bot_bot(set(A)))) ).
% UNION_fun_upd
tff(fact_968_cSUP__union,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),B5))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),B5))) ) ) ) ) ) ) ).
% cSUP_union
tff(fact_969_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A3: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A3))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),A3) ) ).
% pair_in_swap_image
tff(fact_970_cSUP__insert,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),A4: A] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A4)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).
% cSUP_insert
tff(fact_971_cSup__inter__less__eq,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( condit941137186595557371_above(A,A3)
=> ( condit941137186595557371_above(A,B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).
% cSup_inter_less_eq
tff(fact_972_cSUP__subset__mono,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),G: fun(A,B),B5: set(A),F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),B5))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B5))) ) ) ) ) ) ).
% cSUP_subset_mono
tff(fact_973_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: set(B),B5: set(B)] :
( single_valued(A,B,converse(B,A,R3))
=> ( image(B,A,R3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R3,A3)),image(B,A,R3,B5)) ) ) ).
% Image_Int_eq
tff(fact_974_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P2)) = P2 ).
% swap_swap
tff(fact_975_image__update,axiom,
! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B),N2: B] :
( ~ member(A,X,A3)
=> ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F2,X,N2)),A3) = aa(set(A),set(B),image2(A,B,F2),A3) ) ) ).
% image_update
tff(fact_976_bdd__above__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit941137186595557371_above(A,bot_bot(set(A))) ) ).
% bdd_above_empty
tff(fact_977_bdd__above__insert,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A4: A,A3: set(A)] :
( condit941137186595557371_above(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))
<=> condit941137186595557371_above(A,A3) ) ) ).
% bdd_above_insert
tff(fact_978_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) ).
% swap_simp
tff(fact_979_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B)),X: A,Y: B,Z2: B] :
( single_valued(A,B,R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2),R2)
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
tff(fact_980_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( ! [X3: A,Y2: B,Z4: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z4),R2)
=> ( Y2 = Z4 ) ) )
=> single_valued(A,B,R2) ) ).
% single_valuedI
tff(fact_981_single__valued__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B))] :
( single_valued(A,B,R2)
<=> ! [X2: A,Y3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3),R2)
=> ! [Z3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z3),R2)
=> ( Y3 = Z3 ) ) ) ) ).
% single_valued_def
tff(fact_982_single__valued__subset,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
=> ( single_valued(A,B,S2)
=> single_valued(A,B,R2) ) ) ).
% single_valued_subset
tff(fact_983_single__valued__empty,axiom,
! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).
% single_valued_empty
tff(fact_984_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
( single_valued(A,B,R2)
=> ( single_valued(B,C,S2)
=> single_valued(A,C,relcomp(A,B,C,R2,S2)) ) ) ).
% single_valued_relcomp
tff(fact_985_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( single_valued(A,B,R3)
=> single_valued(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R3),S)) ) ).
% single_valued_inter1
tff(fact_986_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( single_valued(A,B,R3)
=> single_valued(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),S),R3)) ) ).
% single_valued_inter2
tff(fact_987_single__valued__Id,axiom,
! [A: $tType] : single_valued(A,A,id2(A)) ).
% single_valued_Id
tff(fact_988_single__valued__Id__on,axiom,
! [A: $tType,A3: set(A)] : single_valued(A,A,id_on(A,A3)) ).
% single_valued_Id_on
tff(fact_989_single__valued__confluent,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( single_valued(A,A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R2))
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y),transitive_rtrancl(A,R2)) ) ) ) ) ).
% single_valued_confluent
tff(fact_990_cSup__le__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A4: A] :
( ( S != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A4)
<=> ! [X2: A] :
( member(A,X2,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A4) ) ) ) ) ) ).
% cSup_le_iff
tff(fact_991_cSup__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B5: set(A),A3: set(A)] :
( ( B5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A3)
=> ( ! [B2: A] :
( member(A,B2,B5)
=> ? [X4: A] :
( member(A,X4,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X4) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B5)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).
% cSup_mono
tff(fact_992_single__valued__below__Id,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),id2(A))
=> single_valued(A,A,R3) ) ).
% single_valued_below_Id
tff(fact_993_bijective__alt,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
( bijective(A,B,R3)
<=> ( single_valued(A,B,R3)
& single_valued(B,A,converse(A,B,R3)) ) ) ).
% bijective_alt
tff(fact_994_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),U: B] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),U)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),U) ) ) ) ) ) ).
% cSUP_le_iff
tff(fact_995_cSUP__mono,axiom,
! [A: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),G: fun(C,B),B5: set(C),F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),B5))
=> ( ! [N3: A] :
( member(A,N3,A3)
=> ? [X4: C] :
( member(C,X4,B5)
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N3)),aa(C,B,G,X4)) ) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),B5))) ) ) ) ) ).
% cSUP_mono
tff(fact_996_cSup__subset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ).
% cSup_subset_mono
tff(fact_997_cSup__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( condit941137186595557371_above(A,X5)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5)) = $ite(X5 = bot_bot(set(A)),A4,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),X5))) ) ) ) ).
% cSup_insert_If
tff(fact_998_cSup__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,X5)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).
% cSup_insert
tff(fact_999_cSup__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,B5)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ) ) ).
% cSup_union_distrib
tff(fact_1000_fun__upd__image,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X: B,Y: A,A3: set(B)] :
aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,X,Y)),A3) = $ite(member(B,X,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(B),set(A),image2(B,A,F2),minus_minus(set(B),A3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F2),A3)) ).
% fun_upd_image
tff(fact_1001_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,sup_sup(A),X,A3) ) ) ) ).
% Sup_fin.eq_fold
tff(fact_1002_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,inf_inf(A),X,A3) ) ) ) ).
% Inf_fin.eq_fold
tff(fact_1003_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A5),R2)
=> ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Ub,A5))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B2)),aa(A,set(B),Ub,A5))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,B2)),aa(A,set(B),Ub,A5))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F2,A5)),aa(A,set(B),F2,B2)) ) )
=> wf(A,R2) ) ).
% wf_bounded_set
tff(fact_1004_psubset__insert__iff,axiom,
! [A: $tType,A3: set(A),X: A,B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B5))
<=> $ite(
member(A,X,B5),
aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5),
$ite(member(A,X,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)) ) ) ).
% psubset_insert_iff
tff(fact_1005_finite__induct__select,axiom,
! [A: $tType,S: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [T4: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T4),S)
=> ( aa(set(A),$o,P,T4)
=> ? [X4: A] :
( member(A,X4,minus_minus(set(A),S,T4))
& aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),T4)) ) ) )
=> aa(set(A),$o,P,S) ) ) ) ).
% finite_induct_select
tff(fact_1006_Max_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B5)) ) ) ) ) ).
% Max.subset_imp
tff(fact_1007_Max__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: set(A),N: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M2),N)
=> ( ( M2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,M2)),lattic643756798349783984er_Max(A,N)) ) ) ) ) ).
% Max_mono
tff(fact_1008_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(A)] : aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).
% image_vimage_eq
tff(fact_1009_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).
% top_apply
tff(fact_1010_UNIV__I,axiom,
! [A: $tType,X: A] : member(A,X,top_top(set(A))) ).
% UNIV_I
tff(fact_1011_converse__UNIV,axiom,
! [B: $tType,A: $tType] : converse(B,A,top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).
% converse_UNIV
tff(fact_1012_Int__UNIV,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = top_top(set(A)) )
<=> ( ( A3 = top_top(set(A)) )
& ( B5 = top_top(set(A)) ) ) ) ).
% Int_UNIV
tff(fact_1013_psubsetI,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != B5 )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5) ) ) ).
% psubsetI
tff(fact_1014_fold__empty,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z2: A] : finite_fold(B,A,F2,Z2,bot_bot(set(B))) = Z2 ).
% fold_empty
tff(fact_1015_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,top_top(set(B))) = top_top(set(A)) ).
% vimage_UNIV
tff(fact_1016_Pow__UNIV,axiom,
! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).
% Pow_UNIV
tff(fact_1017_surj__swap,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).
% surj_swap
tff(fact_1018_Diff__UNIV,axiom,
! [A: $tType,A3: set(A)] : minus_minus(set(A),A3,top_top(set(A))) = bot_bot(set(A)) ).
% Diff_UNIV
tff(fact_1019_Max__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% Max_singleton
tff(fact_1020_range__snd,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(A)) ).
% range_snd
tff(fact_1021_Domain__Id,axiom,
! [A: $tType] : aa(set(product_prod(A,A)),set(A),domain(A,A),id2(A)) = top_top(set(A)) ).
% Domain_Id
tff(fact_1022_Range__Id,axiom,
! [A: $tType] : aa(set(product_prod(A,A)),set(A),range2(A,A),id2(A)) = top_top(set(A)) ).
% Range_Id
tff(fact_1023_Max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),X)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).
% Max.bounded_iff
tff(fact_1024_Max__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,A3)),X)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),X) ) ) ) ) ) ).
% Max_less_iff
tff(fact_1025_lt__ex,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] :
? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X) ) ).
% lt_ex
tff(fact_1026_gt__ex,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] :
? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_1) ) ).
% gt_ex
tff(fact_1027_dense,axiom,
! [A: $tType] :
( dense_order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ? [Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),Y) ) ) ) ).
% dense
tff(fact_1028_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( X != Y ) ) ) ).
% less_imp_neq
tff(fact_1029_order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ).
% order.asym
tff(fact_1030_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),C2) ) ) ) ).
% ord_eq_less_trans
tff(fact_1031_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( ( B3 = C2 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),C2) ) ) ) ).
% ord_less_eq_trans
tff(fact_1032_less__induct,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A4: A] :
( ! [X3: A] :
( ! [Y5: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X3)
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A4) ) ) ).
% less_induct
tff(fact_1033_antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv3
tff(fact_1034_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( ( X != Y )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_cases
tff(fact_1035_dual__order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% dual_order.asym
tff(fact_1036_dual__order_Oirrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),A4) ) ).
% dual_order.irrefl
tff(fact_1037_exists__least__iff,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o)] :
( ? [X_12: A] : aa(A,$o,P,X_12)
<=> ? [N4: A] :
( aa(A,$o,P,N4)
& ! [M4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M4),N4)
=> ~ aa(A,$o,P,M4) ) ) ) ) ).
% exists_least_iff
tff(fact_1038_linorder__less__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,$o)),A4: A,B3: A] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B2)
=> aa(A,$o,aa(A,fun(A,$o),P,A5),B2) )
=> ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),P,A5),A5)
=> ( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,B2),A5)
=> aa(A,$o,aa(A,fun(A,$o),P,A5),B2) )
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) ) ) ) ) ).
% linorder_less_wlog
tff(fact_1039_order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),C2) ) ) ) ).
% order.strict_trans
tff(fact_1040_top_Oextremum__strict,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A4) ) ).
% top.extremum_strict
tff(fact_1041_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A4: A] :
( ( A4 != top_top(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),top_top(A)) ) ) ).
% top.not_eq_extremum
tff(fact_1042_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_1043_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A4) ) ) ) ).
% dual_order.strict_trans
tff(fact_1044_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
tff(fact_1045_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_1046_psubsetD,axiom,
! [A: $tType,A3: set(A),B5: set(A),C2: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> ( member(A,C2,A3)
=> member(A,C2,B5) ) ) ).
% psubsetD
tff(fact_1047_UNIV__eq__I,axiom,
! [A: $tType,A3: set(A)] :
( ! [X3: A] : member(A,X3,A3)
=> ( top_top(set(A)) = A3 ) ) ).
% UNIV_eq_I
tff(fact_1048_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : member(A,X3,top_top(set(A))) ).
% UNIV_witness
tff(fact_1049_psubset__trans,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C6) ) ) ).
% psubset_trans
tff(fact_1050_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_neqE
tff(fact_1051_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_asym
tff(fact_1052_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_neq_iff
tff(fact_1053_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ).
% order_less_asym'
tff(fact_1054_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_less_trans
tff(fact_1055_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( ( A4 = aa(B,A,F2,B3) )
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% ord_eq_less_subst
tff(fact_1056_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( ( aa(A,B,F2,B3) = C2 )
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% ord_less_eq_subst
tff(fact_1057_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X) ) ).
% order_less_irrefl
tff(fact_1058_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,B3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% order_less_subst1
tff(fact_1059_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% order_less_subst2
tff(fact_1060_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_not_sym
tff(fact_1061_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,P: $o] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
=> (P) ) ) ) ).
% order_less_imp_triv
tff(fact_1062_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y )
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_less_linear
tff(fact_1063_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
tff(fact_1064_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
tff(fact_1065_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_imp_not_less
tff(fact_1066_Max__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),lattic643756798349783984er_Max(A,A3))
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X2) ) ) ) ) ) ).
% Max_gr_iff
tff(fact_1067_top__greatest,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),top_top(A)) ) ).
% top_greatest
tff(fact_1068_top_Oextremum__unique,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A4)
<=> ( A4 = top_top(A) ) ) ) ).
% top.extremum_unique
tff(fact_1069_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A4)
=> ( A4 = top_top(A) ) ) ) ).
% top.extremum_uniqueI
tff(fact_1070_leD,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).
% leD
tff(fact_1071_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% leI
tff(fact_1072_nless__le,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
| ( A4 = B3 ) ) ) ) ).
% nless_le
tff(fact_1073_antisym__conv1,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv1
tff(fact_1074_antisym__conv2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv2
tff(fact_1075_dense__ge,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y: A] :
( ! [X3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).
% dense_ge
tff(fact_1076_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,Z2: A] :
( ! [X3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).
% dense_le
tff(fact_1077_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).
% less_le_not_le
tff(fact_1078_not__le__imp__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).
% not_le_imp_less
tff(fact_1079_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
tff(fact_1080_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
tff(fact_1081_order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),C2) ) ) ) ).
% order.strict_trans1
tff(fact_1082_order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),C2) ) ) ) ).
% order.strict_trans2
tff(fact_1083_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ).
% order.strict_iff_not
tff(fact_1084_dense__ge__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
=> ( ! [W2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),W2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W2) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% dense_ge_bounded
tff(fact_1085_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( ! [W2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),W2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z2) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% dense_le_bounded
tff(fact_1086_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
tff(fact_1087_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
tff(fact_1088_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A4) ) ) ) ).
% dual_order.strict_trans1
tff(fact_1089_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A4) ) ) ) ).
% dual_order.strict_trans2
tff(fact_1090_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% dual_order.strict_iff_not
tff(fact_1091_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% order.strict_implies_order
tff(fact_1092_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ).
% dual_order.strict_implies_order
tff(fact_1093_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y ) ) ) ) ).
% order_le_less
tff(fact_1094_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& ( X != Y ) ) ) ) ).
% order_less_le
tff(fact_1095_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_not_le
tff(fact_1096_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_not_less
tff(fact_1097_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% order_less_imp_le
tff(fact_1098_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( ( A4 != B3 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% order_le_neq_trans
tff(fact_1099_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( ( A4 != B3 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% order_neq_le_trans
tff(fact_1100_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_le_less_trans
tff(fact_1101_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_less_le_trans
tff(fact_1102_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(B,A,F2,B3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% order_le_less_subst1
tff(fact_1103_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% order_le_less_subst2
tff(fact_1104_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,F2: fun(B,A),B3: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,B3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(B,A,F2,C2)) ) ) ) ) ).
% order_less_le_subst1
tff(fact_1105_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A4: A,B3: A,F2: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A4)),C2) ) ) ) ) ).
% order_less_le_subst2
tff(fact_1106_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_le_less_linear
tff(fact_1107_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_1108_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).
% less_numeral_extra(4)
tff(fact_1109_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,D3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,A4,C2)),minus_minus(A,B3,D3)) ) ) ) ).
% diff_strict_mono
tff(fact_1110_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( minus_minus(A,A4,B3) = minus_minus(A,C2,D3) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).
% diff_eq_diff_less
tff(fact_1111_diff__strict__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,C2,A4)),minus_minus(A,C2,B3)) ) ) ).
% diff_strict_left_mono
tff(fact_1112_diff__strict__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,A4,C2)),minus_minus(A,B3,C2)) ) ) ).
% diff_strict_right_mono
tff(fact_1113_bot_Oextremum__strict,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),bot_bot(A)) ) ).
% bot.extremum_strict
tff(fact_1114_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A4: A] :
( ( A4 != bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A4) ) ) ).
% bot.not_eq_extremum
tff(fact_1115_not__psubset__empty,axiom,
! [A: $tType,A3: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),bot_bot(set(A))) ).
% not_psubset_empty
tff(fact_1116_psubsetE,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).
% psubsetE
tff(fact_1117_psubset__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
& ( A3 != B5 ) ) ) ).
% psubset_eq
tff(fact_1118_psubset__imp__subset,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% psubset_imp_subset
tff(fact_1119_psubset__subset__trans,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C6) ) ) ).
% psubset_subset_trans
tff(fact_1120_subset__not__subset__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
& ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).
% subset_not_subset_eq
tff(fact_1121_subset__psubset__trans,axiom,
! [A: $tType,A3: set(A),B5: set(A),C6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C6) ) ) ).
% subset_psubset_trans
tff(fact_1122_subset__iff__psubset__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
| ( A3 = B5 ) ) ) ).
% subset_iff_psubset_eq
tff(fact_1123_rangeI,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X: B] : member(A,aa(B,A,F2,X),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).
% rangeI
tff(fact_1124_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F2: fun(B,A),X: B] :
( ( B3 = aa(B,A,F2,X) )
=> member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ) ).
% range_eqI
tff(fact_1125_empty__not__UNIV,axiom,
! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).
% empty_not_UNIV
tff(fact_1126_subset__UNIV,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),top_top(set(A))) ).
% subset_UNIV
tff(fact_1127_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
=> ? [B2: A] : member(A,B2,minus_minus(set(A),B5,A3)) ) ).
% psubset_imp_ex_mem
tff(fact_1128_insert__UNIV,axiom,
! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),top_top(set(A))) = top_top(set(A)) ).
% insert_UNIV
tff(fact_1129_Int__UNIV__left,axiom,
! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B5) = B5 ).
% Int_UNIV_left
tff(fact_1130_Int__UNIV__right,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),top_top(set(A))) = A3 ).
% Int_UNIV_right
tff(fact_1131_Un__UNIV__left,axiom,
! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B5) = top_top(set(A)) ).
% Un_UNIV_left
tff(fact_1132_Un__UNIV__right,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),top_top(set(A))) = top_top(set(A)) ).
% Un_UNIV_right
tff(fact_1133_type__copy__ex__RepI,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),F4: fun(B,$o)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ? [X_12: B] : aa(B,$o,F4,X_12)
<=> ? [B7: A] : aa(B,$o,F4,aa(A,B,Rep,B7)) ) ) ).
% type_copy_ex_RepI
tff(fact_1134_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: A] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ~ ! [X3: B] : S2 != aa(B,A,Abs,X3) ) ).
% type_copy_obj_one_point_absE
tff(fact_1135_less__1__mult,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: A,N2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),N2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N2)) ) ) ) ).
% less_1_mult
tff(fact_1136_infinite__growing,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X5: set(A)] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa3) ) )
=> ~ aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).
% infinite_growing
tff(fact_1137_ex__min__if__finite,axiom,
! [A: $tType] :
( order(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,S)
& ~ ? [Xa3: A] :
( member(A,Xa3,S)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X3) ) ) ) ) ) ).
% ex_min_if_finite
tff(fact_1138_less__cSupD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X5))
=> ? [X3: A] :
( member(A,X3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3) ) ) ) ) ).
% less_cSupD
tff(fact_1139_less__cSupE,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [Y: A,X5: set(A)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
=> ( ( X5 != bot_bot(set(A)) )
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X3) ) ) ) ) ).
% less_cSupE
tff(fact_1140_Max__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> member(A,lattic643756798349783984er_Max(A,A3),A3) ) ) ) ).
% Max_in
tff(fact_1141_range__subsetD,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),B5: set(A),I2: B] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))),B5)
=> member(A,aa(B,A,F2,I2),B5) ) ).
% range_subsetD
tff(fact_1142_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
=> ( ( vimage(B,A,F2,A3) = bot_bot(set(B)) )
<=> ( A3 = bot_bot(set(A)) ) ) ) ).
% surj_vimage_empty
tff(fact_1143_acyclicI__order,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [R2: set(product_prod(A,A)),F2: fun(A,B)] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2),R2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,A5)) )
=> transitive_acyclic(A,R2) ) ) ).
% acyclicI_order
tff(fact_1144_type__definition_OAbs__image,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B)] :
( type_definition(A,B,Rep,Abs,A3)
=> ( aa(set(B),set(A),image2(B,A,Abs),A3) = top_top(set(A)) ) ) ).
% type_definition.Abs_image
tff(fact_1145_type__definition_ORep__range,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B)] :
( type_definition(A,B,Rep,Abs,A3)
=> ( aa(set(A),set(B),image2(A,B,Rep),top_top(set(A))) = A3 ) ) ).
% type_definition.Rep_range
tff(fact_1146_refl__Id,axiom,
! [A: $tType] : refl_on(A,top_top(set(A)),id2(A)) ).
% refl_Id
tff(fact_1147_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [A4: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),X) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),X) = top_top(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),Y) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),Y) = top_top(A) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_1148_union__fold__insert,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = finite_fold(A,set(A),insert2(A),B5,A3) ) ) ).
% union_fold_insert
tff(fact_1149_finite__linorder__min__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [B2: A,A10: set(A)] :
( aa(set(A),$o,finite_finite2(A),A10)
=> ( ! [X4: A] :
( member(A,X4,A10)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),X4) )
=> ( aa(set(A),$o,P,A10)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A10)) ) ) )
=> aa(set(A),$o,P,A3) ) ) ) ) ).
% finite_linorder_min_induct
tff(fact_1150_finite__linorder__max__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [B2: A,A10: set(A)] :
( aa(set(A),$o,finite_finite2(A),A10)
=> ( ! [X4: A] :
( member(A,X4,A10)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B2) )
=> ( aa(set(A),$o,P,A10)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A10)) ) ) )
=> aa(set(A),$o,P,A3) ) ) ) ) ).
% finite_linorder_max_induct
tff(fact_1151_finite__Sup__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A4)
<=> ! [X2: A] :
( member(A,X2,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A4) ) ) ) ) ) ).
% finite_Sup_less_iff
tff(fact_1152_Max__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),M: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ( lattic643756798349783984er_Max(A,A3) = M )
<=> ( member(A,M,A3)
& ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M) ) ) ) ) ) ) ).
% Max_eq_iff
tff(fact_1153_Max__ge__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798349783984er_Max(A,A3))
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).
% Max_ge_iff
tff(fact_1154_eq__Max__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),M: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ( M = lattic643756798349783984er_Max(A,A3) )
<=> ( member(A,M,A3)
& ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M) ) ) ) ) ) ) ).
% eq_Max_iff
tff(fact_1155_Max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),X)
=> ! [A12: A] :
( member(A,A12,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A12),X) ) ) ) ) ) ).
% Max.boundedE
tff(fact_1156_Max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),X) ) ) ) ) ).
% Max.boundedI
tff(fact_1157_Max__insert2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [B2: A] :
( member(A,B2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A4) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = A4 ) ) ) ) ).
% Max_insert2
tff(fact_1158_remove__subset,axiom,
! [A: $tType,X: A,S: set(A)] :
( member(A,X,S)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),S) ) ).
% remove_subset
tff(fact_1159_cSup__eq__Max,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = lattic643756798349783984er_Max(A,X5) ) ) ) ) ).
% cSup_eq_Max
tff(fact_1160_Max__Sup,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).
% Max_Sup
tff(fact_1161_disjoint__alt__simp3,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),minus_minus(set(A),A3,B5)),A3)
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp3
tff(fact_1162_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A4: A,X: B] :
( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) )
=> ( aa(B,A,F2,X) = A4 ) ) ).
% range_eq_singletonD
tff(fact_1163_less__cSup__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,X5)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
<=> ? [X2: A] :
( member(A,X2,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X2) ) ) ) ) ) ).
% less_cSup_iff
tff(fact_1164_underS__Field3,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A4)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).
% underS_Field3
tff(fact_1165_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ~ ? [X4: A] :
( member(A,X4,S)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))) ) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_1166_Sup__fold__sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),A,complete_Sup_Sup(A),A3) = finite_fold(A,A,sup_sup(A),bot_bot(A),A3) ) ) ) ).
% Sup_fold_sup
tff(fact_1167_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A3: set(A),B5: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z2,A3),B5) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
tff(fact_1168_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A3: set(A),X: A,Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( finite_fold(A,B,F2,Z2,A3) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
tff(fact_1169_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
tff(fact_1170_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite673082921795544331dem_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z2,A3)) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
tff(fact_1171_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite673082921795544331dem_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z2),A3) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
tff(fact_1172_SUP__fold__sup,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [A3: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F2),bot_bot(B),A3) ) ) ) ).
% SUP_fold_sup
tff(fact_1173_finite__subset__Union__chain,axiom,
! [A: $tType,A3: set(A),B9: set(set(A)),A14: set(set(A))] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B9))
=> ( ( B9 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B9)
=> ~ ! [B6: set(A)] :
( member(set(A),B6,B9)
=> ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B6) ) ) ) ) ) ).
% finite_subset_Union_chain
tff(fact_1174_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z2,A3)) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
tff(fact_1175_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( finite_fold(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z2),A3) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
tff(fact_1176_merge__true__star,axiom,
aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),top_top(assn)) = top_top(assn) ).
% merge_true_star
tff(fact_1177_assn__basic__inequalities_I5_J,axiom,
top_top(assn) != bot_bot(assn) ).
% assn_basic_inequalities(5)
tff(fact_1178_assn__basic__inequalities_I1_J,axiom,
top_top(assn) != one_one(assn) ).
% assn_basic_inequalities(1)
tff(fact_1179_mod__true,axiom,
! [Ha: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,top_top(assn)),Ha)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Ha) ) ).
% mod_true
tff(fact_1180_fun__comp__eq__conv,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(C,B),G: fun(A,C),Fg: fun(A,B)] :
( ( aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G) = Fg )
<=> ! [X2: A] : aa(C,B,F2,aa(A,C,G,X2)) = aa(A,B,Fg,X2) ) ).
% fun_comp_eq_conv
tff(fact_1181_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Y: fun(A,B),F2: fun(B,C)] :
( ( X = Y )
=> ( aa(fun(A,B),fun(A,C),comp(B,C,A,F2),X) = aa(fun(A,B),fun(A,C),comp(B,C,A,F2),Y) ) ) ).
% comp_cong_right
tff(fact_1182_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: fun(A,B),Y: fun(A,B),F2: fun(C,A)] :
( ( X = Y )
=> ( aa(fun(C,A),fun(C,B),comp(A,B,C,X),F2) = aa(fun(C,A),fun(C,B),comp(A,B,C,Y),F2) ) ) ).
% comp_cong_left
tff(fact_1183_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),X: C,Ha: fun(D,A),K: fun(C,D)] :
( ( aa(B,A,F2,aa(C,B,G,X)) = aa(D,A,Ha,aa(C,D,K,X)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G),X) = aa(C,A,aa(fun(C,D),fun(C,A),comp(D,A,C,Ha),K),X) ) ) ).
% comp_apply_eq
tff(fact_1184_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F3: $tType,Rep: fun(A,B),Abs: fun(B,A),S: fun(B,set(D)),M2: fun(C,B),F2: fun(E,D),S4: fun(C,set(E)),G: fun(F3,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( aa(fun(C,B),fun(C,set(D)),comp(B,set(D),C,S),M2) = aa(fun(C,set(E)),fun(C,set(D)),comp(set(E),set(D),C,image2(E,D,F2)),S4) )
=> ( aa(fun(F3,A),fun(F3,set(D)),comp(A,set(D),F3,aa(fun(A,B),fun(A,set(D)),comp(B,set(D),A,S),Rep)),aa(fun(F3,C),fun(F3,A),comp(C,A,F3,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M2)),G)) = aa(fun(F3,set(E)),fun(F3,set(D)),comp(set(E),set(D),F3,image2(E,D,F2)),aa(fun(F3,C),fun(F3,set(E)),comp(C,set(E),F3,S4),G)) ) ) ) ).
% type_copy_set_map0
tff(fact_1185_top__empty__eq,axiom,
! [A: $tType,X4: A] :
( aa(A,$o,top_top(fun(A,$o)),X4)
<=> member(A,X4,top_top(set(A))) ) ).
% top_empty_eq
tff(fact_1186_top__set__def,axiom,
! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),top_top(fun(A,$o))) ).
% top_set_def
tff(fact_1187_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F2: fun(A,B),G: fun(A,B)] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
<=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
& ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).
% less_fun_def
tff(fact_1188_pred__on_Ochain__empty,axiom,
! [A: $tType,A3: set(A),P: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,A3,P),bot_bot(set(A))) ).
% pred_on.chain_empty
tff(fact_1189_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(B,A),X5: set(B),G1: fun(A,C),G22: fun(A,C)] :
( ( aa(set(B),set(A),image2(B,A,F2),X5) = top_top(set(A)) )
=> ( ! [X3: B] :
( member(B,X3,X5)
=> ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F2),X3) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F2),X3) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
tff(fact_1190_type__copy__map__comp0,axiom,
! [F3: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(C,D),M1: fun(B,D),M22: fun(C,B),F2: fun(D,F3),G: fun(E,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M2 = aa(fun(C,B),fun(C,D),comp(B,D,C,M1),M22) )
=> ( aa(fun(E,C),fun(E,F3),comp(C,F3,E,aa(fun(C,D),fun(C,F3),comp(D,F3,C,F2),M2)),G) = aa(fun(E,A),fun(E,F3),comp(A,F3,E,aa(fun(A,B),fun(A,F3),comp(B,F3,A,aa(fun(B,D),fun(B,F3),comp(D,F3,B,F2),M1)),Rep)),aa(fun(E,C),fun(E,A),comp(C,A,E,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M22)),G)) ) ) ) ).
% type_copy_map_comp0
tff(fact_1191_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,Rep: fun(A,B),Abs: fun(B,A),Rep2: fun(C,D),Abs2: fun(D,C),Rep3: fun(E,F3),Abs3: fun(F3,E),M2: fun(F3,D),M1: fun(B,D),M22: fun(F3,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( type_definition(C,D,Rep2,Abs2,top_top(set(D)))
=> ( type_definition(E,F3,Rep3,Abs3,top_top(set(F3)))
=> ( ( aa(fun(E,F3),fun(E,C),comp(F3,C,E,aa(fun(F3,D),fun(F3,C),comp(D,C,F3,Abs2),M2)),Rep3) = aa(fun(E,A),fun(E,C),comp(A,C,E,aa(fun(A,B),fun(A,C),comp(B,C,A,aa(fun(B,D),fun(B,C),comp(D,C,B,Abs2),M1)),Rep)),aa(fun(E,F3),fun(E,A),comp(F3,A,E,aa(fun(F3,B),fun(F3,A),comp(B,A,F3,Abs),M22)),Rep3)) )
=> ( aa(fun(F3,B),fun(F3,D),comp(B,D,F3,M1),M22) = M2 ) ) ) ) ) ).
% type_copy_map_comp0_undo
tff(fact_1192_wf__bounded__measure,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A5),R2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B2)),aa(A,nat,Ub,A5))
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,B2)),aa(A,nat,Ub,A5))
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,A5)),aa(A,nat,F2,B2)) ) )
=> wf(A,R2) ) ).
% wf_bounded_measure
tff(fact_1193_subset_Ochain__empty,axiom,
! [A: $tType,A3: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),A3,ord_less(set(A))),bot_bot(set(set(A)))) ).
% subset.chain_empty
tff(fact_1194_less__assn__def,axiom,
! [A4: assn,B3: assn] :
( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A4),B3)
<=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A4),B3)
& ( A4 != B3 ) ) ) ).
% less_assn_def
tff(fact_1195_top__assn__def,axiom,
top_top(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,in_range) ).
% top_assn_def
tff(fact_1196_Union__in__chain,axiom,
! [A: $tType,B9: set(set(A)),A14: set(set(A))] :
( aa(set(set(A)),$o,finite_finite2(set(A)),B9)
=> ( ( B9 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B9)
=> member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B9),B9) ) ) ) ).
% Union_in_chain
tff(fact_1197_pred__on_Ochain__extend,axiom,
! [A: $tType,A3: set(A),P: fun(A,fun(A,$o)),C6: set(A),Z2: A] :
( aa(set(A),$o,pred_chain(A,A3,P),C6)
=> ( member(A,Z2,A3)
=> ( ! [X3: A] :
( member(A,X3,C6)
=> aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X3),Z2) )
=> aa(set(A),$o,pred_chain(A,A3,P),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z2),bot_bot(set(A)))),C6)) ) ) ) ).
% pred_on.chain_extend
tff(fact_1198_subset__Zorn__nonempty,axiom,
! [A: $tType,A14: set(set(A))] :
( ( A14 != bot_bot(set(set(A))) )
=> ( ! [C8: set(set(A))] :
( ( C8 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),C8)
=> member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8),A14) ) )
=> ? [X3: set(A)] :
( member(set(A),X3,A14)
& ! [Xa3: set(A)] :
( member(set(A),Xa3,A14)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa3)
=> ( Xa3 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
tff(fact_1199_subset_Ochain__extend,axiom,
! [A: $tType,A3: set(set(A)),C6: set(set(A)),Z2: set(A)] :
( aa(set(set(A)),$o,pred_chain(set(A),A3,ord_less(set(A))),C6)
=> ( member(set(A),Z2,A3)
=> ( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Z2) )
=> aa(set(set(A)),$o,pred_chain(set(A),A3,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z2),bot_bot(set(set(A))))),C6)) ) ) ) ).
% subset.chain_extend
tff(fact_1200_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z2,A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z2),A3) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
tff(fact_1201_in__measure,axiom,
! [A: $tType,X: A,Y: A,F2: fun(A,nat)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measure(A,F2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)) ) ).
% in_measure
tff(fact_1202_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z2: B,Y: B,A4: A] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S)
=> ( finite_fold_graph(A,B,F2,Z2,A3,Y)
=> ( member(A,A4,A3)
=> ? [Y6: B] :
( ( Y = aa(B,B,aa(A,fun(B,B),F2,A4),Y6) )
& finite_fold_graph(A,B,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))),Y6) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_1203_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_top(A)
=> ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).
% top.ordering_top_axioms
tff(fact_1204_Max_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).
% Max.insert_remove
tff(fact_1205_Max_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic643756798349783984er_Max(A,A3) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).
% Max.remove
tff(fact_1206_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B5: set(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))
<=> ( inj_on(A,B,F2,A3)
& inj_on(A,B,F2,B5)
& ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),minus_minus(set(A),A3,B5))),aa(set(A),set(B),image2(A,B,F2),minus_minus(set(A),B5,A3))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_1207_Max__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,A3)) ) ) ) ) ).
% Max_insert
tff(fact_1208_map__prod__surj,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F2: fun(B,A),G: fun(D,C)] :
( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(D),set(C),image2(D,C,G),top_top(set(D))) = top_top(set(C)) )
=> ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F2,G)),top_top(set(product_prod(B,D)))) = top_top(set(product_prod(A,C))) ) ) ) ).
% map_prod_surj
tff(fact_1209_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: A,A3: set(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))
<=> ( inj_on(A,B,F2,A3)
& ~ member(B,aa(A,B,F2,A4),aa(set(A),set(B),image2(A,B,F2),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ).
% inj_on_insert
tff(fact_1210_top1I,axiom,
! [A: $tType,X: A] : aa(A,$o,top_top(fun(A,$o)),X) ).
% top1I
tff(fact_1211_inj__on__empty,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).
% inj_on_empty
tff(fact_1212_max__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).
% max_bot
tff(fact_1213_max__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).
% max_bot2
tff(fact_1214_max__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).
% max_top
tff(fact_1215_max__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).
% max_top2
tff(fact_1216_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A4: C,B3: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A4),B3)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,A4)),aa(D,B,G,B3)) ).
% map_prod_simp
tff(fact_1217_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: fun(A,D),G: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),comp(product_prod(D,C),C,product_prod(A,B),product_snd(D,C)),product_map_prod(A,D,B,C,F2,G)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),G),product_snd(A,B)) ).
% snd_comp_map_prod
tff(fact_1218_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(C,B),G: fun(D,A),X: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F2,G),X)) = aa(D,A,G,aa(product_prod(C,D),D,product_snd(C,D),X)) ).
% snd_map_prod
tff(fact_1219_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A4: A,B3: B,R3: set(product_prod(A,B)),F2: fun(A,C),G: fun(B,D)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),R3)
=> member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F2,A4)),aa(B,D,G,B3)),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F2,G)),R3)) ) ).
% map_prod_imageI
tff(fact_1220_exists__leI,axiom,
! [N2: nat,P: fun(nat,$o)] :
( ( ! [N5: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),N2)
=> ~ aa(nat,$o,P,N5) )
=> aa(nat,$o,P,N2) )
=> ? [N6: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
& aa(nat,$o,P,N6) ) ) ).
% exists_leI
tff(fact_1221_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A4: A] :
( ordering_top(A,Less_eq,Less,Top)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),Top) ) ).
% ordering_top.extremum
tff(fact_1222_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A4: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,Top),A4) ) ).
% ordering_top.extremum_strict
tff(fact_1223_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A4: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A4)
<=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
tff(fact_1224_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A4: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( ( A4 != Top )
<=> aa(A,$o,aa(A,fun(A,$o),Less,A4),Top) ) ) ).
% ordering_top.not_eq_extremum
tff(fact_1225_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A4: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A4)
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
tff(fact_1226_max__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A] :
aa(A,A,aa(A,fun(A,A),ord_max(A),A4),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3),B3,A4) ) ).
% max_def
tff(fact_1227_max__absorb1,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).
% max_absorb1
tff(fact_1228_max__absorb2,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).
% max_absorb2
tff(fact_1229_max__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),minus_minus(A,X,Z2)),minus_minus(A,Y,Z2)) ) ).
% max_diff_distrib_left
tff(fact_1230_map__prod_Ocomp,axiom,
! [A: $tType,E: $tType,C: $tType,D: $tType,F3: $tType,B: $tType,F2: fun(E,C),G: fun(F3,D),Ha: fun(A,E),I2: fun(B,F3)] : aa(fun(product_prod(A,B),product_prod(E,F3)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F3),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F3,D,F2,G)),product_map_prod(A,E,B,F3,Ha,I2)) = product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F2),Ha),aa(fun(B,F3),fun(B,D),comp(F3,D,B,G),I2)) ).
% map_prod.comp
tff(fact_1231_map__prod_Ocompositionality,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F3: $tType,E: $tType,F2: fun(C,A),G: fun(D,B),Ha: fun(E,C),I2: fun(F3,D),Prod: product_prod(E,F3)] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),aa(product_prod(E,F3),product_prod(C,D),product_map_prod(E,C,F3,D,Ha,I2),Prod)) = aa(product_prod(E,F3),product_prod(A,B),product_map_prod(E,A,F3,B,aa(fun(E,C),fun(E,A),comp(C,A,E,F2),Ha),aa(fun(F3,D),fun(F3,B),comp(D,B,F3,G),I2)),Prod) ).
% map_prod.compositionality
tff(fact_1232_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F3,D),G22: fun(B,F3)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F1),F22),aa(fun(B,F3),fun(B,D),comp(F3,D,B,G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F3)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F3),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F3,D,F1,G1)),product_map_prod(A,E,B,F3,F22,G22)) ).
% map_prod_compose
tff(fact_1233_inj__swap,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A3) ).
% inj_swap
tff(fact_1234_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B] : finite_fold_graph(A,B,F2,Z2,bot_bot(set(A)),Z2) ).
% fold_graph.emptyI
tff(fact_1235_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,X: B] :
( finite_fold_graph(A,B,F2,Z2,bot_bot(set(A)),X)
=> ( X = Z2 ) ) ).
% empty_fold_graphE
tff(fact_1236_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X: A,A3: set(A),F2: fun(A,fun(B,B)),Z2: B,Y: B] :
( ~ member(A,X,A3)
=> ( finite_fold_graph(A,B,F2,Z2,A3,Y)
=> finite_fold_graph(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),aa(B,B,aa(A,fun(B,B),F2,X),Y)) ) ) ).
% fold_graph.insertI
tff(fact_1237_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),X: B,B5: set(B)] :
( inj_on(A,B,F2,A3)
=> ( ~ member(B,X,B5)
=> ( ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),B5) = aa(set(A),set(B),image2(A,B,F2),A3) )
=> ~ ! [X8: A,A8: set(A)] :
( ~ member(A,X8,A8)
=> ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X8),A8) )
=> ( ( X = aa(A,B,F2,X8) )
=> ( B5 != aa(set(A),set(B),image2(A,B,F2),A8) ) ) ) ) ) ) ) ).
% inj_img_insertE
tff(fact_1238_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),X: C,S: fun(B,set(C)),Y: B] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( member(C,X,aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,S),Rep),aa(B,A,Abs,Y)))
=> member(C,X,aa(B,set(C),S,Y)) ) ) ).
% type_copy_wit
tff(fact_1239_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F2: fun(C,A),G: fun(D,B),R3: set(product_prod(C,D))] :
( member(product_prod(A,B),C2,aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G)),R3))
=> ~ ! [X3: C,Y2: D] :
( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X3)),aa(D,B,G,Y2)) )
=> ~ member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y2),R3) ) ) ).
% prod_fun_imageE
tff(fact_1240_fold__graph_Osimps,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F2,Z2,A1,A22)
<=> ( ( ( A1 = bot_bot(set(A)) )
& ( A22 = Z2 ) )
| ? [X2: A,A11: set(A),Y3: B] :
( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),A11) )
& ( A22 = aa(B,B,aa(A,fun(B,B),F2,X2),Y3) )
& ~ member(A,X2,A11)
& finite_fold_graph(A,B,F2,Z2,A11,Y3) ) ) ) ).
% fold_graph.simps
tff(fact_1241_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F2,Z2,A1,A22)
=> ( ( ( A1 = bot_bot(set(A)) )
=> ( A22 != Z2 ) )
=> ~ ! [X3: A,A10: set(A)] :
( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),A10) )
=> ! [Y2: B] :
( ( A22 = aa(B,B,aa(A,fun(B,B),F2,X3),Y2) )
=> ( ~ member(A,X3,A10)
=> ~ finite_fold_graph(A,B,F2,Z2,A10,Y2) ) ) ) ) ) ).
% fold_graph.cases
tff(fact_1242_inj__on__iff__surj,axiom,
! [B: $tType,A: $tType,A3: set(A),A9: set(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( ? [F6: fun(A,B)] :
( inj_on(A,B,F6,A3)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A3)),A9) )
<=> ? [G4: fun(B,A)] : aa(set(B),set(A),image2(B,A,G4),A9) = A3 ) ) ).
% inj_on_iff_surj
tff(fact_1243_Sup__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).
% Sup_insert_finite
tff(fact_1244_hom__Max__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ha: fun(A,A),N: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,Ha,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,Ha,X3)),aa(A,A,Ha,Y2))
=> ( aa(set(A),$o,finite_finite2(A),N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,Ha,lattic643756798349783984er_Max(A,N)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,Ha),N)) ) ) ) ) ) ).
% hom_Max_commute
tff(fact_1245_Max_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,B5)),lattic643756798349783984er_Max(A,A3)) = lattic643756798349783984er_Max(A,A3) ) ) ) ) ) ).
% Max.subset
tff(fact_1246_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,A3)) ) ) ) ) ) ).
% Max.insert_not_elem
tff(fact_1247_Max_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic643756798349783984er_Max(A,A3),A3) ) ) ) ) ).
% Max.closed
tff(fact_1248_Max_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B5)) ) ) ) ) ) ) ).
% Max.union
tff(fact_1249_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B,V: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( finite_fold_graph(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3),V)
=> ( ~ member(A,X,A3)
=> ~ ! [Y2: B] :
( ( V = aa(B,B,aa(A,fun(B,B),F2,X),Y2) )
=> ~ finite_fold_graph(A,B,F2,Z2,A3,Y2) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
tff(fact_1250_Max_Oeq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,ord_max(A),X,A3) ) ) ) ).
% Max.eq_fold
tff(fact_1251_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,product_prod(A,C))] :
( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),comp(product_prod(A,C),A,B,product_fst(A,C)),F2)),top_top(set(B))))
=> ( aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),comp(product_prod(A,C),C,B,product_snd(A,C)),F2)),top_top(set(B))))
=> aa(set(product_prod(A,C)),$o,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F2),top_top(set(B)))) ) ) ).
% finite_range_prod
tff(fact_1252_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,C)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2),top_top(set(product_prod(A,B))))
<=> inj_on(B,C,F2,top_top(set(B))) ) ).
% inj_apsnd
tff(fact_1253_total__inv__image,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),R2: set(product_prod(B,B))] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( total_on(B,top_top(set(B)),R2)
=> total_on(A,top_top(set(A)),inv_image(B,A,R2,F2)) ) ) ).
% total_inv_image
tff(fact_1254_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(A,C)] :
( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F2),top_top(set(product_prod(A,B))))
<=> inj_on(A,C,F2,top_top(set(A))) ) ).
% inj_apfst
tff(fact_1255_card__Diff1__less,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A3)) ) ) ).
% card_Diff1_less
tff(fact_1256_card__Diff2__less,axiom,
! [A: $tType,A3: set(A),X: A,Y: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( member(A,Y,A3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,minus_minus(set(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))),finite_card(A,A3)) ) ) ) ).
% card_Diff2_less
tff(fact_1257_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A3))
<=> ( aa(set(A),$o,finite_finite2(A),A3)
& member(A,X,A3) ) ) ).
% card_Diff1_less_iff
tff(fact_1258_Inter__in__chain,axiom,
! [A: $tType,B9: set(set(A)),A14: set(set(A))] :
( aa(set(set(A)),$o,finite_finite2(set(A)),B9)
=> ( ( B9 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A14,ord_less(set(A))),B9)
=> member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9),B9) ) ) ) ).
% Inter_in_chain
tff(fact_1259_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),X: A] :
( inj_on(A,B,F2,A3)
=> ( member(A,X,A3)
=> ( aa(B,A,inv_on(A,B,F2,A3),aa(A,B,F2,X)) = X ) ) ) ).
% inv_on_f_f
tff(fact_1260_card__inverse,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A))] : finite_card(product_prod(A,B),converse(B,A,R3)) = finite_card(product_prod(B,A),R3) ).
% card_inverse
tff(fact_1261_card__eq__UNIV2,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S: set(A)] :
( ( finite_card(A,top_top(set(A))) = finite_card(A,S) )
<=> ( S = top_top(set(A)) ) ) ) ).
% card_eq_UNIV2
tff(fact_1262_card__eq__UNIV,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S: set(A)] :
( ( finite_card(A,S) = finite_card(A,top_top(set(A))) )
<=> ( S = top_top(set(A)) ) ) ) ).
% card_eq_UNIV
tff(fact_1263_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),X)) = aa(C,A,F2,aa(product_prod(C,D),C,product_fst(C,D),X)) ).
% fst_map_prod
tff(fact_1264_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F2: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X)),Y) ).
% apfst_conv
tff(fact_1265_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F2,Y)) ).
% apsnd_conv
tff(fact_1266_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),X: product_prod(C,B),G: fun(C,A)] :
( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),X) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G),X) )
<=> ( aa(C,A,F2,aa(product_prod(C,B),C,product_fst(C,B),X)) = aa(C,A,G,aa(product_prod(C,B),C,product_fst(C,B),X)) ) ) ).
% apfst_eq_conv
tff(fact_1267_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),X: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),X)) = aa(C,A,F2,aa(product_prod(C,B),C,product_fst(C,B),X)) ).
% fst_apfst
tff(fact_1268_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(C,B),X: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F2),X)) = aa(product_prod(C,A),A,product_snd(C,A),X) ).
% snd_apfst
tff(fact_1269_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(B,B)),F2: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),inv_image(B,A,R2,F2))
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,X)),aa(A,B,F2,Y)),R2) ) ).
% in_inv_image
tff(fact_1270_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),X: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),X)) = aa(product_prod(A,C),A,product_fst(A,C),X) ).
% fst_apsnd
tff(fact_1271_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),X: product_prod(A,C),G: fun(C,B)] :
( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),X) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G),X) )
<=> ( aa(C,B,F2,aa(product_prod(A,C),C,product_snd(A,C),X)) = aa(C,B,G,aa(product_prod(A,C),C,product_snd(A,C),X)) ) ) ).
% apsnd_eq_conv
tff(fact_1272_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),X: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F2),X)) = aa(C,A,F2,aa(product_prod(B,C),C,product_snd(B,C),X)) ).
% snd_apsnd
tff(fact_1273_converse__inv__image,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(B,B)),F2: fun(A,B)] : converse(A,A,inv_image(B,A,R3,F2)) = inv_image(B,A,converse(B,B,R3),F2) ).
% converse_inv_image
tff(fact_1274_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: set(A)] :
( ( aa(set(A),A,complete_Inf_Inf(A),A3) = bot_bot(A) )
<=> ! [X2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
=> ? [Xa2: A] :
( member(A,Xa2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X2) ) ) ) ) ).
% Inf_eq_bot_iff
tff(fact_1275_Inf__empty,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).
% Inf_empty
tff(fact_1276_Inf__UNIV,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) = bot_bot(A) ) ) ).
% Inf_UNIV
tff(fact_1277_cInf__singleton,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% cInf_singleton
tff(fact_1278_Inf__insert,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ).
% Inf_insert
tff(fact_1279_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).
% prod.collapse
tff(fact_1280_card__ge__UNIV,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S: set(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,top_top(set(A)))),finite_card(A,S))
<=> ( S = top_top(set(A)) ) ) ) ).
% card_ge_UNIV
tff(fact_1281_card__Diff__insert,axiom,
! [A: $tType,A4: A,A3: set(A),B5: set(A)] :
( member(A,A4,A3)
=> ( ~ member(A,A4,B5)
=> ( finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5))) = minus_minus(nat,finite_card(A,minus_minus(set(A),A3,B5)),one_one(nat)) ) ) ) ).
% card_Diff_insert
tff(fact_1282_img__fst,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S)
=> member(A,A4,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)) ) ).
% img_fst
tff(fact_1283_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X)) = aa(product_prod(B,A),A,product_snd(B,A),X) ).
% fst_swap
tff(fact_1284_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X)) = aa(product_prod(A,B),A,product_fst(A,B),X) ).
% snd_swap
tff(fact_1285_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F2)) = product_snd(A,B) ).
% snd_comp_apfst
tff(fact_1286_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2)) = product_fst(A,B) ).
% fst_comp_apsnd
tff(fact_1287_range__fst,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),top_top(set(product_prod(A,B)))) = top_top(set(A)) ).
% range_fst
tff(fact_1288_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: fun(A,C),G: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),comp(product_prod(C,D),C,product_prod(A,B),product_fst(C,D)),product_map_prod(A,C,B,D,F2,G)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F2),product_fst(A,B)) ).
% fst_comp_map_prod
tff(fact_1289_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F2)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F2),product_fst(A,B)) ).
% fst_comp_apfst
tff(fact_1290_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F2),product_snd(A,B)) ).
% snd_comp_apsnd
tff(fact_1291_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F2: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X))) ).
% apfst_apsnd
tff(fact_1292_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: fun(C,B),G: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F2,aa(product_prod(D,C),C,product_snd(D,C),X))) ).
% apsnd_apfst
tff(fact_1293_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F2: fun(C,B),G: fun(D,A),P2: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),P2)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F2),P2)) ).
% apsnd_apfst_commute
tff(fact_1294_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z2: B] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).
% fstI
tff(fact_1295_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A4: A] :
( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A4 )
=> ( X = A4 ) ) ).
% fst_eqD
tff(fact_1296_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).
% fst_conv
tff(fact_1297_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A4: A,P2: product_prod(A,B)] :
( ( A4 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
<=> ? [B7: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B7) ) ).
% eq_fst_iff
tff(fact_1298_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),A4: A,B3: B,P: fun(A,$o)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),X))
=> aa(A,$o,P,A4) ) ) ).
% fstE
tff(fact_1299_prod__eq__iff,axiom,
! [B: $tType,A: $tType,S2: product_prod(A,B),T5: product_prod(A,B)] :
( ( S2 = T5 )
<=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S2) = aa(product_prod(A,B),A,product_fst(A,B),T5) )
& ( aa(product_prod(A,B),B,product_snd(A,B),S2) = aa(product_prod(A,B),B,product_snd(A,B),T5) ) ) ) ).
% prod_eq_iff
tff(fact_1300_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B),Q3: product_prod(A,B)] :
( ( aa(product_prod(A,B),A,product_fst(A,B),P2) = aa(product_prod(A,B),A,product_fst(A,B),Q3) )
=> ( ( aa(product_prod(A,B),B,product_snd(A,B),P2) = aa(product_prod(A,B),B,product_snd(A,B),Q3) )
=> ( P2 = Q3 ) ) ) ).
% prod_eqI
tff(fact_1301_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
& ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
tff(fact_1302_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
( ! [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_12)
<=> ! [Z3: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z3)),aa(product_prod(A,B),B,product_snd(A,B),Z3)) ) ).
% All_prod_contract
tff(fact_1303_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
( ? [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_12)
<=> ? [Z3: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z3)),aa(product_prod(A,B),B,product_snd(A,B),Z3)) ) ).
% Ex_prod_contract
tff(fact_1304_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,C),X: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G),X)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F2),G)),X) ).
% apfst_compose
tff(fact_1305_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(C,B),G: fun(D,C),X: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G),X)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F2),G)),X) ).
% apsnd_compose
tff(fact_1306_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S: set(A),A3: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
( ! [X3: A] :
( member(A,X3,S)
=> refl_on(B,aa(A,set(B),A3,X3),aa(A,set(product_prod(B,B)),R2,X3)) )
=> refl_on(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),S)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).
% refl_on_INTER
tff(fact_1307_is__singleton__altdef,axiom,
! [A: $tType,A3: set(A)] :
( is_singleton(A,A3)
<=> ( finite_card(A,A3) = one_one(nat) ) ) ).
% is_singleton_altdef
tff(fact_1308_cInf__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X3) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( member(A,X4,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),A4) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = A4 ) ) ) ) ) ).
% cInf_eq_non_empty
tff(fact_1309_cInf__greatest,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ).
% cInf_greatest
tff(fact_1310_Inf__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A),U: A] :
( ! [V3: A] :
( member(A,V3,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
=> ( ( A3 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U) ) ) ) ).
% Inf_less_eq
tff(fact_1311_cInf__lessD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z2)
=> ? [X3: A] :
( member(A,X3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2) ) ) ) ) ).
% cInf_lessD
tff(fact_1312_INF__eq__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),F2: fun(A,B),X: B] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> ( aa(A,B,F2,I3) = X ) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I)) = X ) ) ) ) ).
% INF_eq_const
tff(fact_1313_bot__finite__def,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( bot_bot(A) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).
% bot_finite_def
tff(fact_1314_card__1__singletonE,axiom,
! [A: $tType,A3: set(A)] :
( ( finite_card(A,A3) = one_one(nat) )
=> ~ ! [X3: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_1315_Inf__finite__insert,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ).
% Inf_finite_insert
tff(fact_1316_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,A4: product_prod(A,B)] :
( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> ( ( A4 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A4)),aa(product_prod(A,B),B,product_snd(A,B),A4)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_1317_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% prod.exhaust_sel
tff(fact_1318_surjective__pairing,axiom,
! [B: $tType,A: $tType,T5: product_prod(A,B)] : T5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T5)),aa(product_prod(A,B),B,product_snd(A,B),T5)) ).
% surjective_pairing
tff(fact_1319_card__insert__le,axiom,
! [A: $tType,A3: set(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A3)),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3))) ).
% card_insert_le
tff(fact_1320_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
( member(A,X,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
=> ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),S) ) ).
% in_fst_imageE
tff(fact_1321_trans__inv__image,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),F2: fun(B,A)] :
( trans(A,R2)
=> trans(B,inv_image(A,B,R2,F2)) ) ).
% trans_inv_image
tff(fact_1322_Inter__subset,axiom,
! [A: $tType,A3: set(set(A)),B5: set(A)] :
( ! [X7: set(A)] :
( member(set(A),X7,A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5) )
=> ( ( A3 != bot_bot(set(set(A))) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B5) ) ) ).
% Inter_subset
tff(fact_1323_Inter__empty,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),bot_bot(set(set(A)))) = top_top(set(A)) ).
% Inter_empty
tff(fact_1324_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R3) = aa(set(product_prod(A,B)),set(A),domain(A,B),R3) ).
% fst_eq_Domain
tff(fact_1325_Domain__fst,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R2) ).
% Domain_fst
tff(fact_1326_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set(set(A)),F2: fun(A,B)] :
( ( S != bot_bot(set(set(A))) )
=> ( ! [A10: set(A)] :
( member(set(A),A10,S)
=> inj_on(A,B,F2,A10) )
=> inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ) ).
% inj_on_Inter
tff(fact_1327_trans__INTER,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [X3: A] :
( member(A,X3,S)
=> trans(B,aa(A,set(product_prod(B,B)),R2,X3)) )
=> trans(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).
% trans_INTER
tff(fact_1328_card__partition,axiom,
! [A: $tType,C6: set(set(A)),K: nat] :
( aa(set(set(A)),$o,finite_finite2(set(A)),C6)
=> ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6))
=> ( ! [C3: set(A)] :
( member(set(A),C3,C6)
=> ( finite_card(A,C3) = K ) )
=> ( ! [C1: set(A),C22: set(A)] :
( member(set(A),C1,C6)
=> ( member(set(A),C22,C6)
=> ( ( C1 != C22 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(set(A),C6)) = finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)) ) ) ) ) ) ).
% card_partition
tff(fact_1329_Inter__UNIV,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),top_top(set(set(A)))) = bot_bot(set(A)) ).
% Inter_UNIV
tff(fact_1330_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F2: fun(B,A),A3: set(B)] :
( member(A,Y,aa(set(B),set(A),image2(B,A,F2),A3))
=> member(B,aa(A,B,inv_on(B,A,F2,A3),Y),A3) ) ).
% inv_on_f_range
tff(fact_1331_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F2: fun(B,A),A3: set(B)] :
( member(A,Y,aa(set(B),set(A),image2(B,A,F2),A3))
=> ( aa(B,A,F2,aa(A,B,inv_on(B,A,F2,A3),Y)) = Y ) ) ).
% f_inv_on_f
tff(fact_1332_cINF__greatest,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),M: B,F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F2,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).
% cINF_greatest
tff(fact_1333_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),F2: fun(A,B),C2: B] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),C2) )
=> ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I)) = C2 )
<=> ! [X2: A] :
( member(A,X2,I)
=> ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).
% INF_eq_iff
tff(fact_1334_finite__less__Inf__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(set(A),A,complete_Inf_Inf(A),X5))
<=> ! [X2: A] :
( member(A,X2,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),X2) ) ) ) ) ) ).
% finite_less_Inf_iff
tff(fact_1335_Inf__le__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).
% Inf_le_Sup
tff(fact_1336_finite__Inf__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2),A3) ) )
=> member(A,aa(set(A),A,complete_Inf_Inf(A),A3),A3) ) ) ) ) ).
% finite_Inf_in
tff(fact_1337_card__1__singletonI,axiom,
! [A: $tType,S: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( finite_card(A,S) = one_one(nat) )
=> ( member(A,X,S)
=> ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).
% card_1_singletonI
tff(fact_1338_Sup__finite__empty,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).
% Sup_finite_empty
tff(fact_1339_Inf__finite__empty,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).
% Inf_finite_empty
tff(fact_1340_card__Diff__singleton__if,axiom,
! [A: $tType,A3: set(A),X: A] :
finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = $ite(member(A,X,A3),minus_minus(nat,finite_card(A,A3),one_one(nat)),finite_card(A,A3)) ).
% card_Diff_singleton_if
tff(fact_1341_card__Diff__singleton,axiom,
! [A: $tType,X: A,A3: set(A)] :
( member(A,X,A3)
=> ( finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = minus_minus(nat,finite_card(A,A3),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_1342_fst__image__mp,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B)),B5: set(A),X: A,Y: B] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A3)),B5)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),A3)
=> member(A,X,B5) ) ) ).
% fst_image_mp
tff(fact_1343_fst__in__Field,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R3)),aa(set(product_prod(A,A)),set(A),field2(A),R3)) ).
% fst_in_Field
tff(fact_1344_Inf__fin__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).
% Inf_fin_Inf
tff(fact_1345_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = lattic7752659483105999362nf_fin(A,X5) ) ) ) ) ).
% cInf_eq_Inf_fin
tff(fact_1346_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType,X4: product_prod(A,B)] : basic_fsts(A,B,X4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(product_prod(A,B),A,product_fst(A,B),X4)),bot_bot(set(A))) ).
% prod_set_defs(1)
tff(fact_1347_prod_Oswap__def,axiom,
! [A: $tType,B: $tType,P2: product_prod(B,A)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2)) ).
% prod.swap_def
tff(fact_1348_equiv__proj,axiom,
! [A: $tType,A3: set(A),R3: set(product_prod(A,A)),Z2: product_prod(A,A)] :
( equiv_equiv(A,A3,R3)
=> ( member(product_prod(A,A),Z2,R3)
=> ( aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),comp(A,set(A),product_prod(A,A),equiv_proj(A,A,R3)),product_fst(A,A)),Z2) = aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),comp(A,set(A),product_prod(A,A),equiv_proj(A,A,R3)),product_snd(A,A)),Z2) ) ) ) ).
% equiv_proj
tff(fact_1349_INF__le__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [A3: set(A),F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).
% INF_le_SUP
tff(fact_1350_card__Diff1__le,axiom,
! [A: $tType,A3: set(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A3)) ).
% card_Diff1_le
tff(fact_1351_exI__realizer,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Y: A,X: B] :
( aa(B,$o,aa(A,fun(B,$o),P,Y),X)
=> aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))) ) ).
% exI_realizer
tff(fact_1352_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),P2: A,Q: fun(B,$o),Q3: B] :
( aa(A,$o,P,P2)
=> ( aa(B,$o,Q,Q3)
=> ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q3)))
& aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q3))) ) ) ) ).
% conjI_realizer
tff(fact_1353_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F2: fun(C,A),P2: product_prod(C,B)] :
( ( Q3 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),P2) )
=> ~ ! [X3: C,Y2: B] :
( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y2) )
=> ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X3)),Y2) ) ) ) ).
% apfst_convE
tff(fact_1354_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A)] :
( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
=> ~ ! [X3: B,Y2: A] : ~ aa(B,$o,aa(A,fun(B,$o),P,Y2),X3) ) ).
% exE_realizer'
tff(fact_1355_snd__sndOp,axiom,
! [C: $tType,A: $tType,B: $tType,Bc: product_prod(B,A),P: fun(B,fun(C,$o)),Q: fun(C,fun(A,$o))] : aa(product_prod(B,A),A,product_snd(B,A),Bc) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(C,A)),fun(product_prod(B,A),A),comp(product_prod(C,A),A,product_prod(B,A),product_snd(C,A)),bNF_sndOp(B,C,A,P,Q)),Bc) ).
% snd_sndOp
tff(fact_1356_fst__fstOp,axiom,
! [C: $tType,B: $tType,A: $tType,Bc: product_prod(A,B),P: fun(A,fun(C,$o)),Q: fun(C,fun(B,$o))] : aa(product_prod(A,B),A,product_fst(A,B),Bc) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),bNF_fstOp(A,C,B,P,Q)),Bc) ).
% fst_fstOp
tff(fact_1357_card__Un__disjoint,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A3)),finite_card(A,B5)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_1358_card__Suc__Diff1,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) = finite_card(A,A3) ) ) ) ).
% card_Suc_Diff1
tff(fact_1359_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A4: A,B3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2) )
<=> ( B3 = C2 ) ) ) ).
% add_left_cancel
tff(fact_1360_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B3: A,A4: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4) )
<=> ( B3 = C2 ) ) ) ).
% add_right_cancel
tff(fact_1361_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% add_le_cancel_left
tff(fact_1362_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% add_le_cancel_right
tff(fact_1363_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% add_less_cancel_left
tff(fact_1364_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% add_less_cancel_right
tff(fact_1365_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),B3) = A4 ) ).
% add_diff_cancel
tff(fact_1366_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A4,B3)),B3) = A4 ) ).
% diff_add_cancel
tff(fact_1367_add__diff__cancel__left,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [C2: A,A4: A,B3: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) = minus_minus(A,A4,B3) ) ).
% add_diff_cancel_left
tff(fact_1368_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),A4) = B3 ) ).
% add_diff_cancel_left'
tff(fact_1369_add__diff__cancel__right,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A4: A,C2: A,B3: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = minus_minus(A,A4,B3) ) ).
% add_diff_cancel_right
tff(fact_1370_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),B3) = A4 ) ).
% add_diff_cancel_right'
tff(fact_1371_card__insert__disjoint,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(nat,nat,suc,finite_card(A,A3)) ) ) ) ).
% card_insert_disjoint
tff(fact_1372_Suc__diff,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
=> ( aa(nat,nat,suc,minus_minus(nat,N2,M)) = minus_minus(nat,N2,minus_minus(nat,M,one_one(nat))) ) ) ) ).
% Suc_diff
tff(fact_1373_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& ( K = L ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_1374_group__cancel_Oadd1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A4) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% group_cancel.add1
tff(fact_1375_group__cancel_Oadd2,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [B5: A,K: A,B3: A,A4: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B3) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% group_cancel.add2
tff(fact_1376_add_Oassoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ).
% add.assoc
tff(fact_1377_add_Oleft__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2) )
<=> ( B3 = C2 ) ) ) ).
% add.left_cancel
tff(fact_1378_add_Oright__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B3: A,A4: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4) )
<=> ( B3 = C2 ) ) ) ).
% add.right_cancel
tff(fact_1379_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) ) ).
% ab_semigroup_add_class.add.commute
tff(fact_1380_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [B3: A,A4: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_1381_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A4: A,B3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
tff(fact_1382_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B3: A,A4: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
tff(fact_1383_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J2)
& ( K = L ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_1384_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_1385_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_1386_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D3)) ) ) ) ).
% add_mono
tff(fact_1387_add__left__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).
% add_left_mono
tff(fact_1388_less__eqE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ~ ! [C3: A] : B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C3) ) ) ).
% less_eqE
tff(fact_1389_add__right__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).
% add_right_mono
tff(fact_1390_le__iff__add,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
<=> ? [C5: A] : B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C5) ) ) ).
% le_iff_add
tff(fact_1391_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% add_le_imp_le_left
tff(fact_1392_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% add_le_imp_le_right
tff(fact_1393_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1394_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( ( I2 = J2 )
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1395_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J2)
& ( K = L ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1396_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D3)) ) ) ) ).
% add_strict_mono
tff(fact_1397_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).
% add_strict_left_mono
tff(fact_1398_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).
% add_strict_right_mono
tff(fact_1399_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% add_less_imp_less_left
tff(fact_1400_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% add_less_imp_less_right
tff(fact_1401_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A4: A,E4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),E4)),C2) ) ).
% combine_common_factor
tff(fact_1402_distrib__right,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% distrib_right
tff(fact_1403_distrib__left,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ).
% distrib_left
tff(fact_1404_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% comm_semiring_class.distrib
tff(fact_1405_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ).
% ring_class.ring_distribs(1)
tff(fact_1406_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% ring_class.ring_distribs(2)
tff(fact_1407_group__cancel_Osub1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,K: A,A4: A,B3: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A4) )
=> ( minus_minus(A,A3,B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),minus_minus(A,A4,B3)) ) ) ) ).
% group_cancel.sub1
tff(fact_1408_diff__eq__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus(A,A4,B3) = C2 )
<=> ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3) ) ) ) ).
% diff_eq_eq
tff(fact_1409_eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,C2: A,B3: A] :
( ( A4 = minus_minus(A,C2,B3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = C2 ) ) ) ).
% eq_diff_eq
tff(fact_1410_add__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),minus_minus(A,B3,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),C2) ) ).
% add_diff_eq
tff(fact_1411_diff__diff__eq2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A] : minus_minus(A,A4,minus_minus(A,B3,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2),B3) ) ).
% diff_diff_eq2
tff(fact_1412_diff__add__eq,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A4,B3)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2),B3) ) ).
% diff_add_eq
tff(fact_1413_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A,C2: A] : minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = minus_minus(A,minus_minus(A,A4,C2),B3) ) ).
% diff_add_eq_diff_diff_swap
tff(fact_1414_add__implies__diff,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [C2: A,B3: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3) = A4 )
=> ( C2 = minus_minus(A,A4,B3) ) ) ) ).
% add_implies_diff
tff(fact_1415_diff__diff__eq,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A4: A,B3: A,C2: A] : minus_minus(A,minus_minus(A,A4,B3),C2) = minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ).
% diff_diff_eq
tff(fact_1416_mlex__bound,axiom,
! [A4: nat,A3: nat,B3: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),A3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),N)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N)) ) ) ).
% mlex_bound
tff(fact_1417_mlex__fst__decrI,axiom,
! [A4: nat,A6: nat,B3: nat,N: nat,B4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),A6)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),N)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),N)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ) ).
% mlex_fst_decrI
tff(fact_1418_mlex__snd__decrI,axiom,
! [A4: nat,A6: nat,B3: nat,B4: nat,N: nat] :
( ( A4 = A6 )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),B4)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ).
% mlex_snd_decrI
tff(fact_1419_mlex__leI,axiom,
! [A4: nat,A6: nat,B3: nat,B4: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A4),A6)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),B4)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ).
% mlex_leI
tff(fact_1420_Inf__nat__def1,axiom,
! [K2: set(nat)] :
( ( K2 != bot_bot(set(nat)) )
=> member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K2),K2) ) ).
% Inf_nat_def1
tff(fact_1421_max__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% max_add_distrib_left
tff(fact_1422_max__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).
% max_add_distrib_right
tff(fact_1423_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1424_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J2: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L)) ) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1425_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D3)) ) ) ) ).
% add_le_less_mono
tff(fact_1426_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D3)) ) ) ) ).
% add_less_le_mono
tff(fact_1427_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( ( minus_minus(A,B3,A4) = C2 )
<=> ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1428_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),minus_minus(A,B3,A4)) = B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1429_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( minus_minus(A,C2,minus_minus(A,B3,A4)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4),B3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1430_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B3,A4)),C2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1431_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B3,A4)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2),A4) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1432_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),minus_minus(A,B3,A4)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1433_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),minus_minus(A,B3,A4)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3),A4) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1434_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),minus_minus(A,B3,A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A4)),B3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1435_le__add__diff,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2),A4)) ) ) ).
% le_add_diff
tff(fact_1436_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B3,A4)),A4) = B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1437_le__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),minus_minus(A,C2,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) ) ) ).
% le_diff_eq
tff(fact_1438_diff__le__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,A4,B3)),C2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).
% diff_le_eq
tff(fact_1439_less__add__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))) ) ).
% less_add_one
tff(fact_1440_add__mono1,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),one_one(A))) ) ) ).
% add_mono1
tff(fact_1441_diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,A4,B3)),C2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).
% diff_less_eq
tff(fact_1442_less__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),minus_minus(A,C2,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),C2) ) ) ).
% less_diff_eq
tff(fact_1443_square__diff__square__factored,axiom,
! [A: $tType] :
( comm_ring(A)
=> ! [X: A,Y: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),minus_minus(A,X,Y)) ) ).
% square_diff_square_factored
tff(fact_1444_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3) )
<=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B3,A4)),E4)),D3) ) ) ) ).
% eq_add_iff2
tff(fact_1445_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,B3)),E4)),C2) = D3 ) ) ) ).
% eq_add_iff1
tff(fact_1446_Suc__n__minus__m__eq,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M)
=> ( aa(nat,nat,suc,minus_minus(nat,N2,M)) = minus_minus(nat,N2,minus_minus(nat,M,one_one(nat))) ) ) ) ).
% Suc_n_minus_m_eq
tff(fact_1447_nat__in__between__eq_I1_J,axiom,
! [A4: nat,B3: nat] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B3)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),aa(nat,nat,suc,A4)) )
<=> ( B3 = aa(nat,nat,suc,A4) ) ) ).
% nat_in_between_eq(1)
tff(fact_1448_nat__in__between__eq_I2_J,axiom,
! [A4: nat,B3: nat] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A4),B3)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),aa(nat,nat,suc,A4)) )
<=> ( B3 = A4 ) ) ).
% nat_in_between_eq(2)
tff(fact_1449_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set(product_prod(A,A)),F2: fun(nat,A)] :
( wf(A,R2)
=> ~ ! [K3: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F2,aa(nat,nat,suc,K3))),aa(nat,A,F2,K3)),R2) ) ).
% wf_no_infinite_down_chainE
tff(fact_1450_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ~ ? [F6: fun(nat,A)] :
! [I4: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,aa(nat,nat,suc,I4))),aa(nat,A,F6,I4)),R2) ) ).
% wf_iff_no_infinite_down_chain
tff(fact_1451_Inf__INT__eq,axiom,
! [A: $tType,S: set(fun(A,$o)),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),S),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).
% Inf_INT_eq
tff(fact_1452_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B3,A4)),E4)),D3)) ) ) ).
% le_add_iff2
tff(fact_1453_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,B3)),E4)),C2)),D3) ) ) ).
% le_add_iff1
tff(fact_1454_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B3,A4)),E4)),D3)) ) ) ).
% less_add_iff2
tff(fact_1455_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,E4: A,C2: A,B3: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),E4)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E4)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,B3)),E4)),C2)),D3) ) ) ).
% less_add_iff1
tff(fact_1456_square__diff__one__factored,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),minus_minus(A,X,one_one(A))) ) ).
% square_diff_one_factored
tff(fact_1457_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set(A),K: nat] :
( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
<=> ? [B7: A,B10: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B10) )
& ~ member(A,B7,B10)
& ( finite_card(A,B10) = K )
& aa(set(A),$o,finite_finite2(A),B10) ) ) ).
% card_Suc_eq_finite
tff(fact_1458_card__insert__if,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(member(A,X,A3),finite_card(A,A3),aa(nat,nat,suc,finite_card(A,A3))) ) ) ).
% card_insert_if
tff(fact_1459_card__le__Suc__iff,axiom,
! [A: $tType,N2: nat,A3: set(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,N2)),finite_card(A,A3))
<=> ? [A7: A,B10: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),B10) )
& ~ member(A,A7,B10)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),finite_card(A,B10))
& aa(set(A),$o,finite_finite2(A),B10) ) ) ).
% card_le_Suc_iff
tff(fact_1460_card_Oremove,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( finite_card(A,A3) = aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% card.remove
tff(fact_1461_card_Oinsert__remove,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).
% card.insert_remove
tff(fact_1462_mult__Suc__right,axiom,
! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ).
% mult_Suc_right
tff(fact_1463_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = one_one(nat) )
<=> ( ( M = one_one(nat) )
& ( N2 = one_one(nat) ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_1464_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
<=> ( ( M = one_one(nat) )
& ( N2 = one_one(nat) ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_1465_nat__less__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),I2)
=> ( 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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2))
<=> 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),minus_minus(nat,I2,J2)),U)),M)),N2) ) ) ).
% nat_less_add_iff1
tff(fact_1466_nat__less__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,J2,I2)),U)),N2)) ) ) ).
% nat_less_add_iff2
tff(fact_1467_nat__diff__add__eq2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2)) = minus_minus(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,J2,I2)),U)),N2)) ) ) ).
% nat_diff_add_eq2
tff(fact_1468_nat__diff__add__eq1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),I2)
=> ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2)) = 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),minus_minus(nat,I2,J2)),U)),M),N2) ) ) ).
% nat_diff_add_eq1
tff(fact_1469_nat__le__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,J2,I2)),U)),N2)) ) ) ).
% nat_le_add_iff2
tff(fact_1470_nat__mult__max__right,axiom,
! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).
% nat_mult_max_right
tff(fact_1471_nat__mult__max__left,axiom,
! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N2)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) ).
% nat_mult_max_left
tff(fact_1472_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2) )
<=> ( M = N2 ) ) ).
% Suc_mult_cancel1
tff(fact_1473_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J2)) ) ).
% mult_le_mono2
tff(fact_1474_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K)) ) ).
% mult_le_mono1
tff(fact_1475_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_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),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),L)) ) ) ).
% mult_le_mono
tff(fact_1476_le__square,axiom,
! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).
% le_square
tff(fact_1477_le__cube,axiom,
! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).
% le_cube
tff(fact_1478_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)) ).
% add_mult_distrib
tff(fact_1479_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) ).
% add_mult_distrib2
tff(fact_1480_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J2: 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),I2),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),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),I2),J2)),U)),K) ).
% left_add_mult_distrib
tff(fact_1481_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,M,N2)),K) = minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)) ).
% diff_mult_distrib
tff(fact_1482_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),minus_minus(nat,M,N2)) = minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) ).
% diff_mult_distrib2
tff(fact_1483_nat__mult__1,axiom,
! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N2) = N2 ).
% nat_mult_1
tff(fact_1484_nat__mult__1__right,axiom,
! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),one_one(nat)) = N2 ).
% nat_mult_1_right
tff(fact_1485_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2) ) ).
% Suc_mult_less_cancel1
tff(fact_1486_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ).
% Suc_mult_le_cancel1
tff(fact_1487_mult__Suc,axiom,
! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ).
% mult_Suc
tff(fact_1488_nat__eq__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),I2)
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2) )
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,I2,J2)),U)),M) = N2 ) ) ) ).
% nat_eq_add_iff1
tff(fact_1489_nat__eq__add__iff2,axiom,
! [I2: nat,J2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N2) )
<=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,J2,I2)),U)),N2) ) ) ) ).
% nat_eq_add_iff2
tff(fact_1490_nat__le__add__iff1,axiom,
! [J2: nat,I2: nat,U: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),I2)
=> ( 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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),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,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,I2,J2)),U)),M)),N2) ) ) ).
% nat_le_add_iff1
tff(fact_1491_discrete,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),B3) ) ) ).
% discrete
tff(fact_1492_fstOp__def,axiom,
! [B: $tType,C: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(B,fun(C,$o)),Ac: product_prod(A,C)] : aa(product_prod(A,C),product_prod(A,B),bNF_fstOp(A,B,C,P,Q),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P,Q,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).
% fstOp_def
tff(fact_1493_sndOp__def,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(C,fun(A,$o)),Q: fun(A,fun(B,$o)),Ac: product_prod(C,B)] : aa(product_prod(C,B),product_prod(A,B),bNF_sndOp(C,A,B,P,Q),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P,Q,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).
% sndOp_def
tff(fact_1494_card__insert__le__m1,axiom,
! [A: $tType,N2: nat,Y: set(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Y)),minus_minus(nat,N2,one_one(nat)))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),Y))),N2) ) ) ).
% card_insert_le_m1
tff(fact_1495_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( A4 != B3 )
& ( C2 != D3 ) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).
% crossproduct_noteq
tff(fact_1496_crossproduct__eq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [W: A,Y: A,X: A,Z2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
<=> ( ( W = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
tff(fact_1497_dbl__dec__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl_dec(A,X) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X),one_one(A)) ) ).
% dbl_dec_def
tff(fact_1498_card__insert__disjoint_H,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( minus_minus(nat,finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,A3) ) ) ) ).
% card_insert_disjoint'
tff(fact_1499_le__zero__eq,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),zero_zero(A))
<=> ( N2 = zero_zero(A) ) ) ) ).
% le_zero_eq
tff(fact_1500_not__gr__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N2: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N2)
<=> ( N2 = zero_zero(A) ) ) ) ).
% not_gr_zero
tff(fact_1501_add_Oright__neutral,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),zero_zero(A)) = A4 ) ).
% add.right_neutral
tff(fact_1502_double__zero__sym,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A4) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% double_zero_sym
tff(fact_1503_add__cancel__left__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [B3: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) = A4 )
<=> ( B3 = zero_zero(A) ) ) ) ).
% add_cancel_left_left
tff(fact_1504_add__cancel__left__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = A4 )
<=> ( B3 = zero_zero(A) ) ) ) ).
% add_cancel_left_right
tff(fact_1505_add__cancel__right__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A4: A,B3: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4) )
<=> ( B3 = zero_zero(A) ) ) ) ).
% add_cancel_right_left
tff(fact_1506_add__cancel__right__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A4: A,B3: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) )
<=> ( B3 = zero_zero(A) ) ) ) ).
% add_cancel_right_right
tff(fact_1507_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_1508_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Y: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_1509_add__0,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A4) = A4 ) ).
% add_0
tff(fact_1510_mult__cancel__right,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [A4: A,C2: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
<=> ( ( C2 = zero_zero(A) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_right
tff(fact_1511_mult__cancel__left,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
<=> ( ( C2 = zero_zero(A) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_left
tff(fact_1512_mult__eq__0__iff,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = zero_zero(A) )
<=> ( ( A4 = zero_zero(A) )
| ( B3 = zero_zero(A) ) ) ) ) ).
% mult_eq_0_iff
tff(fact_1513_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),zero_zero(A)) = zero_zero(A) ) ).
% mult_zero_right
tff(fact_1514_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A4) = zero_zero(A) ) ).
% mult_zero_left
tff(fact_1515_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : minus_minus(A,A4,A4) = zero_zero(A) ) ).
% diff_self
tff(fact_1516_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : minus_minus(A,A4,zero_zero(A)) = A4 ) ).
% diff_0_right
tff(fact_1517_zero__diff,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A4: A] : minus_minus(A,zero_zero(A),A4) = zero_zero(A) ) ).
% zero_diff
tff(fact_1518_diff__zero,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A4: A] : minus_minus(A,A4,zero_zero(A)) = A4 ) ).
% diff_zero
tff(fact_1519_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A4: A] : minus_minus(A,A4,A4) = zero_zero(A) ) ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1520_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K) )
<=> ( ( M = N2 )
| ( K = zero_zero(nat) ) ) ) ).
% mult_cancel2
tff(fact_1521_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2) )
<=> ( ( M = N2 )
| ( K = zero_zero(nat) ) ) ) ).
% mult_cancel1
tff(fact_1522_mult__0__right,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).
% mult_0_right
tff(fact_1523_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
| ( N2 = zero_zero(nat) ) ) ) ).
% mult_is_0
tff(fact_1524_Sup__nat__empty,axiom,
aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).
% Sup_nat_empty
tff(fact_1525_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).
% dbl_dec_simps(3)
tff(fact_1526_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_1527_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_1528_le__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).
% le_add_same_cancel2
tff(fact_1529_le__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).
% le_add_same_cancel1
tff(fact_1530_add__le__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% add_le_same_cancel2
tff(fact_1531_add__le__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% add_le_same_cancel1
tff(fact_1532_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),minus_minus(A,A4,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ).
% diff_ge_0_iff_ge
tff(fact_1533_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),minus_minus(A,A4,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ).
% zero_comp_diff_simps(1)
tff(fact_1534_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_1535_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_1536_less__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).
% less_add_same_cancel2
tff(fact_1537_less__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).
% less_add_same_cancel1
tff(fact_1538_add__less__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% add_less_same_cancel2
tff(fact_1539_add__less__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% add_less_same_cancel1
tff(fact_1540_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),minus_minus(A,A4,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ).
% zero_comp_diff_simps(2)
tff(fact_1541_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),minus_minus(A,A4,B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ).
% diff_gt_0_iff_gt
tff(fact_1542_diff__add__zero,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) = zero_zero(A) ) ).
% diff_add_zero
tff(fact_1543_mult__cancel__right2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A4: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A4 = one_one(A) ) ) ) ) ).
% mult_cancel_right2
tff(fact_1544_mult__cancel__right1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B3: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
<=> ( ( C2 = zero_zero(A) )
| ( B3 = one_one(A) ) ) ) ) ).
% mult_cancel_right1
tff(fact_1545_mult__cancel__left2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A4 = one_one(A) ) ) ) ) ).
% mult_cancel_left2
tff(fact_1546_mult__cancel__left1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B3: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
<=> ( ( C2 = zero_zero(A) )
| ( B3 = one_one(A) ) ) ) ) ).
% mult_cancel_left1
tff(fact_1547_image__add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [S: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).
% image_add_0
tff(fact_1548_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( minus_minus(A,one_one(A),one_one(A)) = zero_zero(A) ) ) ).
% diff_numeral_special(9)
tff(fact_1549_max__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),zero_zero(A)) = one_one(A) ) ) ).
% max_0_1(2)
tff(fact_1550_max__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),one_one(A)) = one_one(A) ) ) ).
% max_0_1(1)
tff(fact_1551_card_Oempty,axiom,
! [A: $tType] : finite_card(A,bot_bot(set(A))) = zero_zero(nat) ).
% card.empty
tff(fact_1552_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
<=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% one_eq_mult_iff
tff(fact_1553_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% mult_eq_1_iff
tff(fact_1554_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2) ) ) ).
% nat_mult_less_cancel_disj
tff(fact_1555_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).
% nat_0_less_mult_iff
tff(fact_1556_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2) ) ) ).
% mult_less_cancel2
tff(fact_1557_card__0__eq,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( finite_card(A,A3) = zero_zero(nat) )
<=> ( A3 = bot_bot(set(A)) ) ) ) ).
% card_0_eq
tff(fact_1558_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N2) ) ) ).
% one_le_mult_iff
tff(fact_1559_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ).
% mult_le_cancel2
tff(fact_1560_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ).
% nat_mult_le_cancel_disj
tff(fact_1561_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
% bot_nat_def
tff(fact_1562_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_1563_zero__le,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ).
% zero_le
tff(fact_1564_gr__zeroI,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N2: A] :
( ( N2 != zero_zero(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N2) ) ) ).
% gr_zeroI
tff(fact_1565_not__less__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),N2),zero_zero(A)) ) ).
% not_less_zero
tff(fact_1566_gr__implies__not__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [M: A,N2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),N2)
=> ( N2 != zero_zero(A) ) ) ) ).
% gr_implies_not_zero
tff(fact_1567_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N2)
<=> ( N2 != zero_zero(A) ) ) ) ).
% zero_less_iff_neq_zero
tff(fact_1568_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A4) = A4 ) ).
% comm_monoid_add_class.add_0
tff(fact_1569_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),zero_zero(A)) = A4 ) ).
% add.comm_neutral
tff(fact_1570_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A4) = A4 ) ).
% add.group_left_neutral
tff(fact_1571_mult__right__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
<=> ( A4 = B3 ) ) ) ) ).
% mult_right_cancel
tff(fact_1572_mult__left__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
<=> ( A4 = B3 ) ) ) ) ).
% mult_left_cancel
tff(fact_1573_no__zero__divisors,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) != zero_zero(A) ) ) ) ) ).
% no_zero_divisors
tff(fact_1574_divisors__zero,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = zero_zero(A) )
=> ( ( A4 = zero_zero(A) )
| ( B3 = zero_zero(A) ) ) ) ) ).
% divisors_zero
tff(fact_1575_mult__not__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) != zero_zero(A) )
=> ( ( A4 != zero_zero(A) )
& ( B3 != zero_zero(A) ) ) ) ) ).
% mult_not_zero
tff(fact_1576_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_1577_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
<=> ( minus_minus(A,A4,B3) = zero_zero(A) ) ) ) ).
% eq_iff_diff_eq_0
tff(fact_1578_mult__0,axiom,
! [N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N2) = zero_zero(nat) ).
% mult_0
tff(fact_1579_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2) )
<=> ( ( K = zero_zero(nat) )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
tff(fact_1580_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [R2: A,A4: A,B3: A,C2: A,D3: A] :
( ( R2 != zero_zero(A) )
=> ( ( ( A4 = B3 )
& ( C2 != D3 ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).
% add_scale_eq_noteq
tff(fact_1581_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),lattic643756798349783984er_Max(nat,X5)) ).
% Sup_nat_def
tff(fact_1582_add__decreasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),B3) ) ) ) ).
% add_decreasing
tff(fact_1583_add__increasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) ) ) ) ).
% add_increasing
tff(fact_1584_add__decreasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)),B3) ) ) ) ).
% add_decreasing2
tff(fact_1585_add__increasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) ) ) ) ).
% add_increasing2
tff(fact_1586_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% add_nonneg_nonneg
tff(fact_1587_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),zero_zero(A)) ) ) ) ).
% add_nonpos_nonpos
tff(fact_1588_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_1589_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_1590_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ordere2520102378445227354miring(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1591_zero__le__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) ) ) ) ).
% zero_le_mult_iff
tff(fact_1592_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)),zero_zero(A)) ) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1593_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A)) ) ) ) ).
% mult_nonpos_nonneg
tff(fact_1594_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A)) ) ) ) ).
% mult_nonneg_nonpos
tff(fact_1595_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ) ).
% mult_nonneg_nonneg
tff(fact_1596_split__mult__neg__le,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A4: A,B3: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A)) ) ) ).
% split_mult_neg_le
tff(fact_1597_mult__le__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ) ) ).
% mult_le_0_iff
tff(fact_1598_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).
% mult_right_mono
tff(fact_1599_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).
% mult_right_mono_neg
tff(fact_1600_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% mult_left_mono
tff(fact_1601_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ) ).
% mult_nonpos_nonpos
tff(fact_1602_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% mult_left_mono_neg
tff(fact_1603_split__mult__pos__le,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A4: A,B3: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ).
% split_mult_pos_le
tff(fact_1604_zero__le__square,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4)) ) ).
% zero_le_square
tff(fact_1605_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_mono'
tff(fact_1606_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_mono
tff(fact_1607_not__one__le__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).
% not_one_le_zero
tff(fact_1608_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1609_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).
% zero_less_one_class.zero_le_one
tff(fact_1610_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,A4,B3)),zero_zero(A)) ) ) ).
% le_iff_diff_le_0
tff(fact_1611_pos__add__strict,axiom,
! [A: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) ) ) ) ).
% pos_add_strict
tff(fact_1612_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ~ ! [C3: A] :
( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C3) )
=> ( C3 = zero_zero(A) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_1613_add__pos__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% add_pos_pos
tff(fact_1614_add__neg__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),zero_zero(A)) ) ) ) ).
% add_neg_neg
tff(fact_1615_mult__neg__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ) ).
% mult_neg_neg
tff(fact_1616_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4)),zero_zero(A)) ) ).
% not_square_less_zero
tff(fact_1617_mult__less__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ) ).
% mult_less_0_iff
tff(fact_1618_mult__neg__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A)) ) ) ) ).
% mult_neg_pos
tff(fact_1619_mult__pos__neg,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A)) ) ) ) ).
% mult_pos_neg
tff(fact_1620_mult__pos__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ) ).
% mult_pos_pos
tff(fact_1621_mult__pos__neg2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)),zero_zero(A)) ) ) ) ).
% mult_pos_neg2
tff(fact_1622_zero__less__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) ) ) ) ) ).
% zero_less_mult_iff
tff(fact_1623_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ).
% zero_less_mult_pos
tff(fact_1624_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ).
% zero_less_mult_pos2
tff(fact_1625_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ).
% mult_less_cancel_left_neg
tff(fact_1626_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% mult_less_cancel_left_pos
tff(fact_1627_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% mult_strict_left_mono_neg
tff(fact_1628_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% mult_strict_left_mono
tff(fact_1629_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_1630_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).
% mult_strict_right_mono_neg
tff(fact_1631_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).
% mult_strict_right_mono
tff(fact_1632_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_1633_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord2810124833399127020strict(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1634_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).
% less_numeral_extra(1)
tff(fact_1635_zero__less__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).
% zero_less_one
tff(fact_1636_not__one__less__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).
% not_one_less_zero
tff(fact_1637_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,A4,B3)),zero_zero(A)) ) ) ).
% less_iff_diff_less_0
tff(fact_1638_inj__on__mult,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A4: A,A3: set(A)] :
( ( A4 != zero_zero(A) )
=> inj_on(A,A,aa(A,fun(A,A),times_times(A),A4),A3) ) ) ).
% inj_on_mult
tff(fact_1639_nat__compl__induct,axiom,
! [P: fun(nat,$o),N2: nat] :
( aa(nat,$o,P,zero_zero(nat))
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N3)
=> aa(nat,$o,P,Nn) )
=> aa(nat,$o,P,aa(nat,nat,suc,N3)) )
=> aa(nat,$o,P,N2) ) ) ).
% nat_compl_induct
tff(fact_1640_nat__compl__induct_H,axiom,
! [P: fun(nat,$o),N2: nat] :
( aa(nat,$o,P,zero_zero(nat))
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N3)
=> aa(nat,$o,P,Nn) )
=> aa(nat,$o,P,aa(nat,nat,suc,N3)) )
=> aa(nat,$o,P,N2) ) ) ).
% nat_compl_induct'
tff(fact_1641_Suc__to__right,axiom,
! [N2: nat,M: nat] :
( ( aa(nat,nat,suc,N2) = M )
=> ( N2 = minus_minus(nat,M,aa(nat,nat,suc,zero_zero(nat))) ) ) ).
% Suc_to_right
tff(fact_1642_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K)) ) ) ).
% mult_less_mono1
tff(fact_1643_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J2)) ) ) ).
% mult_less_mono2
tff(fact_1644_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2) )
<=> ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
tff(fact_1645_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2) ) ) ).
% nat_mult_less_cancel1
tff(fact_1646_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( aa(nat,$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_1647_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) )
=> ( ( N2 = one_one(nat) )
| ( M = zero_zero(nat) ) ) ) ).
% mult_eq_self_implies_10
tff(fact_1648_add__strict__increasing2,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) ) ) ) ).
% add_strict_increasing2
tff(fact_1649_add__strict__increasing,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2)) ) ) ) ).
% add_strict_increasing
tff(fact_1650_add__pos__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% add_pos_nonneg
tff(fact_1651_add__nonpos__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),zero_zero(A)) ) ) ) ).
% add_nonpos_neg
tff(fact_1652_add__nonneg__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) ) ) ) ).
% add_nonneg_pos
tff(fact_1653_add__neg__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),zero_zero(A)) ) ) ) ).
% add_neg_nonpos
tff(fact_1654_mult__less__le__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_1655_mult__le__less__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_1656_mult__right__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% mult_right_le_imp_le
tff(fact_1657_mult__left__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% mult_left_le_imp_le
tff(fact_1658_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% mult_le_cancel_left_pos
tff(fact_1659_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ).
% mult_le_cancel_left_neg
tff(fact_1660_mult__less__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ) ).
% mult_less_cancel_right
tff(fact_1661_mult__strict__mono_H,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_strict_mono'
tff(fact_1662_mult__right__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% mult_right_less_imp_less
tff(fact_1663_mult__less__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ) ).
% mult_less_cancel_left
tff(fact_1664_mult__strict__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ) ).
% mult_strict_mono
tff(fact_1665_mult__left__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% mult_left_less_imp_less
tff(fact_1666_mult__le__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A4: A,C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ) ).
% mult_le_cancel_right
tff(fact_1667_mult__le__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ) ).
% mult_le_cancel_left
tff(fact_1668_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).
% sum_squares_ge_zero
tff(fact_1669_mult__left__le,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),A4) ) ) ) ).
% mult_left_le
tff(fact_1670_mult__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),one_one(A)) ) ) ) ) ).
% mult_le_one
tff(fact_1671_mult__right__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X) ) ) ) ) ).
% mult_right_le_one_le
tff(fact_1672_mult__left__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X) ) ) ) ) ).
% mult_left_le_one_le
tff(fact_1673_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).
% not_sum_squares_lt_zero
tff(fact_1674_zero__less__two,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).
% zero_less_two
tff(fact_1675_card__eq__0__iff,axiom,
! [A: $tType,A3: set(A)] :
( ( finite_card(A,A3) = zero_zero(nat) )
<=> ( ( A3 = bot_bot(set(A)) )
| ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).
% card_eq_0_iff
tff(fact_1676_one__less__mult,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ) ) ).
% one_less_mult
tff(fact_1677_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) ) ) ).
% n_less_m_mult_n
tff(fact_1678_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M)) ) ) ).
% n_less_n_mult_m
tff(fact_1679_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ).
% nat_mult_le_cancel1
tff(fact_1680_mult__le__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A)) ) ) ) ) ).
% mult_le_cancel_left1
tff(fact_1681_mult__le__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4) ) ) ) ) ).
% mult_le_cancel_left2
tff(fact_1682_mult__le__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A)) ) ) ) ) ).
% mult_le_cancel_right1
tff(fact_1683_mult__le__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4) ) ) ) ) ).
% mult_le_cancel_right2
tff(fact_1684_mult__less__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A)) ) ) ) ) ).
% mult_less_cancel_left1
tff(fact_1685_mult__less__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4) ) ) ) ) ).
% mult_less_cancel_left2
tff(fact_1686_mult__less__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A)) ) ) ) ) ).
% mult_less_cancel_right1
tff(fact_1687_mult__less__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4) ) ) ) ) ).
% mult_less_cancel_right2
tff(fact_1688_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X: A,A4: A,Y: A,U: A,V: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A4) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_1689_card__gt__0__iff,axiom,
! [A: $tType,A3: set(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,A3))
<=> ( ( A3 != bot_bot(set(A)) )
& aa(set(A),$o,finite_finite2(A),A3) ) ) ).
% card_gt_0_iff
tff(fact_1690_card__1__singleton__iff,axiom,
! [A: $tType,A3: set(A)] :
( ( finite_card(A,A3) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ? [X2: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).
% card_1_singleton_iff
tff(fact_1691_card__eq__SucD,axiom,
! [A: $tType,A3: set(A),K: nat] :
( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
=> ? [B2: A,B6: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B6) )
& ~ member(A,B2,B6)
& ( finite_card(A,B6) = K )
& ( ( K = zero_zero(nat) )
=> ( B6 = bot_bot(set(A)) ) ) ) ) ).
% card_eq_SucD
tff(fact_1692_card__Suc__eq,axiom,
! [A: $tType,A3: set(A),K: nat] :
( ( finite_card(A,A3) = aa(nat,nat,suc,K) )
<=> ? [B7: A,B10: set(A)] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B10) )
& ~ member(A,B7,B10)
& ( finite_card(A,B10) = K )
& ( ( K = zero_zero(nat) )
=> ( B10 = bot_bot(set(A)) ) ) ) ) ).
% card_Suc_eq
tff(fact_1693_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),minus_minus(nat,K,aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).
% nz_le_conv_less
tff(fact_1694_mult__eq__if,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,M,one_one(nat))),N2))) ).
% mult_eq_if
tff(fact_1695_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X: A,A4: A,Y: A,U: A,V: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A4) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_1696_inj__mult__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [A4: A] :
( inj_on(A,A,aa(A,fun(A,A),times_times(A),A4),top_top(set(A)))
<=> ( A4 != zero_zero(A) ) ) ) ).
% inj_mult_left
tff(fact_1697_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( ! [Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),X)),Y) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% field_le_mult_one_interval
tff(fact_1698_size__prod__simp,axiom,
! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : aa(product_prod(A,B),nat,basic_BNF_size_prod(A,B,F2,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(A,nat,F2,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).
% size_prod_simp
tff(fact_1699_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_1700_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_1701_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mult_le_cancel_iff2
tff(fact_1702_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mult_le_cancel_iff1
tff(fact_1703_divides__aux__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q3: A,R2: A] :
( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2))
<=> ( R2 = zero_zero(A) ) ) ) ).
% divides_aux_eq
tff(fact_1704_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),nat),comp(product_prod(C,D),nat,product_prod(A,B),basic_BNF_size_prod(C,D,F2,Fa)),product_map_prod(A,C,B,D,G,Ga)) = basic_BNF_size_prod(A,B,aa(fun(A,C),fun(A,nat),comp(C,nat,A,F2),G),aa(fun(B,D),fun(B,nat),comp(D,nat,B,Fa),Ga)) ).
% prod.size_gen_o_map
tff(fact_1705_divides__aux__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Qr: product_prod(A,A)] :
( unique5940410009612947441es_aux(A,Qr)
<=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).
% divides_aux_def
tff(fact_1706_mult__less__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).
% mult_less_iff1
tff(fact_1707_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_1708_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,zero_zero(A)) = one_one(A) ) ) ).
% dbl_inc_simps(2)
tff(fact_1709_enumerate__Suc_H,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N2: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = infini527867602293511546merate(A,minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),infini527867602293511546merate(A,S,zero_zero(nat))),bot_bot(set(A)))),N2) ) ).
% enumerate_Suc'
tff(fact_1710_power__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B3: A,M: nat,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M) ) ) ) ) ).
% power_decreasing_iff
tff(fact_1711_dvd__partition,axiom,
! [A: $tType,C6: set(set(A)),K: nat] :
( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6))
=> ( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> dvd_dvd(nat,K,finite_card(A,X3)) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,C6)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> dvd_dvd(nat,K,finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6))) ) ) ) ).
% dvd_partition
tff(fact_1712_cInf__le__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A3)
=> ( condit1013018076250108175_below(A,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).
% cInf_le_cSup
tff(fact_1713_sum__le__card__Max,axiom,
! [A: $tType,A3: set(A),F2: fun(A,nat)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,A3)),lattic643756798349783984er_Max(nat,aa(set(A),set(nat),image2(A,nat,F2),A3)))) ) ).
% sum_le_card_Max
tff(fact_1714_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,A,gbinomial(A,A4),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),minus_minus(nat,K,one_one(nat)))),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),K)) ) ) ) ).
% gbinomial_reduce_nat
tff(fact_1715_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N2) = one_one(A) ) ).
% power_one
tff(fact_1716_bdd__below__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit1013018076250108175_below(A,bot_bot(set(A))) ) ).
% bdd_below_empty
tff(fact_1717_bdd__below__insert,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A4: A,A3: set(A)] :
( condit1013018076250108175_below(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))
<=> condit1013018076250108175_below(A,A3) ) ) ).
% bdd_below_insert
tff(fact_1718_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 != zero_zero(A) )
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4))
<=> dvd_dvd(A,B3,C2) ) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_1719_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 != zero_zero(A) )
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2))
<=> dvd_dvd(A,B3,C2) ) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_1720_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [A4: A,C2: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( ( C2 = zero_zero(A) )
| dvd_dvd(A,A4,B3) ) ) ) ).
% dvd_mult_cancel_right
tff(fact_1721_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [C2: A,A4: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( ( C2 = zero_zero(A) )
| dvd_dvd(A,A4,B3) ) ) ) ).
% dvd_mult_cancel_left
tff(fact_1722_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A4: A,C2: A,B3: A] :
( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),B3))
<=> dvd_dvd(A,A4,B3) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_1723_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)))
<=> dvd_dvd(A,A4,B3) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_1724_unit__prod,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( dvd_dvd(A,B3,one_one(A))
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),one_one(A)) ) ) ) ).
% unit_prod
tff(fact_1725_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,M: nat,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2) )
<=> ( M = N2 ) ) ) ) ).
% power_inject_exp
tff(fact_1726_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> ( ( K = zero_zero(nat) )
| dvd_dvd(nat,M,N2) ) ) ).
% nat_mult_dvd_cancel_disj
tff(fact_1727_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A4: A] : aa(nat,A,gbinomial(A,A4),zero_zero(nat)) = one_one(A) ) ).
% gbinomial_0(1)
tff(fact_1728_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B3: A,X: nat,Y: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).
% power_strict_increasing_iff
tff(fact_1729_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B3: A,M: nat,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M) ) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_1730_power__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B3: A,X: nat,Y: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).
% power_increasing_iff
tff(fact_1731_is__unit__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,N2: nat] :
( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2),one_one(A))
<=> ( dvd_dvd(A,A4,one_one(A))
| ( N2 = zero_zero(nat) ) ) ) ) ).
% is_unit_power_iff
tff(fact_1732_power__commuting__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N2: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) ) ) ) ).
% power_commuting_commutes
tff(fact_1733_power__mult__distrib,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N2)) ) ).
% power_mult_distrib
tff(fact_1734_power__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ).
% power_commutes
tff(fact_1735_dvd__triv__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A] : dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)) ) ).
% dvd_triv_right
tff(fact_1736_dvd__mult__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2)
=> dvd_dvd(A,B3,C2) ) ) ).
% dvd_mult_right
tff(fact_1737_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( dvd_dvd(A,A4,B3)
=> ( dvd_dvd(A,C2,D3)
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ).
% mult_dvd_mono
tff(fact_1738_dvd__triv__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A] : dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% dvd_triv_left
tff(fact_1739_dvd__mult__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2)
=> dvd_dvd(A,A4,C2) ) ) ).
% dvd_mult_left
tff(fact_1740_dvd__mult2,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,B3)
=> dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).
% dvd_mult2
tff(fact_1741_dvd__mult,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A,C2: A,B3: A] :
( dvd_dvd(A,A4,C2)
=> dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).
% dvd_mult
tff(fact_1742_dvd__def,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,A4)
<=> ? [K5: A] : A4 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K5) ) ) ).
% dvd_def
tff(fact_1743_dvdI,axiom,
! [A: $tType] :
( dvd(A)
=> ! [A4: A,B3: A,K: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K) )
=> dvd_dvd(A,B3,A4) ) ) ).
% dvdI
tff(fact_1744_dvdE,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,A4)
=> ~ ! [K3: A] : A4 != aa(A,A,aa(A,fun(A,A),times_times(A),B3),K3) ) ) ).
% dvdE
tff(fact_1745_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,B3)
=> ( dvd_dvd(A,B3,one_one(A))
=> dvd_dvd(A,A4,one_one(A)) ) ) ) ).
% dvd_unit_imp_unit
tff(fact_1746_unit__imp__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> dvd_dvd(A,B3,A4) ) ) ).
% unit_imp_dvd
tff(fact_1747_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: A] : dvd_dvd(A,one_one(A),A4) ) ).
% one_dvd
tff(fact_1748_power__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,M: nat,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M)),N2) ) ).
% power_mult
tff(fact_1749_dvd__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,M: nat,N2: nat] :
( ( X != zero_zero(A) )
=> ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2))
<=> ( dvd_dvd(A,X,one_one(A))
| aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ) ) ).
% dvd_power_iff
tff(fact_1750_dvd__power,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N2: nat,X: A] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
| ( X = one_one(A) ) )
=> dvd_dvd(A,X,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) ) ) ).
% dvd_power
tff(fact_1751_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A4),K)),aa(nat,A,gbinomial(A,A4),aa(nat,nat,suc,K))) ) ).
% gbinomial_Suc_Suc
tff(fact_1752_one__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ).
% one_le_power
tff(fact_1753_not__is__unit__0,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ~ dvd_dvd(A,zero_zero(A),one_one(A)) ) ).
% not_is_unit_0
tff(fact_1754_left__right__inverse__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N2: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N2)) = one_one(A) ) ) ) ).
% left_right_inverse_power
tff(fact_1755_power__Suc2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),A4) ) ).
% power_Suc2
tff(fact_1756_power__Suc,axiom,
! [A: $tType] :
( power(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ).
% power_Suc
tff(fact_1757_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4) )
<=> ( B3 = C2 ) ) ) ) ).
% unit_mult_right_cancel
tff(fact_1758_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) )
<=> ( B3 = C2 ) ) ) ) ).
% unit_mult_left_cancel
tff(fact_1759_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2)
<=> dvd_dvd(A,B3,C2) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_1760_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> dvd_dvd(A,A4,C2) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_1761_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2)
<=> dvd_dvd(A,A4,C2) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_1762_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> dvd_dvd(A,A4,C2) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_1763_is__unit__mult__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),one_one(A))
<=> ( dvd_dvd(A,A4,one_one(A))
& dvd_dvd(A,B3,one_one(A)) ) ) ) ).
% is_unit_mult_iff
tff(fact_1764_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),zero_zero(nat)) = one_one(A) ) ).
% power_0
tff(fact_1765_power__add,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,M: nat,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ).
% power_add
tff(fact_1766_gbinomial__addition__formula,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,A4),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),K)) ) ).
% gbinomial_addition_formula
tff(fact_1767_power__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),one_one(A)) ) ) ) ).
% power_le_one
tff(fact_1768_power__less__power__Suc,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2))) ) ) ).
% power_less_power_Suc
tff(fact_1769_power__gt1__lemma,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2))) ) ) ).
% power_gt1_lemma
tff(fact_1770_unity__coeff__ex,axiom,
! [A: $tType] :
( ( dvd(A)
& semiring_0(A) )
=> ! [P: fun(A,$o),L: A] :
( ? [X2: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X2))
<=> ? [X2: A] :
( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),zero_zero(A)))
& aa(A,$o,P,X2) ) ) ) ).
% unity_coeff_ex
tff(fact_1771_power__0__left,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N2) = $ite(N2 = zero_zero(nat),one_one(A),zero_zero(A)) ) ).
% power_0_left
tff(fact_1772_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ~ ( ( A4 != zero_zero(A) )
=> ! [C3: A] : B3 != aa(A,A,aa(A,fun(A,A),times_times(A),A4),C3) ) ) ) ).
% unit_dvdE
tff(fact_1773_power__gt1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,N2))) ) ) ).
% power_gt1
tff(fact_1774_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,N: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N)) ) ) ) ).
% power_strict_increasing
tff(fact_1775_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,M: nat,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2) ) ) ) ).
% power_less_imp_less_exp
tff(fact_1776_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,N: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N)) ) ) ) ).
% power_increasing
tff(fact_1777_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T5: A] :
( dvd_dvd(A,D3,D4)
=> ! [X4: A,K4: A] :
( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T5))
<=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T5)) ) ) ) ).
% inf_period(3)
tff(fact_1778_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T5: A] :
( dvd_dvd(A,D3,D4)
=> ! [X4: A,K4: A] :
( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T5))
<=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,X4,aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T5)) ) ) ) ).
% inf_period(4)
tff(fact_1779_cInf__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B5: set(A),A3: set(A)] :
( ( B5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,A3)
=> ( ! [B2: A] :
( member(A,B2,B5)
=> ? [X4: A] :
( member(A,X4,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B2) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ).
% cInf_mono
tff(fact_1780_le__cInf__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A4: A] :
( ( S != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(set(A),A,complete_Inf_Inf(A),S))
<=> ! [X2: A] :
( member(A,X2,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X2) ) ) ) ) ) ).
% le_cInf_iff
tff(fact_1781_cInf__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X5)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y)
<=> ? [X2: A] :
( member(A,X2,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y) ) ) ) ) ) ).
% cInf_less_iff
tff(fact_1782_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> dvd_dvd(nat,M,N2) ) ) ).
% dvd_mult_cancel
tff(fact_1783_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))
<=> dvd_dvd(nat,M,N2) ) ) ).
% nat_mult_dvd_cancel1
tff(fact_1784_power__Suc__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ) ).
% power_Suc_less
tff(fact_1785_power__Suc__le__self,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,N2))),A4) ) ) ) ).
% power_Suc_le_self
tff(fact_1786_power__Suc__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,N2))),one_one(A)) ) ) ) ).
% power_Suc_less_one
tff(fact_1787_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,N: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ) ) ).
% power_strict_decreasing
tff(fact_1788_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,N: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ) ) ).
% power_decreasing
tff(fact_1789_power__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,M: nat,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ) ).
% power_le_imp_le_exp
tff(fact_1790_self__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ) ).
% self_le_power
tff(fact_1791_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ) ) ).
% one_less_power
tff(fact_1792_cINF__mono,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [B5: set(A),F2: fun(C,B),A3: set(C),G: fun(A,B)] :
( ( B5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F2),A3))
=> ( ! [M3: A] :
( member(A,M3,B5)
=> ? [X4: C] :
( member(C,X4,A3)
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F2,X4)),aa(A,B,G,M3)) ) )
=> aa(B,$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,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B5))) ) ) ) ) ).
% cINF_mono
tff(fact_1793_le__cINF__iff,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),U: B] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X2)) ) ) ) ) ) ).
% le_cINF_iff
tff(fact_1794_cInf__superset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B5)),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ) ) ) ).
% cInf_superset_mono
tff(fact_1795_cInf__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X5)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).
% cInf_insert
tff(fact_1796_cInf__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A4: A] :
( condit1013018076250108175_below(A,X5)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),X5)) = $ite(X5 = bot_bot(set(A)),A4,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),X5))) ) ) ) ).
% cInf_insert_If
tff(fact_1797_cInf__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,B5)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ) ) ).
% cInf_union_distrib
tff(fact_1798_dbl__inc__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).
% dbl_inc_def
tff(fact_1799_dvd__mult__cancel1,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
=> ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2),M)
<=> ( N2 = one_one(nat) ) ) ) ).
% dvd_mult_cancel1
tff(fact_1800_dvd__mult__cancel2,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
=> ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M),M)
<=> ( N2 = one_one(nat) ) ) ) ).
% dvd_mult_cancel2
tff(fact_1801_dvd__minus__add,axiom,
! [Q3: nat,N2: nat,R2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
=> ( dvd_dvd(nat,M,minus_minus(nat,N2,Q3))
<=> dvd_dvd(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M),Q3))) ) ) ) ).
% dvd_minus_add
tff(fact_1802_cINF__superset__mono,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),G: fun(A,B),B5: set(A),F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),B5))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [X3: A] :
( member(A,X3,B5)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X3)),aa(A,B,F2,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ) ).
% cINF_superset_mono
tff(fact_1803_less__eq__cInf__inter,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A3: set(A),B5: set(A)] :
( condit1013018076250108175_below(A,A3)
=> ( condit1013018076250108175_below(A,B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))) ) ) ) ) ).
% less_eq_cInf_inter
tff(fact_1804_cINF__insert,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),A4: A] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A4)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).
% cINF_insert
tff(fact_1805_power__eq__if,axiom,
! [A: $tType] :
( power(A)
=> ! [P2: A,M: nat] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),minus_minus(nat,M,one_one(nat))))) ) ).
% power_eq_if
tff(fact_1806_power__minus__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N2: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),minus_minus(nat,N2,one_one(nat)))),A4) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2) ) ) ) ).
% power_minus_mult
tff(fact_1807_cINF__union,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),B5))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B5))) ) ) ) ) ) ) ).
% cINF_union
tff(fact_1808_sum_Oinsert,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A3: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).
% sum.insert
tff(fact_1809_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty
tff(fact_1810_member__le__sum,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [I2: A,A3: set(A),F2: fun(A,B)] :
( member(A,I2,A3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I2),bot_bot(set(A)))))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ) ).
% member_le_sum
tff(fact_1811_sum_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A3: set(A),B5: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B5)) ) ) ) ) ) ).
% sum.union_disjoint
tff(fact_1812_sum__diff1,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [A3: set(A),F2: fun(A,B),A4: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = $ite(member(A,A4,A3),minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3),aa(A,B,F2,A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ).
% sum_diff1
tff(fact_1813_sum_Oremove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A3: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% sum.remove
tff(fact_1814_sum_Oinsert__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A3: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% sum.insert_remove
tff(fact_1815_sum__strict__mono,axiom,
! [B: $tType,A: $tType] :
( strict7427464778891057005id_add(B)
=> ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).
% sum_strict_mono
tff(fact_1816_sum_Oinsert__if,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A3: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(member(A,X,A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3))) ) ) ) ).
% sum.insert_if
tff(fact_1817_sum__pos,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [I: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I)) ) ) ) ) ).
% sum_pos
tff(fact_1818_sum_OUnion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [C6: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> aa(set(A),$o,finite_finite2(A),X3) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,C6)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),C6) ) ) ) ) ).
% sum.Union_disjoint
tff(fact_1819_sum__diff1__nat,axiom,
! [A: $tType,F2: fun(A,nat),A3: set(A),A4: A] :
aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = $ite(member(A,A4,A3),minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3),aa(A,nat,F2,A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ).
% sum_diff1_nat
tff(fact_1820_bezout__add__strong__nat,axiom,
! [A4: nat,B3: nat] :
( ( A4 != zero_zero(nat) )
=> ? [D2: nat,X3: nat,Y2: nat] :
( dvd_dvd(nat,D2,A4)
& dvd_dvd(nat,D2,B3)
& ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),D2) ) ) ) ).
% bezout_add_strong_nat
tff(fact_1821_bezout1__nat,axiom,
! [A4: nat,B3: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( dvd_dvd(nat,D2,A4)
& dvd_dvd(nat,D2,B3)
& ( ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)) = D2 )
| ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y2)) = D2 ) ) ) ).
% bezout1_nat
tff(fact_1822_bezout__add__nat,axiom,
! [A4: nat,B3: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( dvd_dvd(nat,D2,A4)
& dvd_dvd(nat,D2,B3)
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),D2) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y2)),D2) ) ) ) ).
% bezout_add_nat
tff(fact_1823_bezout__lemma__nat,axiom,
! [D3: nat,A4: nat,B3: nat,X: nat,Y: nat] :
( dvd_dvd(nat,D3,A4)
=> ( dvd_dvd(nat,D3,B3)
=> ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y)),D3) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y)),D3) ) )
=> ? [X3: nat,Y2: nat] :
( dvd_dvd(nat,D3,A4)
& dvd_dvd(nat,D3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)),Y2)),D3) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y2)),D3) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_1824_dvd__productE,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [P2: A,A4: A,B3: A] :
( dvd_dvd(A,P2,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> ~ ! [X3: A,Y2: A] :
( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y2) )
=> ( dvd_dvd(A,X3,A4)
=> ~ dvd_dvd(A,Y2,B3) ) ) ) ) ).
% dvd_productE
tff(fact_1825_division__decomp,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
=> ? [B11: A,C9: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),times_times(A),B11),C9) )
& dvd_dvd(A,B11,B3)
& dvd_dvd(A,C9,C2) ) ) ) ).
% division_decomp
tff(fact_1826_card__Min__le__sum,axiom,
! [A: $tType,A3: set(A),F2: fun(A,nat)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,A3)),lattic643756798350308766er_Min(nat,aa(set(A),set(nat),image2(A,nat,F2),A3)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ) ).
% card_Min_le_sum
tff(fact_1827_Min__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% Min_singleton
tff(fact_1828_Min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A3))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X2) ) ) ) ) ) ).
% Min.bounded_iff
tff(fact_1829_Min__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),lattic643756798350308766er_Min(A,A3))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X2) ) ) ) ) ) ).
% Min_gr_iff
tff(fact_1830_Min__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> member(A,lattic643756798350308766er_Min(A,A3),A3) ) ) ) ).
% Min_in
tff(fact_1831_Min__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),M: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ( lattic643756798350308766er_Min(A,A3) = M )
<=> ( member(A,M,A3)
& ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X2) ) ) ) ) ) ) ).
% Min_eq_iff
tff(fact_1832_Min__le__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,A3)),X)
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),X) ) ) ) ) ) ).
% Min_le_iff
tff(fact_1833_eq__Min__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),M: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ( M = lattic643756798350308766er_Min(A,A3) )
<=> ( member(A,M,A3)
& ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X2) ) ) ) ) ) ) ).
% eq_Min_iff
tff(fact_1834_Min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A3))
=> ! [A12: A] :
( member(A,A12,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A12) ) ) ) ) ) ).
% Min.boundedE
tff(fact_1835_Min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A5) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).
% Min.boundedI
tff(fact_1836_Min__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798350308766er_Min(A,A3)),X)
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),X) ) ) ) ) ) ).
% Min_less_iff
tff(fact_1837_Min__insert2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [B2: A] :
( member(A,B2,A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B2) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = A4 ) ) ) ) ).
% Min_insert2
tff(fact_1838_cInf__eq__Min,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = lattic643756798350308766er_Min(A,X5) ) ) ) ) ).
% cInf_eq_Min
tff(fact_1839_Min__Inf,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).
% Min_Inf
tff(fact_1840_Min__antimono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: set(A),N: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M2),N)
=> ( ( M2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,N)),lattic643756798350308766er_Min(A,M2)) ) ) ) ) ).
% Min_antimono
tff(fact_1841_Min_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,B5)),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).
% Min.subset_imp
tff(fact_1842_minus__Min__eq__Max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).
% minus_Min_eq_Max
tff(fact_1843_minus__Max__eq__Min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).
% minus_Max_eq_Min
tff(fact_1844_Gcd__0__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
( ( gcd_Gcd(A,A3) = zero_zero(A) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).
% Gcd_0_iff
tff(fact_1845_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = zero_zero(A) )
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))
& aa(set(A),$o,finite_finite2(A),A3) ) ) ) ).
% Gcd_fin_0_iff
tff(fact_1846_sum__bounded__above__strict,axiom,
! [A: $tType,B: $tType] :
( ( ordere8940638589300402666id_add(B)
& semiring_1(B) )
=> ! [A3: set(A),F2: fun(A,B),K2: B] :
( ! [I3: A] :
( member(A,I3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),K2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,A3))
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K2)) ) ) ) ).
% sum_bounded_above_strict
tff(fact_1847_prod__le__power,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(B)
=> ! [A3: set(A),F2: fun(A,B),N2: B,K: nat] :
( ! [I3: A] :
( member(A,I3,A3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),N2) ) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A3)),K)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),N2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),N2),K)) ) ) ) ) ).
% prod_le_power
tff(fact_1848_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [B5: set(A),A3: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),B5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [B2: A] :
( member(A,B2,minus_minus(set(A),B5,A3))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B2)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A5)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B5)) ) ) ) ) ) ).
% prod_mono2
tff(fact_1849_neg__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,uminus_uminus(A),B3) )
<=> ( A4 = B3 ) ) ) ).
% neg_equal_iff_equal
tff(fact_1850_add_Oinverse__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A4)) = A4 ) ).
% add.inverse_inverse
tff(fact_1851_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B5))
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ).
% Compl_subset_Compl_iff
tff(fact_1852_Compl__anti__mono,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).
% Compl_anti_mono
tff(fact_1853_neg__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% neg_le_iff_le
tff(fact_1854_neg__equal__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( ( aa(A,A,uminus_uminus(A),A4) = A4 )
<=> ( A4 = zero_zero(A) ) ) ) ).
% neg_equal_zero
tff(fact_1855_equal__neg__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( ( A4 = aa(A,A,uminus_uminus(A),A4) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% equal_neg_zero
tff(fact_1856_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] :
( ( aa(A,A,uminus_uminus(A),A4) = zero_zero(A) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% neg_equal_0_iff_equal
tff(fact_1857_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] :
( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A4) )
<=> ( zero_zero(A) = A4 ) ) ) ).
% neg_0_equal_iff_equal
tff(fact_1858_add_Oinverse__neutral,axiom,
! [A: $tType] :
( group_add(A)
=> ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).
% add.inverse_neutral
tff(fact_1859_neg__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% neg_less_iff_less
tff(fact_1860_minus__add__distrib,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),aa(A,A,uminus_uminus(A),B3)) ) ).
% minus_add_distrib
tff(fact_1861_minus__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) = B3 ) ).
% minus_add_cancel
tff(fact_1862_add__minus__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),B3)) = B3 ) ).
% add_minus_cancel
tff(fact_1863_mult__minus__left,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A4)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% mult_minus_left
tff(fact_1864_minus__mult__minus,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A4)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) ) ).
% minus_mult_minus
tff(fact_1865_mult__minus__right,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% mult_minus_right
tff(fact_1866_minus__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,uminus_uminus(A),minus_minus(A,A4,B3)) = minus_minus(A,B3,A4) ) ).
% minus_diff_eq
tff(fact_1867_Compl__disjoint,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = bot_bot(set(A)) ).
% Compl_disjoint
tff(fact_1868_Compl__disjoint2,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = bot_bot(set(A)) ).
% Compl_disjoint2
tff(fact_1869_inter__compl__diff__conv,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = minus_minus(set(A),A3,B5) ).
% inter_compl_diff_conv
tff(fact_1870_Diff__Compl,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : minus_minus(set(A),A3,aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) ).
% Diff_Compl
tff(fact_1871_Compl__Diff__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),minus_minus(set(A),A3,B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),B5) ).
% Compl_Diff_eq
tff(fact_1872_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% neg_0_le_iff_le
tff(fact_1873_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% neg_le_0_iff_le
tff(fact_1874_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% less_eq_neg_nonpos
tff(fact_1875_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% neg_less_eq_nonneg
tff(fact_1876_less__neg__neg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% less_neg_neg
tff(fact_1877_neg__less__pos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A4)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4) ) ) ).
% neg_less_pos
tff(fact_1878_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% neg_0_less_iff_less
tff(fact_1879_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4) ) ) ).
% neg_less_0_iff_less
tff(fact_1880_ab__left__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),A4) = zero_zero(A) ) ).
% ab_left_minus
tff(fact_1881_add_Oright__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,uminus_uminus(A),A4)) = zero_zero(A) ) ).
% add.right_inverse
tff(fact_1882_diff__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A] : minus_minus(A,zero_zero(A),A4) = aa(A,A,uminus_uminus(A),A4) ) ).
% diff_0
tff(fact_1883_diff__minus__eq__add,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) ) ).
% diff_minus_eq_add
tff(fact_1884_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),B3) = minus_minus(A,B3,A4) ) ).
% uminus_add_conv_diff
tff(fact_1885_mult__minus1,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z2) = aa(A,A,uminus_uminus(A),Z2) ) ).
% mult_minus1
tff(fact_1886_mult__minus1__right,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z2) ) ).
% mult_minus1_right
tff(fact_1887_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).
% boolean_algebra.compl_one
tff(fact_1888_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).
% boolean_algebra.compl_zero
tff(fact_1889_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).
% boolean_algebra.conj_cancel_right
tff(fact_1890_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).
% boolean_algebra.conj_cancel_left
tff(fact_1891_inf__compl__bot__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).
% inf_compl_bot_right
tff(fact_1892_inf__compl__bot__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y)) = bot_bot(A) ) ).
% inf_compl_bot_left2
tff(fact_1893_inf__compl__bot__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = bot_bot(A) ) ).
% inf_compl_bot_left1
tff(fact_1894_subset__Compl__singleton,axiom,
! [A: $tType,A3: set(A),B3: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
<=> ~ member(A,B3,A3) ) ).
% subset_Compl_singleton
tff(fact_1895_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty
tff(fact_1896_prod_Oinfinite,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).
% prod.infinite
tff(fact_1897_of__nat__mult,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M: nat,N2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).
% of_nat_mult
tff(fact_1898_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) ) ) ).
% of_nat_1
tff(fact_1899_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: nat] :
( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N2) )
<=> ( N2 = one_one(nat) ) ) ) ).
% of_nat_1_eq_iff
tff(fact_1900_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),N2) = one_one(A) )
<=> ( N2 = one_one(nat) ) ) ) ).
% of_nat_eq_1_iff
tff(fact_1901_Gcd__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_empty
tff(fact_1902_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_1903_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_fin.empty
tff(fact_1904_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_1905_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
( dvd_dvd(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A3),one_one(A))
<=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = one_one(A) ) ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_1906_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% dbl_inc_simps(4)
tff(fact_1907_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% add_neg_numeral_special(7)
tff(fact_1908_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).
% add_neg_numeral_special(8)
tff(fact_1909_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% diff_numeral_special(12)
tff(fact_1910_minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)) = one_one(A) ) ).
% minus_one_mult_self
tff(fact_1911_left__minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),A4)) = A4 ) ).
% left_minus_one_mult_self
tff(fact_1912_of__nat__Suc,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).
% of_nat_Suc
tff(fact_1913_prod_Oinsert,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).
% prod.insert
tff(fact_1914_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% dbl_dec_simps(2)
tff(fact_1915_minus__equation__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,uminus_uminus(A),A4) = B3 )
<=> ( aa(A,A,uminus_uminus(A),B3) = A4 ) ) ) ).
% minus_equation_iff
tff(fact_1916_equation__minus__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( A4 = aa(A,A,uminus_uminus(A),B3) )
<=> ( B3 = aa(A,A,uminus_uminus(A),A4) ) ) ) ).
% equation_minus_iff
tff(fact_1917_prod_Oneutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).
% prod.neutral
tff(fact_1918_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A3: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) != one_one(A) )
=> ~ ! [A5: B] :
( member(B,A5,A3)
=> ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_1919_mult__of__nat__commute,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).
% mult_of_nat_commute
tff(fact_1920_le__imp__neg__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A4)) ) ) ).
% le_imp_neg_le
tff(fact_1921_minus__le__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),A4) ) ) ).
% minus_le_iff
tff(fact_1922_le__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,uminus_uminus(A),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,uminus_uminus(A),A4)) ) ) ).
% le_minus_iff
tff(fact_1923_minus__less__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A4)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),A4) ) ) ).
% minus_less_iff
tff(fact_1924_less__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,uminus_uminus(A),B3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,uminus_uminus(A),A4)) ) ) ).
% less_minus_iff
tff(fact_1925_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A4)) ) ).
% add.inverse_distrib_swap
tff(fact_1926_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,K: A,A4: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A4) )
=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A4)) ) ) ) ).
% group_cancel.neg1
tff(fact_1927_square__eq__iff,axiom,
! [A: $tType] :
( idom(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),B3) )
<=> ( ( A4 = B3 )
| ( A4 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).
% square_eq_iff
tff(fact_1928_minus__mult__commute,axiom,
! [A: $tType] :
( ring(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A4)),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),B3)) ) ).
% minus_mult_commute
tff(fact_1929_one__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% one_neq_neg_one
tff(fact_1930_minus__diff__commute,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B3: A,A4: A] : minus_minus(A,aa(A,A,uminus_uminus(A),B3),A4) = minus_minus(A,aa(A,A,uminus_uminus(A),A4),B3) ) ).
% minus_diff_commute
tff(fact_1931_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A3)) ).
% vimage_Compl
tff(fact_1932_Gcd__1,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
( member(A,one_one(A),A3)
=> ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).
% Gcd_1
tff(fact_1933_minus__assn__def,axiom,
! [A4: assn,B3: assn] : minus_minus(assn,A4,B3) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),aa(assn,assn,uminus_uminus(assn),B3)) ).
% minus_assn_def
tff(fact_1934_prod__ge__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(B)
=> ! [A3: set(A),F2: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).
% prod_ge_1
tff(fact_1935_prod_OUnion__comp,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [B5: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,B5)
=> aa(set(A),$o,finite_finite2(A),X3) )
=> ( ! [A15: set(A)] :
( member(set(A),A15,B5)
=> ! [A24: set(A)] :
( member(set(A),A24,B5)
=> ( ( A15 != A24 )
=> ! [X3: A] :
( member(A,X3,A15)
=> ( member(A,X3,A24)
=> ( aa(A,B,G,X3) = one_one(B) ) ) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B5) ) ) ) ) ).
% prod.Union_comp
tff(fact_1936_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,uminus_uminus(A),A4) = B3 )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = zero_zero(A) ) ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_1937_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( A4 = aa(A,A,uminus_uminus(A),B3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = zero_zero(A) ) ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_1938_add_Oinverse__unique,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),A4) = B3 ) ) ) ).
% add.inverse_unique
tff(fact_1939_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),A4) = zero_zero(A) ) ).
% ab_group_add_class.ab_left_minus
tff(fact_1940_add__eq__0__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = zero_zero(A) )
<=> ( B3 = aa(A,A,uminus_uminus(A),A4) ) ) ) ).
% add_eq_0_iff
tff(fact_1941_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% le_minus_one_simps(4)
tff(fact_1942_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).
% le_minus_one_simps(2)
tff(fact_1943_zero__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ( zero_zero(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% zero_neq_neg_one
tff(fact_1944_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).
% less_minus_one_simps(2)
tff(fact_1945_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% less_minus_one_simps(4)
tff(fact_1946_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,uminus_uminus(A),B3)) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1947_diff__conv__add__uminus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,uminus_uminus(A),B3)) ) ).
% diff_conv_add_uminus
tff(fact_1948_group__cancel_Osub2,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B5: A,K: A,B3: A,A4: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B3) )
=> ( minus_minus(A,A4,B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),minus_minus(A,A4,B3)) ) ) ) ).
% group_cancel.sub2
tff(fact_1949_square__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [X: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
<=> ( ( X = one_one(A) )
| ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% square_eq_1_iff
tff(fact_1950_inf__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A4)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B3)) = bot_bot(A) ) ).
% inf_cancel_left2
tff(fact_1951_inf__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A4)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B3)) = bot_bot(A) ) ).
% inf_cancel_left1
tff(fact_1952_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3))
<=> ( A3 = bot_bot(set(A)) ) ) ).
% subset_Compl_self_eq
tff(fact_1953_Compl__UNIV__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).
% Compl_UNIV_eq
tff(fact_1954_Compl__empty__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).
% Compl_empty_eq
tff(fact_1955_Compl__partition2,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = top_top(set(A)) ).
% Compl_partition2
tff(fact_1956_Compl__partition,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = top_top(set(A)) ).
% Compl_partition
tff(fact_1957_Compl__eq__Diff__UNIV,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = minus_minus(set(A),top_top(set(A)),A3) ).
% Compl_eq_Diff_UNIV
tff(fact_1958_Compl__Int,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).
% Compl_Int
tff(fact_1959_Compl__Un,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).
% Compl_Un
tff(fact_1960_gbinomial__negated__upper,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,A4),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,minus_minus(A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K),A4),one_one(A))),K)) ) ).
% gbinomial_negated_upper
tff(fact_1961_gbinomial__index__swap,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(nat,A,gbinomial(A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K)),one_one(A))),N2)) ) ).
% gbinomial_index_swap
tff(fact_1962_prod_OUnion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [C6: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> aa(set(A),$o,finite_finite2(A),X3) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,C6)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,C6)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),C6) ) ) ) ) ).
% prod.Union_disjoint
tff(fact_1963_Gcd__remove0__nat,axiom,
! [M2: set(nat)] :
( aa(set(nat),$o,finite_finite2(nat),M2)
=> ( gcd_Gcd(nat,M2) = gcd_Gcd(nat,minus_minus(set(nat),M2,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).
% Gcd_remove0_nat
tff(fact_1964_Gcd__eq__1__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A,A3: set(A)] :
( dvd_dvd(A,A4,one_one(A))
=> ( member(A,A4,A3)
=> ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).
% Gcd_eq_1_I
tff(fact_1965_prod__le__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(B)
=> ! [A3: set(A),F2: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3))
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),one_one(B)) ) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),one_one(B)) ) ) ).
% prod_le_1
tff(fact_1966_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R3: fun(A,fun(A,$o)),S: set(B),Ha: fun(B,A),G: fun(B,A)] :
( aa(A,$o,aa(A,fun(A,$o),R3,one_one(A)),one_one(A))
=> ( ! [X12: A,Y12: A,X23: A,Y23: A] :
( ( aa(A,$o,aa(A,fun(A,$o),R3,X12),X23)
& aa(A,$o,aa(A,fun(A,$o),R3,Y12),Y23) )
=> aa(A,$o,aa(A,fun(A,$o),R3,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
=> ( aa(set(B),$o,finite_finite2(B),S)
=> ( ! [X3: B] :
( member(B,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),R3,aa(B,A,Ha,X3)),aa(B,A,G,X3)) )
=> aa(A,$o,aa(A,fun(A,$o),R3,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Ha),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S)) ) ) ) ) ) ).
% prod.related
tff(fact_1967_prod_Oinsert__if,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(member(A,X,A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3))) ) ) ) ).
% prod.insert_if
tff(fact_1968_prod_Oreindex__bij__witness__not__neutral,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [S4: set(A),T6: set(B),S: set(A),I2: fun(B,A),J2: fun(A,B),T3: set(B),G: fun(A,C),Ha: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),S4)
=> ( aa(set(B),$o,finite_finite2(B),T6)
=> ( ! [A5: A] :
( member(A,A5,minus_minus(set(A),S,S4))
=> ( aa(B,A,I2,aa(A,B,J2,A5)) = A5 ) )
=> ( ! [A5: A] :
( member(A,A5,minus_minus(set(A),S,S4))
=> member(B,aa(A,B,J2,A5),minus_minus(set(B),T3,T6)) )
=> ( ! [B2: B] :
( member(B,B2,minus_minus(set(B),T3,T6))
=> ( aa(A,B,J2,aa(B,A,I2,B2)) = B2 ) )
=> ( ! [B2: B] :
( member(B,B2,minus_minus(set(B),T3,T6))
=> member(A,aa(B,A,I2,B2),minus_minus(set(A),S,S4)) )
=> ( ! [A5: A] :
( member(A,A5,S4)
=> ( aa(A,C,G,A5) = one_one(C) ) )
=> ( ! [B2: B] :
( member(B,B2,T6)
=> ( aa(B,C,Ha,B2) = one_one(C) ) )
=> ( ! [A5: A] :
( member(A,A5,S)
=> ( aa(B,C,Ha,aa(A,B,J2,A5)) = aa(A,C,G,A5) ) )
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Ha),T3) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_1969_gbinomial__minus,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A4)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(nat,A,semiring_1_of_nat(A),K)),one_one(A))),K)) ) ).
% gbinomial_minus
tff(fact_1970_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% le_minus_one_simps(3)
tff(fact_1971_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).
% le_minus_one_simps(1)
tff(fact_1972_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).
% less_minus_one_simps(1)
tff(fact_1973_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% less_minus_one_simps(3)
tff(fact_1974_inf__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).
% inf_shunt
tff(fact_1975_power__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A4)),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ).
% power_minus
tff(fact_1976_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ) ).
% disjoint_eq_subset_Compl
tff(fact_1977_Compl__insert,axiom,
! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = minus_minus(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% Compl_insert
tff(fact_1978_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(B,A)),A3: set(B),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R2,A3)),B5)
<=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,converse(B,A,R2),aa(set(A),set(A),uminus_uminus(set(A)),B5)))) ) ).
% Image_subset_eq
tff(fact_1979_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I: set(A),I2: A,F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( member(A,I2,I)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2))
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I)) ) ) ) ) ) ).
% less_1_prod2
tff(fact_1980_rel__restrict__compl,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),rel_restrict(A,R3,A3)),rel_restrict(A,R3,aa(set(A),set(A),uminus_uminus(set(A)),A3))) = bot_bot(set(product_prod(A,A))) ).
% rel_restrict_compl
tff(fact_1981_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I)) ) ) ) ) ).
% less_1_prod
tff(fact_1982_prod_Oreindex__nontrivial,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [A3: set(A),Ha: fun(A,B),G: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> ( ( X3 != Y2 )
=> ( ( aa(A,B,Ha,X3) = aa(A,B,Ha,Y2) )
=> ( aa(B,C,G,aa(A,B,Ha,X3)) = one_one(C) ) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image2(A,B,Ha),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),Ha)),A3) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_1983_prod_Osubset__diff,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [B5: set(A),A3: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B5)) ) ) ) ) ).
% prod.subset_diff
tff(fact_1984_prod_Osame__carrier,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [C6: set(A),A3: set(A),B5: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> ( ! [A5: A] :
( member(A,A5,minus_minus(set(A),C6,A3))
=> ( aa(A,B,G,A5) = one_one(B) ) )
=> ( ! [B2: A] :
( member(A,B2,minus_minus(set(A),C6,B5))
=> ( aa(A,B,Ha,B2) = one_one(B) ) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),B5) )
<=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),C6) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_1985_prod_Osame__carrierI,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [C6: set(A),A3: set(A),B5: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C6)
=> ( ! [A5: A] :
( member(A,A5,minus_minus(set(A),C6,A3))
=> ( aa(A,B,G,A5) = one_one(B) ) )
=> ( ! [B2: A] :
( member(A,B2,minus_minus(set(A),C6,B5))
=> ( aa(A,B,Ha,B2) = one_one(B) ) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),C6) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),B5) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_1986_prod_Omono__neutral__left,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T3: set(A),S: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),T3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) ) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_1987_prod_Omono__neutral__right,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T3: set(A),S: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),T3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_1988_prod_Omono__neutral__cong__left,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T3: set(A),S: set(A),Ha: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),T3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,Ha,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,Ha,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),T3) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_1989_prod_Omono__neutral__cong__right,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T3: set(A),S: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),T3)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,Ha,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),S) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_1990_prod_Ounion__inter,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),B5: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B5)) ) ) ) ) ).
% prod.union_inter
tff(fact_1991_prod_OInt__Diff,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,B5))) ) ) ) ).
% prod.Int_Diff
tff(fact_1992_prod_Omono__neutral__cong,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T3: set(A),S: set(A),Ha: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),T3)
=> ( aa(set(A),$o,finite_finite2(A),S)
=> ( ! [I3: A] :
( member(A,I3,minus_minus(set(A),T3,S))
=> ( aa(A,B,Ha,I3) = one_one(B) ) )
=> ( ! [I3: A] :
( member(A,I3,minus_minus(set(A),S,T3))
=> ( aa(A,B,G,I3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3))
=> ( aa(A,B,G,X3) = aa(A,B,Ha,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),T3) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_1993_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A3) ) ).
% prod.eq_fold
tff(fact_1994_sum__bounded__above,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [A3: set(A),F2: fun(A,B),K2: B] :
( ! [I3: A] :
( member(A,I3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),K2) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K2)) ) ) ).
% sum_bounded_above
tff(fact_1995_sum__bounded__below,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [A3: set(A),K2: B,F2: fun(A,B)] :
( ! [I3: A] :
( member(A,I3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K2),aa(A,B,F2,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3))),K2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).
% sum_bounded_below
tff(fact_1996_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A4),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),K)) ) ).
% gbinomial_absorb_comp
tff(fact_1997_gbinomial__mult__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,gbinomial(A,A4),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A4),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A4),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1
tff(fact_1998_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A4),K)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A4),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A4),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1'
tff(fact_1999_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
=> ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_2000_prod__mono__strict,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(B)
=> ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [I3: A] :
( member(A,I3,A3)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
=> ( ( A3 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).
% prod_mono_strict
tff(fact_2001_prod_Oinsert__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% prod.insert_remove
tff(fact_2002_prod_Oremove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% prod.remove
tff(fact_2003_prod_Ounion__inter__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),B5: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B5)) ) ) ) ) ) ).
% prod.union_inter_neutral
tff(fact_2004_prod_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),B5: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B5)) ) ) ) ) ) ).
% prod.union_disjoint
tff(fact_2005_prod_Ounion__diff2,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),B5: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),B5,A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))) ) ) ) ) ).
% prod.union_diff2
tff(fact_2006_Suc__times__gbinomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(nat,A,gbinomial(A,A4),K)) ) ).
% Suc_times_gbinomial
tff(fact_2007_gbinomial__absorption,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A4),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),K)) ) ).
% gbinomial_absorption
tff(fact_2008_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,M: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A4),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A4),K)),aa(nat,A,gbinomial(A,minus_minus(A,A4,aa(nat,A,semiring_1_of_nat(A),K))),minus_minus(nat,M,K))) ) ) ) ).
% gbinomial_trinomial_revision
tff(fact_2009_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),minus_minus(nat,N2,K)) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2010_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
=> ? [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N3)),X)) ) ) ).
% ex_less_of_nat_mult
tff(fact_2011_gbinomial__absorption_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,A,gbinomial(A,A4),K) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),minus_minus(nat,K,one_one(nat)))) ) ) ) ).
% gbinomial_absorption'
tff(fact_2012_pochhammer__minus,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B3,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).
% pochhammer_minus
tff(fact_2013_pochhammer__minus_H,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B3,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K)) ) ).
% pochhammer_minus'
tff(fact_2014_power__minus_H,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,N2: nat] :
( nO_MATCH(A,A,one_one(A),X)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) ) ) ) ).
% power_minus'
tff(fact_2015_zmult__zless__mono2__lemma,axiom,
! [I2: int,J2: int,K: nat] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J2)
=> ( aa(nat,$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)),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J2)) ) ) ).
% zmult_zless_mono2_lemma
tff(fact_2016_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B5) )
<=> ( A3 = B5 ) ) ).
% Compl_eq_Compl_iff
tff(fact_2017_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set(A)] :
( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
<=> ~ member(A,C2,A3) ) ).
% Compl_iff
tff(fact_2018_ComplI,axiom,
! [A: $tType,C2: A,A3: set(A)] :
( ~ member(A,C2,A3)
=> member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).
% ComplI
tff(fact_2019_times__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [B3: A,C2: A,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,C2)),A4) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4),C2) ) ).
% times_divide_eq_left
tff(fact_2020_divide__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A,C2: A] : divide_divide(A,divide_divide(A,A4,B3),C2) = divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).
% divide_divide_eq_left
tff(fact_2021_divide__divide__eq__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A,C2: A] : divide_divide(A,A4,divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),B3) ) ).
% divide_divide_eq_right
tff(fact_2022_times__divide__eq__right,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ).
% times_divide_eq_right
tff(fact_2023_div__by__1,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A4: A] : divide_divide(A,A4,one_one(A)) = A4 ) ).
% div_by_1
tff(fact_2024_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) = $ite(K = zero_zero(nat),zero_zero(nat),divide_divide(nat,M,N2)) ).
% nat_mult_div_cancel_disj
tff(fact_2025_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),B3) = A4 ) ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_2026_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),A4) = B3 ) ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2027_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A4: A,B3: A] :
divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A4,B3)) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2028_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = divide_divide(A,A4,B3) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2029_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,A4,B3) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2030_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,A4,B3) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2031_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = divide_divide(A,A4,B3) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_2032_div__self,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( divide_divide(A,A4,A4) = one_one(A) ) ) ) ).
% div_self
tff(fact_2033_divide__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] :
( ( divide_divide(A,A4,B3) = one_one(A) )
<=> ( ( B3 != zero_zero(A) )
& ( A4 = B3 ) ) ) ) ).
% divide_eq_1_iff
tff(fact_2034_one__eq__divide__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] :
( ( one_one(A) = divide_divide(A,A4,B3) )
<=> ( ( B3 != zero_zero(A) )
& ( A4 = B3 ) ) ) ) ).
% one_eq_divide_iff
tff(fact_2035_divide__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( divide_divide(A,A4,A4) = one_one(A) ) ) ) ).
% divide_self
tff(fact_2036_divide__self__if,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] :
divide_divide(A,A4,A4) = $ite(A4 = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% divide_self_if
tff(fact_2037_divide__eq__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( ( divide_divide(A,B3,A4) = one_one(A) )
<=> ( ( A4 != zero_zero(A) )
& ( A4 = B3 ) ) ) ) ).
% divide_eq_eq_1
tff(fact_2038_eq__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( ( one_one(A) = divide_divide(A,B3,A4) )
<=> ( ( A4 != zero_zero(A) )
& ( A4 = B3 ) ) ) ) ).
% eq_divide_eq_1
tff(fact_2039_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( ( divide_divide(A,one_one(A),A4) = zero_zero(A) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% one_divide_eq_0_iff
tff(fact_2040_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( ( zero_zero(A) = divide_divide(A,one_one(A),A4) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% zero_eq_1_divide_iff
tff(fact_2041_divide__minus1,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A] : divide_divide(A,X,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).
% divide_minus1
tff(fact_2042_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,B3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,B3,A4)) = B3 ) ) ) ).
% dvd_mult_div_cancel
tff(fact_2043_dvd__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,B3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,A4)),A4) = B3 ) ) ) ).
% dvd_div_mult_self
tff(fact_2044_unit__div__1__div__1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A4)) = A4 ) ) ) ).
% unit_div_1_div_1
tff(fact_2045_unit__div__1__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
=> dvd_dvd(A,divide_divide(A,one_one(A),A4),one_one(A)) ) ) ).
% unit_div_1_unit
tff(fact_2046_unit__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( dvd_dvd(A,B3,one_one(A))
=> dvd_dvd(A,divide_divide(A,A4,B3),one_one(A)) ) ) ) ).
% unit_div
tff(fact_2047_pochhammer__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A] : comm_s3205402744901411588hammer(A,A4,zero_zero(nat)) = one_one(A) ) ).
% pochhammer_0
tff(fact_2048_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% zero_le_divide_1_iff
tff(fact_2049_divide__le__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) ) ) ).
% divide_le_0_1_iff
tff(fact_2050_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4) ) ) ).
% zero_less_divide_1_iff
tff(fact_2051_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% less_divide_eq_1_pos
tff(fact_2052_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ).
% less_divide_eq_1_neg
tff(fact_2053_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A4)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ).
% divide_less_eq_1_pos
tff(fact_2054_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A4)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ).
% divide_less_eq_1_neg
tff(fact_2055_divide__less__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% divide_less_0_1_iff
tff(fact_2056_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,B3,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = divide_divide(A,one_one(A),A4) ) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2057_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2058_unit__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,A4)),A4) = B3 ) ) ) ).
% unit_div_mult_self
tff(fact_2059_unit__mult__div__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,one_one(A),A4)) = divide_divide(A,B3,A4) ) ) ) ).
% unit_mult_div_div
tff(fact_2060_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% le_divide_eq_1_pos
tff(fact_2061_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A4))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ).
% le_divide_eq_1_neg
tff(fact_2062_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A4)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ).
% divide_le_eq_1_pos
tff(fact_2063_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A4)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ).
% divide_le_eq_1_neg
tff(fact_2064_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N2) = one_one(int) )
=> ( ( M = one_one(int) )
| ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
tff(fact_2065_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N2) = one_one(int) )
<=> ( ( ( M = one_one(int) )
& ( N2 = one_one(int) ) )
| ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
& ( N2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
% zmult_eq_1_iff
tff(fact_2066_double__complement,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = A3 ).
% double_complement
tff(fact_2067_ComplD,axiom,
! [A: $tType,C2: A,A3: set(A)] :
( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
=> ~ member(A,C2,A3) ) ).
% ComplD
tff(fact_2068_zdvd__mono,axiom,
! [K: int,M: int,T5: int] :
( ( K != zero_zero(int) )
=> ( dvd_dvd(int,M,T5)
<=> dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),T5)) ) ) ).
% zdvd_mono
tff(fact_2069_zdvd__mult__cancel,axiom,
! [K: int,M: int,N2: int] :
( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),N2))
=> ( ( K != zero_zero(int) )
=> dvd_dvd(int,M,N2) ) ) ).
% zdvd_mult_cancel
tff(fact_2070_zdvd__reduce,axiom,
! [K: int,N2: int,M: int] :
( dvd_dvd(int,K,aa(int,int,aa(int,fun(int,int),plus_plus(int),N2),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)))
<=> dvd_dvd(int,K,N2) ) ).
% zdvd_reduce
tff(fact_2071_zdvd__period,axiom,
! [A4: int,D3: int,X: int,T5: int,C2: int] :
( dvd_dvd(int,A4,D3)
=> ( dvd_dvd(int,A4,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T5))
<=> dvd_dvd(int,A4,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D3))),T5)) ) ) ).
% zdvd_period
tff(fact_2072_unique__quotient__lemma__neg,axiom,
! [B3: int,Q5: int,R5: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R5)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R5)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q5) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_2073_unique__quotient__lemma,axiom,
! [B3: int,Q5: int,R5: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R5)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R5)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R5),B3)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q3) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_2074_zdiv__mono2__neg__lemma,axiom,
! [B3: int,Q3: int,R2: int,B4: int,Q5: int,R5: int] :
( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5)),zero_zero(int))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R5)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q3) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_2075_zdiv__mono2__lemma,axiom,
! [B3: int,Q3: int,R2: int,B4: int,Q5: int,R5: int] :
( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R5),B4)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q5) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_2076_int__div__pos__eq,axiom,
! [A4: int,B3: int,Q3: int,R2: int] :
( ( A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
=> ( divide_divide(int,A4,B3) = Q3 ) ) ) ) ).
% int_div_pos_eq
tff(fact_2077_int__div__neg__eq,axiom,
! [A4: int,B3: int,Q3: int,R2: int] :
( ( A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
=> ( divide_divide(int,A4,B3) = Q3 ) ) ) ) ).
% int_div_neg_eq
tff(fact_2078_q__pos__lemma,axiom,
! [B4: int,Q5: int,R5: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R5),B4)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).
% q_pos_lemma
tff(fact_2079_split__zdiv,axiom,
! [P: fun(int,$o),N2: int,K: int] :
( aa(int,$o,P,divide_divide(int,N2,K))
<=> ( ( ( K = zero_zero(int) )
=> aa(int,$o,P,zero_zero(int)) )
& ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
=> ! [I4: 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)
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,P,I4) ) )
& ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
=> ! [I4: int,J3: int] :
( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
& aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,P,I4) ) ) ) ) ).
% split_zdiv
tff(fact_2080_incr__mult__lemma,axiom,
! [D3: int,P: fun(int,$o),K: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> ( ! [X3: int] :
( aa(int,$o,P,X3)
=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3)) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
=> ! [X4: int] :
( aa(int,$o,P,X4)
=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_2081_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
=> ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N2) = one_one(int) )
<=> ( ( M = one_one(int) )
& ( N2 = one_one(int) ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_2082_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J2)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J2)) ) ) ).
% zmult_zless_mono2
tff(fact_2083_minusinfinity,axiom,
! [D3: int,P1: fun(int,$o),P: fun(int,$o)] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> ( ! [X3: int,K3: int] :
( aa(int,$o,P1,X3)
<=> aa(int,$o,P1,minus_minus(int,X3,aa(int,int,aa(int,fun(int,int),times_times(int),K3),D3))) )
=> ( ? [Z6: int] :
! [X3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z6)
=> ( aa(int,$o,P,X3)
<=> aa(int,$o,P1,X3) ) )
=> ( ? [X_13: int] : aa(int,$o,P1,X_13)
=> ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).
% minusinfinity
tff(fact_2084_plusinfinity,axiom,
! [D3: int,P5: fun(int,$o),P: fun(int,$o)] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> ( ! [X3: int,K3: int] :
( aa(int,$o,P5,X3)
<=> aa(int,$o,P5,minus_minus(int,X3,aa(int,int,aa(int,fun(int,int),times_times(int),K3),D3))) )
=> ( ? [Z6: int] :
! [X3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z6),X3)
=> ( aa(int,$o,P,X3)
<=> aa(int,$o,P5,X3) ) )
=> ( ? [X_13: int] : aa(int,$o,P5,X_13)
=> ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).
% plusinfinity
tff(fact_2085_decr__mult__lemma,axiom,
! [D3: int,P: fun(int,$o),K: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> ( ! [X3: int] :
( aa(int,$o,P,X3)
=> aa(int,$o,P,minus_minus(int,X3,D3)) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
=> ! [X4: int] :
( aa(int,$o,P,X4)
=> aa(int,$o,P,minus_minus(int,X4,aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) ) ) ) ) ).
% decr_mult_lemma
tff(fact_2086_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_2087_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_2088_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_2089_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_2090_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),minus_minus(int,Z1,Z22)) = 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_2091_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),minus_minus(int,Z1,Z22)),W) = 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_2092_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,C2: A] : divide_divide(A,divide_divide(A,A4,B3),C2) = divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).
% divide_divide_eq_left'
tff(fact_2093_divide__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W: A] : divide_divide(A,divide_divide(A,X,Y),divide_divide(A,Z2,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),W),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ).
% divide_divide_times_eq
tff(fact_2094_times__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,X,Y)),divide_divide(A,Z2,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).
% times_divide_times_eq
tff(fact_2095_distrib__left__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( semiring(B)
=> ! [X: A,Y: A,A4: B,B3: B,C2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),A4)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A4),aa(B,B,aa(B,fun(B,B),plus_plus(B),B3),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A4),B3)),aa(B,B,aa(B,fun(B,B),times_times(B),A4),C2)) ) ) ) ).
% distrib_left_NO_MATCH
tff(fact_2096_distrib__right__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( semiring(B)
=> ! [X: A,Y: A,C2: B,A4: B,B3: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),C2)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A4),B3)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A4),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B3),C2)) ) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2097_right__diff__distrib__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( ring(B)
=> ! [X: A,Y: A,A4: B,B3: B,C2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),A4)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A4),minus_minus(B,B3,C2)) = minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A4),B3),aa(B,B,aa(B,fun(B,B),times_times(B),A4),C2)) ) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_2098_left__diff__distrib__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( ring(B)
=> ! [X: A,Y: A,C2: B,A4: B,B3: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),C2)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),minus_minus(B,A4,B3)),C2) = minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A4),C2),aa(B,B,aa(B,fun(B,B),times_times(B),B3),C2)) ) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_2099_frac__eq__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( ( divide_divide(A,X,Y) = divide_divide(A,W,Z2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).
% frac_eq_eq
tff(fact_2100_divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,C2: A,A4: A] :
( ( divide_divide(A,B3,C2) = A4 )
<=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),A4 = zero_zero(A)) ) ) ).
% divide_eq_eq
tff(fact_2101_eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = divide_divide(A,B3,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = B3,A4 = zero_zero(A)) ) ) ).
% eq_divide_eq
tff(fact_2102_divide__eq__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B3: A,A4: A] :
( ( C2 != zero_zero(A) )
=> ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) )
=> ( divide_divide(A,B3,C2) = A4 ) ) ) ) ).
% divide_eq_imp
tff(fact_2103_eq__divide__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = B3 )
=> ( A4 = divide_divide(A,B3,C2) ) ) ) ) ).
% eq_divide_imp
tff(fact_2104_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B3: A,A4: A] :
( ( C2 != zero_zero(A) )
=> ( ( divide_divide(A,B3,C2) = A4 )
<=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) ) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2105_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( ( A4 = divide_divide(A,B3,C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = B3 ) ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2106_right__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( ( divide_divide(A,A4,B3) = one_one(A) )
<=> ( A4 = B3 ) ) ) ) ).
% right_inverse_eq
tff(fact_2107_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,D3: A,C2: A] :
( dvd_dvd(A,B3,A4)
=> ( dvd_dvd(A,D3,C2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),divide_divide(A,C2,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D3)) ) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_2108_dvd__mult__imp__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,C2: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),B3)
=> dvd_dvd(A,A4,divide_divide(A,B3,C2)) ) ) ).
% dvd_mult_imp_div
tff(fact_2109_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,C2: A,A4: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2),A4)
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A4,B3),C2) ) ) ) ).
% dvd_div_mult2_eq
tff(fact_2110_div__div__eq__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B3: A,A4: A] :
( dvd_dvd(A,C2,B3)
=> ( dvd_dvd(A,B3,A4)
=> ( divide_divide(A,A4,divide_divide(A,B3,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),C2) ) ) ) ) ).
% div_div_eq_right
tff(fact_2111_div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B3: A,A4: A] :
( dvd_dvd(A,C2,B3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ) ) ).
% div_mult_swap
tff(fact_2112_dvd__div__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B3: A,A4: A] :
( dvd_dvd(A,C2,B3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B3,C2)),A4) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4),C2) ) ) ) ).
% dvd_div_mult
tff(fact_2113_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,A4,divide_divide(A,C2,B3))
<=> dvd_dvd(A,A4,C2) ) ) ) ).
% dvd_div_unit_iff
tff(fact_2114_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,divide_divide(A,A4,B3),C2)
<=> dvd_dvd(A,A4,C2) ) ) ) ).
% div_unit_dvd_iff
tff(fact_2115_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( ( divide_divide(A,B3,A4) = divide_divide(A,C2,A4) )
<=> ( B3 = C2 ) ) ) ) ).
% unit_div_cancel
tff(fact_2116_power__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,one_one(A),A4)),N2) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) ) ).
% power_one_over
tff(fact_2117_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A4,B3),C2) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2118_divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),A4)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)) ) ) ) ).
% divide_less_eq
tff(fact_2119_less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))) ) ) ) ).
% less_divide_eq
tff(fact_2120_neg__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) ) ) ) ).
% neg_divide_less_eq
tff(fact_2121_neg__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,B3,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% neg_less_divide_eq
tff(fact_2122_pos__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% pos_divide_less_eq
tff(fact_2123_pos__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,B3,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) ) ) ) ).
% pos_less_divide_eq
tff(fact_2124_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,X,Y)),Z2) ) ) ) ).
% mult_imp_div_pos_less
tff(fact_2125_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),divide_divide(A,X,Y)) ) ) ) ).
% mult_imp_less_div_pos
tff(fact_2126_divide__strict__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A4)),divide_divide(A,C2,B3)) ) ) ) ) ).
% divide_strict_left_mono
tff(fact_2127_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A4)),divide_divide(A,C2,B3)) ) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_2128_divide__less__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A4)),one_one(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
| ( A4 = zero_zero(A) ) ) ) ) ).
% divide_less_eq_1
tff(fact_2129_less__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A4))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ) ) ).
% less_divide_eq_1
tff(fact_2130_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,Z2: A,B3: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,Z2)),B3) = $ite(Z2 = zero_zero(A),B3,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z2)),Z2)) ) ).
% add_divide_eq_if_simps(2)
tff(fact_2131_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,Z2: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),divide_divide(A,B3,Z2)) = $ite(Z2 = zero_zero(A),A4,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),Z2)),B3),Z2)) ) ).
% add_divide_eq_if_simps(1)
tff(fact_2132_add__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Y)),divide_divide(A,W,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% add_frac_eq
tff(fact_2133_add__frac__num,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,X: A,Z2: A] :
( ( Y != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Y)),Z2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Y) ) ) ) ).
% add_frac_num
tff(fact_2134_add__num__frac,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A] :
( ( Y != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,X,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Y) ) ) ) ).
% add_num_frac
tff(fact_2135_add__divide__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,Y,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y),Z2) ) ) ) ).
% add_divide_eq_iff
tff(fact_2136_divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).
% divide_add_eq_iff
tff(fact_2137_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,Z2: A] :
minus_minus(A,A4,divide_divide(A,B3,Z2)) = $ite(Z2 = zero_zero(A),A4,divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),Z2),B3),Z2)) ) ).
% add_divide_eq_if_simps(4)
tff(fact_2138_diff__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,divide_divide(A,X,Y),divide_divide(A,W,Z2)) = divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% diff_frac_eq
tff(fact_2139_diff__divide__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,X,divide_divide(A,Y,Z2)) = divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2),Y),Z2) ) ) ) ).
% diff_divide_eq_iff
tff(fact_2140_divide__diff__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,divide_divide(A,X,Z2),Y) = divide_divide(A,minus_minus(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).
% divide_diff_eq_iff
tff(fact_2141_gt__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B3) ) ) ).
% gt_half_sum
tff(fact_2142_less__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).
% less_half_sum
tff(fact_2143_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,C2: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( ( C2 = aa(A,A,uminus_uminus(A),divide_divide(A,A4,B3)) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) = aa(A,A,uminus_uminus(A),A4) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_2144_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,A4: A,C2: A] :
( ( B3 != zero_zero(A) )
=> ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A4,B3)) = C2 )
<=> ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_2145_minus__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,C2: A,A4: A] :
( ( aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)) = A4 )
<=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),A4 = zero_zero(A)) ) ) ).
% minus_divide_eq_eq
tff(fact_2146_eq__minus__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2) = aa(A,A,uminus_uminus(A),B3),A4 = zero_zero(A)) ) ) ).
% eq_minus_divide_eq
tff(fact_2147_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] :
( ( divide_divide(A,A4,B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( B3 != zero_zero(A) )
& ( A4 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_2148_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( A4 != zero_zero(A) )
=> ( ( C2 != zero_zero(A) )
=> ( dvd_dvd(A,A4,B3)
=> ( dvd_dvd(A,C2,D3)
=> ( ( divide_divide(A,B3,A4) = divide_divide(A,D3,C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),D3) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_2149_dvd__div__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B3: A,A4: A] :
( ( C2 != zero_zero(A) )
=> ( dvd_dvd(A,C2,B3)
=> ( dvd_dvd(A,A4,divide_divide(A,B3,C2))
<=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),B3) ) ) ) ) ).
% dvd_div_iff_mult
tff(fact_2150_div__dvd__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( ( B3 != zero_zero(A) )
=> ( dvd_dvd(A,B3,A4)
=> ( dvd_dvd(A,divide_divide(A,A4,B3),C2)
<=> dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ) ).
% div_dvd_iff_mult
tff(fact_2151_dvd__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( ( A4 != zero_zero(A) )
=> ( dvd_dvd(A,A4,B3)
=> ( ( divide_divide(A,B3,A4) = C2 )
<=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4) ) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_2152_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( ( divide_divide(A,A4,B3) = zero_zero(A) )
<=> ( A4 = zero_zero(A) ) ) ) ) ).
% unit_div_eq_0_iff
tff(fact_2153_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,C2: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( dvd_dvd(A,C2,one_one(A))
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A4,B3),C2) ) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_2154_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A4: A,B3: A] :
( dvd_dvd(A,C2,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,B3,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ) ) ).
% unit_div_mult_swap
tff(fact_2155_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),B3) ) ) ) ).
% unit_div_commute
tff(fact_2156_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B3: A,A4: A] :
( dvd_dvd(A,C2,one_one(A))
=> ( dvd_dvd(A,B3,A4)
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,divide_divide(A,A4,B3),C2) ) ) ) ) ).
% div_mult_unit2
tff(fact_2157_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( ( A4 = divide_divide(A,C2,B3) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = C2 ) ) ) ) ).
% unit_eq_div2
tff(fact_2158_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A,C2: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( ( divide_divide(A,A4,B3) = C2 )
<=> ( A4 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).
% unit_eq_div1
tff(fact_2159_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) = divide_divide(nat,M,N2) ) ) ).
% nat_mult_div_cancel1
tff(fact_2160_pochhammer__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N2: nat] :
comm_s3205402744901411588hammer(A,zero_zero(A),N2) = $ite(N2 = zero_zero(nat),one_one(A),zero_zero(A)) ) ).
% pochhammer_0_left
tff(fact_2161_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E4)
=> ~ ! [N3: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),E4) ) ) ).
% nat_approx_posE
tff(fact_2162_divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),A4)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)) ) ) ) ).
% divide_le_eq
tff(fact_2163_le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))) ) ) ) ).
% le_divide_eq
tff(fact_2164_divide__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A4)),divide_divide(A,C2,B3)) ) ) ) ) ).
% divide_left_mono
tff(fact_2165_neg__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) ) ) ) ).
% neg_divide_le_eq
tff(fact_2166_neg__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),divide_divide(A,B3,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% neg_le_divide_eq
tff(fact_2167_pos__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% pos_divide_le_eq
tff(fact_2168_pos__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),divide_divide(A,B3,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) ) ) ) ).
% pos_le_divide_eq
tff(fact_2169_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,X,Y)),Z2) ) ) ) ).
% mult_imp_div_pos_le
tff(fact_2170_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),divide_divide(A,X,Y)) ) ) ) ).
% mult_imp_le_div_pos
tff(fact_2171_divide__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A4)),divide_divide(A,C2,B3)) ) ) ) ) ).
% divide_left_mono_neg
tff(fact_2172_divide__le__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A4)),one_one(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) )
| ( A4 = zero_zero(A) ) ) ) ) ).
% divide_le_eq_1
tff(fact_2173_le__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A4))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ) ) ).
% le_divide_eq_1
tff(fact_2174_frac__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W,Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A)) ) ) ) ) ).
% frac_le_eq
tff(fact_2175_frac__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W,Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A)) ) ) ) ) ).
% frac_less_eq
tff(fact_2176_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% pos_minus_divide_less_eq
tff(fact_2177_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).
% pos_less_minus_divide_eq
tff(fact_2178_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).
% neg_minus_divide_less_eq
tff(fact_2179_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% neg_less_minus_divide_eq
tff(fact_2180_minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)) ) ) ) ).
% minus_divide_less_eq
tff(fact_2181_less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))) ) ) ) ).
% less_minus_divide_eq
tff(fact_2182_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).
% minus_divide_add_eq_iff
tff(fact_2183_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,Z2: A,B3: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A4,Z2))),B3) = $ite(Z2 = zero_zero(A),B3,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z2)),Z2)) ) ).
% add_divide_eq_if_simps(3)
tff(fact_2184_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,Z2: A,B3: A] :
minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A4,Z2)),B3) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B3),divide_divide(A,minus_minus(A,aa(A,A,uminus_uminus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z2)),Z2)) ) ).
% add_divide_eq_if_simps(6)
tff(fact_2185_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,Z2: A,B3: A] :
minus_minus(A,divide_divide(A,A4,Z2),B3) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B3),divide_divide(A,minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z2)),Z2)) ) ).
% add_divide_eq_if_simps(5)
tff(fact_2186_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,X,Z2)),Y) = divide_divide(A,minus_minus(A,aa(A,A,uminus_uminus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_2187_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ~ ( ( A4 != zero_zero(A) )
=> ! [B2: A] :
( ( B2 != zero_zero(A) )
=> ( dvd_dvd(A,B2,one_one(A))
=> ( ( divide_divide(A,one_one(A),A4) = B2 )
=> ( ( divide_divide(A,one_one(A),B2) = A4 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B2) = one_one(A) )
=> ( divide_divide(A,C2,A4) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_2188_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( dvd_dvd(A,B3,one_one(A))
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_2189_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( dvd_dvd(A,B3,one_one(A))
=> ( divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_2190_pochhammer__rec,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A,N2: nat] : comm_s3205402744901411588hammer(A,A4,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),N2)) ) ).
% pochhammer_rec
tff(fact_2191_pochhammer__rec_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N2: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N2))),comm_s3205402744901411588hammer(A,Z2,N2)) ) ).
% pochhammer_rec'
tff(fact_2192_pochhammer__Suc,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A,N2: nat] : comm_s3205402744901411588hammer(A,A4,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A4,N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).
% pochhammer_Suc
tff(fact_2193_pochhammer__product_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N2: nat,M: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N2)),M)) ) ).
% pochhammer_product'
tff(fact_2194_scaling__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [U: A,V: A,R2: A,S2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),minus_minus(A,V,U)),S2))),V) ) ) ) ) ).
% scaling_mono
tff(fact_2195_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% pos_minus_divide_le_eq
tff(fact_2196_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).
% pos_le_minus_divide_eq
tff(fact_2197_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).
% neg_minus_divide_le_eq
tff(fact_2198_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)) ) ) ) ).
% neg_le_minus_divide_eq
tff(fact_2199_minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2))),A4)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)) ) ) ) ).
% minus_divide_le_eq
tff(fact_2200_le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,uminus_uminus(A),B3)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))) ) ) ) ).
% le_minus_divide_eq
tff(fact_2201_pochhammer__product,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: nat,N2: nat,Z2: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( comm_s3205402744901411588hammer(A,Z2,N2) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M)),minus_minus(nat,N2,M))) ) ) ) ).
% pochhammer_product
tff(fact_2202_prod__diff1,axiom,
! [B: $tType,A: $tType] :
( semidom_divide(B)
=> ! [A3: set(A),F2: fun(A,B),A4: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( aa(A,B,F2,A4) != zero_zero(B) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) = $ite(member(A,A4,A3),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),aa(A,B,F2,A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ) ).
% prod_diff1
tff(fact_2203_sum__bounded__above__divide,axiom,
! [A: $tType,B: $tType] :
( linordered_field(B)
=> ! [A3: set(A),F2: fun(A,B),K2: B] :
( ! [I3: A] :
( member(A,I3,A3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),divide_divide(B,K2,aa(nat,B,semiring_1_of_nat(B),finite_card(A,A3)))) )
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),K2) ) ) ) ) ).
% sum_bounded_above_divide
tff(fact_2204_prod__Un,axiom,
! [B: $tType,A: $tType] :
( field(B)
=> ! [A3: set(A),B5: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))
=> ( aa(A,B,F2,X3) != zero_zero(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))) ) ) ) ) ) ).
% prod_Un
tff(fact_2205_gbinomial__factors,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A4),K)) ) ).
% gbinomial_factors
tff(fact_2206_gbinomial__rec,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A4),K)),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).
% gbinomial_rec
tff(fact_2207_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,R2,aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).
% pochhammer_absorb_comp
tff(fact_2208_div__add__self2__no__field,axiom,
! [A: $tType,B: $tType] :
( ( euclid4440199948858584721cancel(B)
& field(A) )
=> ! [X: A,B3: B,A4: B] :
( nO_MATCH(A,B,X,B3)
=> ( ( B3 != zero_zero(B) )
=> ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A4),B3),B3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A4,B3)),one_one(B)) ) ) ) ) ).
% div_add_self2_no_field
tff(fact_2209_div__add__self1__no__field,axiom,
! [A: $tType,B: $tType] :
( ( euclid4440199948858584721cancel(B)
& field(A) )
=> ! [X: A,B3: B,A4: B] :
( nO_MATCH(A,B,X,B3)
=> ( ( B3 != zero_zero(B) )
=> ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),B3),A4),B3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A4,B3)),one_one(B)) ) ) ) ) ).
% div_add_self1_no_field
tff(fact_2210_div__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,C2: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A4,B3)) ) ) ) ).
% div_mult_self4
tff(fact_2211_div__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,C2: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A4,B3)) ) ) ) ).
% div_mult_self3
tff(fact_2212_div__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,A4: A,C2: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A4,B3)) ) ) ) ).
% div_mult_self2
tff(fact_2213_div__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,A4: A,C2: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A4,B3)) ) ) ) ).
% div_mult_self1
tff(fact_2214_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2),N2) = M ) ) ).
% div_mult_self_is_m
tff(fact_2215_div__mult__mult1__if,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A4: A,B3: A] :
divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A4,B3)) ) ).
% div_mult_mult1_if
tff(fact_2216_div__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = divide_divide(A,A4,B3) ) ) ) ).
% div_mult_mult2
tff(fact_2217_div__mult__mult1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A4: A,B3: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = divide_divide(A,A4,B3) ) ) ) ).
% div_mult_mult1
tff(fact_2218_div__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A4: A] : divide_divide(A,A4,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A4) ) ).
% div_minus1_right
tff(fact_2219_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),M),N2) = M ) ) ).
% div_mult_self1_is_m
tff(fact_2220_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q3: nat] : divide_divide(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) = divide_divide(nat,divide_divide(nat,M,N2),Q3) ).
% div_mult2_eq
tff(fact_2221_zdiv__zmult2__eq,axiom,
! [C2: int,A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
=> ( divide_divide(int,A4,aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = divide_divide(int,divide_divide(int,A4,B3),C2) ) ) ).
% zdiv_zmult2_eq
tff(fact_2222_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),N2))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M,N2)),I2) ) ).
% less_mult_imp_div_less
tff(fact_2223_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N2)),N2)),M) ).
% div_times_less_eq_dividend
tff(fact_2224_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),divide_divide(nat,M,N2))),M) ).
% times_div_less_eq_dividend
tff(fact_2225_div__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A,M: nat,N2: nat] : divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2))) = divide_divide(A,divide_divide(A,A4,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).
% div_mult2_eq'
tff(fact_2226_div__less__iff__less__mult,axiom,
! [Q3: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M,Q3)),N2)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) ) ) ).
% div_less_iff_less_mult
tff(fact_2227_div__add__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,B3)),one_one(A)) ) ) ) ).
% div_add_self2
tff(fact_2228_div__add__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,B3)),one_one(A)) ) ) ) ).
% div_add_self1
tff(fact_2229_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),divide_divide(nat,N2,Q3))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),N2) ) ) ).
% less_eq_div_iff_mult_less_eq
tff(fact_2230_div__nat__eqI,axiom,
! [N2: nat,Q3: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)),M)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,Q3)))
=> ( divide_divide(nat,M,N2) = Q3 ) ) ) ).
% div_nat_eqI
tff(fact_2231_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),divide_divide(nat,M,N2)))) ) ).
% dividend_less_times_div
tff(fact_2232_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N2)),N2))) ) ).
% dividend_less_div_times
tff(fact_2233_split__div,axiom,
! [P: fun(nat,$o),M: nat,N2: nat] :
( aa(nat,$o,P,divide_divide(nat,M,N2))
<=> ( ( ( N2 = zero_zero(nat) )
=> aa(nat,$o,P,zero_zero(nat)) )
& ( ( N2 != zero_zero(nat) )
=> ! [I4: nat,J3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),N2)
=> ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),I4)),J3) )
=> aa(nat,$o,P,I4) ) ) ) ) ) ).
% split_div
tff(fact_2234_split__div_H,axiom,
! [P: fun(nat,$o),M: nat,N2: nat] :
( aa(nat,$o,P,divide_divide(nat,M,N2))
<=> ( ( ( N2 = zero_zero(nat) )
& aa(nat,$o,P,zero_zero(nat)) )
| ? [Q6: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q6)),M)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,Q6)))
& aa(nat,$o,P,Q6) ) ) ) ).
% split_div'
tff(fact_2235_power__diff__power__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,M: nat,N2: nat] :
( ( A4 != zero_zero(A) )
=> ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),minus_minus(nat,M,N2)),divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),minus_minus(nat,N2,M)))) ) ) ) ).
% power_diff_power_eq
tff(fact_2236_bits__div__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A] : divide_divide(A,A4,one_one(A)) = A4 ) ).
% bits_div_by_1
tff(fact_2237_int__ops_I7_J,axiom,
! [A4: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),B3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A4)),aa(nat,int,semiring_1_of_nat(int),B3)) ).
% int_ops(7)
tff(fact_2238_gbinomial__pochhammer,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,A4),K) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A4),K)),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer
tff(fact_2239_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(nat,A,gbinomial(A,A4),K) = divide_divide(A,comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A4,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer'
tff(fact_2240_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
<=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R2) )
& $ite(
aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
& aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
$ite(
aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
& aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
Q3 = zero_zero(int) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_2241_pochhammer__same,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_ring_1(A)
& semiri3467727345109120633visors(A) )
=> ! [N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N2)),N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2)),semiring_char_0_fact(A,N2)) ) ).
% pochhammer_same
tff(fact_2242_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_2243_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_2244_fact__Suc__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).
% fact_Suc_0
tff(fact_2245_fact__Suc,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2))),semiring_char_0_fact(A,N2)) ) ).
% fact_Suc
tff(fact_2246_fact__ge__1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N2)) ) ).
% fact_ge_1
tff(fact_2247_pochhammer__fact,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_semiring_1(A) )
=> ! [N2: nat] : semiring_char_0_fact(A,N2) = comm_s3205402744901411588hammer(A,one_one(A),N2) ) ).
% pochhammer_fact
tff(fact_2248_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L != zero_zero(int) )
=> ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_2249_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N2: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2))) ) ).
% fact_fact_dvd_fact
tff(fact_2250_choose__dvd,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,minus_minus(nat,N2,K))),semiring_char_0_fact(A,N2)) ) ) ).
% choose_dvd
tff(fact_2251_fact__num__eq__if,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M: nat] :
semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,minus_minus(nat,M,one_one(nat))))) ) ).
% fact_num_eq_if
tff(fact_2252_fact__reduce,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( semiring_char_0_fact(A,N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),semiring_char_0_fact(A,minus_minus(nat,N2,one_one(nat)))) ) ) ) ).
% fact_reduce
tff(fact_2253_split__pos__lemma,axiom,
! [K: int,P: fun(int,fun(int,$o)),N2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
=> ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,N2,K)),modulo_modulo(int,N2,K))
<=> ! [I4: 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)
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).
% split_pos_lemma
tff(fact_2254_split__neg__lemma,axiom,
! [K: int,P: fun(int,fun(int,$o)),N2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
=> ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,N2,K)),modulo_modulo(int,N2,K))
<=> ! [I4: int,J3: int] :
( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
& aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).
% split_neg_lemma
tff(fact_2255_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),minus_minus(int,X,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,minus_minus(int,X,Z2))),one_one(int))),D3))),Z2) ) ).
% decr_lemma
tff(fact_2256_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),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),abs_abs(int,minus_minus(int,X,Z2))),one_one(int))),D3))) ) ).
% incr_lemma
tff(fact_2257_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_2258_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_2259_power__mult__numeral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,M: num,N2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).
% power_mult_numeral
tff(fact_2260_abs__idempotent,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : abs_abs(A,abs_abs(A,A4)) = abs_abs(A,A4) ) ).
% abs_idempotent
tff(fact_2261_numeral__times__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ) ).
% numeral_times_numeral
tff(fact_2262_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [V: num,W: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Z2) ) ).
% mult_numeral_left_semiring_numeral
tff(fact_2263_power__add__numeral2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,M: num,N2: num,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),N2))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2)))),B3) ) ).
% power_add_numeral2
tff(fact_2264_power__add__numeral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2))) ) ).
% power_add_numeral
tff(fact_2265_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_2266_abs__eq__0,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( ( abs_abs(A,A4) = zero_zero(A) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% abs_eq_0
tff(fact_2267_abs__0__eq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( ( zero_zero(A) = abs_abs(A,A4) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% abs_0_eq
tff(fact_2268_abs__add__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A4)),abs_abs(A,B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A4)),abs_abs(A,B3)) ) ).
% abs_add_abs
tff(fact_2269_abs__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A4)),abs_abs(A,A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4) ) ).
% abs_mult_self_eq
tff(fact_2270_abs__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
% abs_1
tff(fact_2271_abs__minus__cancel,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : abs_abs(A,aa(A,A,uminus_uminus(A),A4)) = abs_abs(A,A4) ) ).
% abs_minus_cancel
tff(fact_2272_distrib__right__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [A4: A,B3: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V))) ) ).
% distrib_right_numeral
tff(fact_2273_distrib__left__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [V: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).
% distrib_left_numeral
tff(fact_2274_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [A4: A,B3: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A4,B3)),aa(num,A,numeral_numeral(A),V)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V))) ) ).
% left_diff_distrib_numeral
tff(fact_2275_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [V: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),minus_minus(A,B3,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).
% right_diff_distrib_numeral
tff(fact_2276_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).
% mult_neg_numeral_simps(3)
tff(fact_2277_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).
% mult_neg_numeral_simps(2)
tff(fact_2278_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ) ).
% mult_neg_numeral_simps(1)
tff(fact_2279_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Y) ) ).
% semiring_norm(172)
tff(fact_2280_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).
% semiring_norm(171)
tff(fact_2281_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).
% semiring_norm(170)
tff(fact_2282_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( abs_abs(A,A4) = A4 ) ) ) ).
% abs_of_nonneg
tff(fact_2283_abs__le__self__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% abs_le_self_iff
tff(fact_2284_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),zero_zero(A))
<=> ( A4 = zero_zero(A) ) ) ) ).
% abs_le_zero_iff
tff(fact_2285_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),B3) = zero_zero(A) ) ).
% mod_mult_self2_is_0
tff(fact_2286_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,A4: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4),B3) = zero_zero(A) ) ).
% mod_mult_self1_is_0
tff(fact_2287_mod__by__1,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A4: A] : modulo_modulo(A,A4,one_one(A)) = zero_zero(A) ) ).
% mod_by_1
tff(fact_2288_bits__mod__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A] : modulo_modulo(A,A4,one_one(A)) = zero_zero(A) ) ).
% bits_mod_by_1
tff(fact_2289_zero__less__abs__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),abs_abs(A,A4))
<=> ( A4 != zero_zero(A) ) ) ) ).
% zero_less_abs_iff
tff(fact_2290_mod__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,C2: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),B3) = modulo_modulo(A,A4,B3) ) ).
% mod_mult_self1
tff(fact_2291_mod__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,B3: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),B3) = modulo_modulo(A,A4,B3) ) ).
% mod_mult_self2
tff(fact_2292_mod__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,B3: A,A4: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A4),B3) = modulo_modulo(A,A4,B3) ) ).
% mod_mult_self3
tff(fact_2293_mod__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B3: A,C2: A,A4: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A4),B3) = modulo_modulo(A,A4,B3) ) ).
% mod_mult_self4
tff(fact_2294_abs__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).
% abs_neg_one
tff(fact_2295_max__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(5)
tff(fact_2296_max__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(6)
tff(fact_2297_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,W: num,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W))) ) ) ).
% divide_le_eq_numeral1(1)
tff(fact_2298_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).
% le_divide_eq_numeral1(1)
tff(fact_2299_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,W: num,A4: A] :
( ( divide_divide(A,B3,aa(num,A,numeral_numeral(A),W)) = A4 )
<=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W)),A4 = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2300_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,W: num] :
( ( A4 = divide_divide(A,B3,aa(num,A,numeral_numeral(A),W)) )
<=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W)) = B3,A4 = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2301_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).
% less_divide_eq_numeral1(1)
tff(fact_2302_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,W: num,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),W))) ) ) ).
% divide_less_eq_numeral1(1)
tff(fact_2303_abs__of__nonpos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A))
=> ( abs_abs(A,A4) = aa(A,A,uminus_uminus(A),A4) ) ) ) ).
% abs_of_nonpos
tff(fact_2304_mod__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A4: A] : modulo_modulo(A,A4,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).
% mod_minus1_right
tff(fact_2305_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).
% le_divide_eq_numeral1(2)
tff(fact_2306_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,W: num,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).
% divide_le_eq_numeral1(2)
tff(fact_2307_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,W: num,A4: A] :
( ( divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A4 )
<=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A4 = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_2308_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A,W: num] :
( ( A4 = divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
<=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B3,A4 = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_2309_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,W: num,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).
% divide_less_eq_numeral1(2)
tff(fact_2310_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).
% less_divide_eq_numeral1(2)
tff(fact_2311_abs__ge__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),abs_abs(A,A4)) ) ).
% abs_ge_self
tff(fact_2312_abs__le__D1,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% abs_le_D1
tff(fact_2313_abs__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A4: A,B3: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A4)),abs_abs(A,B3)) ) ).
% abs_mult
tff(fact_2314_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_2315_abs__minus__commute,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : abs_abs(A,minus_minus(A,A4,B3)) = abs_abs(A,minus_minus(A,B3,A4)) ) ).
% abs_minus_commute
tff(fact_2316_mod__mult__right__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,B3: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ).
% mod_mult_right_eq
tff(fact_2317_mod__mult__left__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,C2: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A4,C2)),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ).
% mod_mult_left_eq
tff(fact_2318_mult__mod__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A4,B3)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).
% mult_mod_right
tff(fact_2319_mod__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,C2: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A4,B3)),C2) ) ).
% mod_mult_mult2
tff(fact_2320_mod__mult__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,C2: A,A6: A,B3: A,B4: A] :
( ( modulo_modulo(A,A4,C2) = modulo_modulo(A,A6,C2) )
=> ( ( modulo_modulo(A,B3,C2) = modulo_modulo(A,B4,C2) )
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A6),B4),C2) ) ) ) ) ).
% mod_mult_cong
tff(fact_2321_mod__mult__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: A,C2: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A4,C2)),modulo_modulo(A,B3,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ).
% mod_mult_eq
tff(fact_2322_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A,K: num,L: num] : divide_divide(A,divide_divide(A,A4,aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = divide_divide(A,A4,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).
% div_mult2_numeral_eq
tff(fact_2323_abs__ge__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),abs_abs(A,A4)) ) ).
% abs_ge_zero
tff(fact_2324_abs__of__pos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( abs_abs(A,A4) = A4 ) ) ) ).
% abs_of_pos
tff(fact_2325_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,A4)),zero_zero(A)) ) ).
% abs_not_less_zero
tff(fact_2326_abs__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A4)),abs_abs(A,B3))) ) ).
% abs_triangle_ineq
tff(fact_2327_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,abs_abs(A,A4),abs_abs(A,B3))),abs_abs(A,minus_minus(A,A4,B3))) ) ).
% abs_triangle_ineq2
tff(fact_2328_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,abs_abs(A,A4),abs_abs(A,B3)))),abs_abs(A,minus_minus(A,A4,B3))) ) ).
% abs_triangle_ineq3
tff(fact_2329_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,abs_abs(A,A4),abs_abs(A,B3))),abs_abs(A,minus_minus(A,B3,A4))) ) ).
% abs_triangle_ineq2_sym
tff(fact_2330_abs__mult__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,C2: A,B3: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,A4)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,B3)),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A4)),abs_abs(A,B3))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ) ).
% abs_mult_less
tff(fact_2331_abs__leI,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),B3) ) ) ) ).
% abs_leI
tff(fact_2332_abs__le__D2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),B3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),B3) ) ) ).
% abs_le_D2
tff(fact_2333_abs__le__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,A4)),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),B3) ) ) ) ).
% abs_le_iff
tff(fact_2334_abs__ge__minus__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A4)),abs_abs(A,A4)) ) ).
% abs_ge_minus_self
tff(fact_2335_mod__eqE,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo(A,A4,C2) = modulo_modulo(A,B3,C2) )
=> ~ ! [D2: A] : B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ).
% mod_eqE
tff(fact_2336_one__le__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N2)) ) ).
% one_le_numeral
tff(fact_2337_not__numeral__less__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),N2)),one_one(A)) ) ).
% not_numeral_less_one
tff(fact_2338_one__plus__numeral__commute,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).
% one_plus_numeral_commute
tff(fact_2339_zmod__eq__0__iff,axiom,
! [M: int,D3: int] :
( ( modulo_modulo(int,M,D3) = zero_zero(int) )
<=> ? [Q6: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q6) ) ).
% zmod_eq_0_iff
tff(fact_2340_zmod__eq__0D,axiom,
! [M: int,D3: int] :
( ( modulo_modulo(int,M,D3) = zero_zero(int) )
=> ? [Q7: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q7) ) ).
% zmod_eq_0D
tff(fact_2341_numeral__times__minus__swap,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [W: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).
% numeral_times_minus_swap
tff(fact_2342_abs__zmult__eq__1,axiom,
! [M: int,N2: int] :
( ( abs_abs(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),N2)) = one_one(int) )
=> ( abs_abs(int,M) = one_one(int) ) ) ).
% abs_zmult_eq_1
tff(fact_2343_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N2: num] : aa(num,A,numeral_numeral(A),N2) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% numeral_neq_neg_one
tff(fact_2344_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N2: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)) ) ).
% one_neq_neg_numeral
tff(fact_2345_dense__eq0__I,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs(A)
& dense_linorder(A) )
=> ! [X: A] :
( ! [E2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X)),E2) )
=> ( X = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_2346_abs__mult__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,Y)),X) = abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).
% abs_mult_pos
tff(fact_2347_abs__eq__mult,axiom,
! [A: $tType] :
( ordered_ring_abs(A)
=> ! [A4: A,B3: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),zero_zero(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
=> ( abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A4)),abs_abs(A,B3)) ) ) ) ).
% abs_eq_mult
tff(fact_2348_abs__minus__le__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),abs_abs(A,A4))),zero_zero(A)) ) ).
% abs_minus_le_zero
tff(fact_2349_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A,C2: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,minus_minus(A,A4,C2))),abs_abs(A,minus_minus(A,B3,D3)))) ) ).
% abs_diff_triangle_ineq
tff(fact_2350_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,A4,B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A4)),abs_abs(A,B3))) ) ).
% abs_triangle_ineq4
tff(fact_2351_abs__of__neg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( abs_abs(A,A4) = aa(A,A,uminus_uminus(A),A4) ) ) ) ).
% abs_of_neg
tff(fact_2352_cSup__abs__le,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A4: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X3)),A4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(set(A),A,complete_Sup_Sup(A),S))),A4) ) ) ) ).
% cSup_abs_le
tff(fact_2353_cInf__abs__ge,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A4: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X3)),A4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(set(A),A,complete_Inf_Inf(A),S))),A4) ) ) ) ).
% cInf_abs_ge
tff(fact_2354_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( modulo_modulo(A,A4,B3) = zero_zero(A) ) ) ) ).
% unit_imp_mod_eq_0
tff(fact_2355_div__mult1__eq,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A4: A,B3: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,B3,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),modulo_modulo(A,B3,C2)),C2)) ) ).
% div_mult1_eq
tff(fact_2356_mult__div__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [B3: A,A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A4,B3))),modulo_modulo(A,A4,B3)) = A4 ) ).
% mult_div_mod_eq
tff(fact_2357_mod__mult__div__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,B3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A4,B3))) = A4 ) ).
% mod_mult_div_eq
tff(fact_2358_mod__div__mult__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,B3)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3)) = A4 ) ).
% mod_div_mult_eq
tff(fact_2359_div__mult__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3)),modulo_modulo(A,A4,B3)) = A4 ) ).
% div_mult_mod_eq
tff(fact_2360_mod__div__decomp,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3)),modulo_modulo(A,A4,B3)) ) ).
% mod_div_decomp
tff(fact_2361_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3)),modulo_modulo(A,A4,B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2) ) ).
% cancel_div_mod_rules(1)
tff(fact_2362_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [B3: A,A4: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A4,B3))),modulo_modulo(A,A4,B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),C2) ) ).
% cancel_div_mod_rules(2)
tff(fact_2363_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3)) = modulo_modulo(A,A4,B3) ) ).
% minus_div_mult_eq_mod
tff(fact_2364_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,modulo_modulo(A,A4,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3) ) ).
% minus_mod_eq_div_mult
tff(fact_2365_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,modulo_modulo(A,A4,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A4,B3)) ) ).
% minus_mod_eq_mult_div
tff(fact_2366_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A4: A,B3: A] : minus_minus(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),divide_divide(A,A4,B3))) = modulo_modulo(A,A4,B3) ) ).
% minus_mult_div_eq_mod
tff(fact_2367_neg__numeral__le__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).
% neg_numeral_le_one
tff(fact_2368_neg__one__le__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).
% neg_one_le_numeral
tff(fact_2369_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% neg_numeral_le_neg_one
tff(fact_2370_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% not_numeral_le_neg_one
tff(fact_2371_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).
% not_one_le_neg_numeral
tff(fact_2372_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,C2: A,W: num] :
( ( divide_divide(A,B3,C2) = aa(num,A,numeral_numeral(A),W) )
<=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2373_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B3: A,C2: A] :
( ( aa(num,A,numeral_numeral(A),W) = divide_divide(A,B3,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B3,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2374_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).
% not_neg_one_less_neg_numeral
tff(fact_2375_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).
% not_one_less_neg_numeral
tff(fact_2376_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% not_numeral_less_neg_one
tff(fact_2377_neg__one__less__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).
% neg_one_less_numeral
tff(fact_2378_neg__numeral__less__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).
% neg_numeral_less_one
tff(fact_2379_div__mod__decomp__int,axiom,
! [A3: int,N2: int] : A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A3,N2)),N2)),modulo_modulo(int,A3,N2)) ).
% div_mod_decomp_int
tff(fact_2380_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),abs_abs(A,X))) ) ).
% abs_add_one_gt_zero
tff(fact_2381_cSup__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E4: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,X3,L))),E4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S),L))),E4) ) ) ) ).
% cSup_asclose
tff(fact_2382_cInf__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E4: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,X3,L))),E4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,aa(set(A),A,complete_Inf_Inf(A),S),L))),E4) ) ) ) ).
% cInf_asclose
tff(fact_2383_mod__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A,M: nat,N2: nat] : modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,divide_divide(A,A4,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N2)))),modulo_modulo(A,A4,aa(nat,A,semiring_1_of_nat(A),M))) ) ).
% mod_mult2_eq'
tff(fact_2384_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),aa(num,A,numeral_numeral(A),W))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_2385_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_2386_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B3: A,C2: A,W: num] :
( ( divide_divide(A,B3,C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
<=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_2387_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B3: A,C2: A] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = divide_divide(A,B3,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_2388_zdvd__mult__cancel1,axiom,
! [M: int,N2: int] :
( ( M != zero_zero(int) )
=> ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),N2),M)
<=> ( abs_abs(int,N2) = one_one(int) ) ) ) ).
% zdvd_mult_cancel1
tff(fact_2389_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),modulo_modulo(A,divide_divide(A,A4,B3),C2))),modulo_modulo(A,A4,B3)) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2390_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C2)),aa(num,A,numeral_numeral(A),W))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_2391_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_2392_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B3,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_2393_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B3: A,C2: A,W: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_2394_split__zmod,axiom,
! [P: fun(int,$o),N2: int,K: int] :
( aa(int,$o,P,modulo_modulo(int,N2,K))
<=> ( ( ( K = zero_zero(int) )
=> aa(int,$o,P,N2) )
& ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
=> ! [I4: 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)
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,P,J3) ) )
& ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
=> ! [I4: int,J3: int] :
( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
& aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
& ( N2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
=> aa(int,$o,P,J3) ) ) ) ) ).
% split_zmod
tff(fact_2395_int__mod__neg__eq,axiom,
! [A4: int,B3: int,Q3: int,R2: int] :
( ( A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
=> ( modulo_modulo(int,A4,B3) = R2 ) ) ) ) ).
% int_mod_neg_eq
tff(fact_2396_int__mod__pos__eq,axiom,
! [A4: int,B3: int,Q3: int,R2: int] :
( ( A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R2) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
=> ( modulo_modulo(int,A4,B3) = R2 ) ) ) ) ).
% int_mod_pos_eq
tff(fact_2397_zmod__zmult2__eq,axiom,
! [C2: int,A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
=> ( modulo_modulo(int,A4,aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),modulo_modulo(int,divide_divide(int,A4,B3),C2))),modulo_modulo(int,A4,B3)) ) ) ).
% zmod_zmult2_eq
tff(fact_2398_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).
% neg_numeral_le_ceiling
tff(fact_2399_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).
% ceiling_less_neg_numeral
tff(fact_2400_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).
% floor_le_neg_numeral
tff(fact_2401_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).
% neg_numeral_less_floor
tff(fact_2402_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
<=> ( ? [K5: int] :
( ( A1 = K5 )
& ( A22 = zero_zero(int) )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K5) ) )
| ? [L2: int,K5: int,Q6: int] :
( ( A1 = K5 )
& ( A22 = L2 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
& ( L2 != zero_zero(int) )
& ( K5 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L2) ) )
| ? [R6: int,L2: int,K5: int,Q6: int] :
( ( A1 = K5 )
& ( A22 = L2 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R6) )
& ( aa(int,int,sgn_sgn(int),R6) = aa(int,int,sgn_sgn(int),L2) )
& aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_abs(int,R6)),abs_abs(int,L2))
& ( K5 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L2)),R6) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_2403_sgn__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).
% sgn_1
tff(fact_2404_Suc__mod__mult__self4,axiom,
! [N2: nat,K: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K)),M)),N2) = modulo_modulo(nat,aa(nat,nat,suc,M),N2) ).
% Suc_mod_mult_self4
tff(fact_2405_Suc__mod__mult__self3,axiom,
! [K: nat,N2: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)),M)),N2) = modulo_modulo(nat,aa(nat,nat,suc,M),N2) ).
% Suc_mod_mult_self3
tff(fact_2406_Suc__mod__mult__self2,axiom,
! [M: nat,N2: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))),N2) = modulo_modulo(nat,aa(nat,nat,suc,M),N2) ).
% Suc_mod_mult_self2
tff(fact_2407_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2))),N2) = modulo_modulo(nat,aa(nat,nat,suc,M),N2) ).
% Suc_mod_mult_self1
tff(fact_2408_divide__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] : divide_divide(A,A4,aa(A,A,sgn_sgn(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,sgn_sgn(A),B3)) ) ).
% divide_sgn
tff(fact_2409_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_2410_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_2411_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,A,sgn_sgn(A),A4) = one_one(A) ) ) ) ).
% sgn_pos
tff(fact_2412_abs__sgn__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( abs_abs(A,aa(A,A,sgn_sgn(A),A4)) = one_one(A) ) ) ) ).
% abs_sgn_eq_1
tff(fact_2413_Suc__times__numeral__mod__eq,axiom,
! [K: num,N2: nat] :
( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
=> ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N2)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_2414_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R2: int] :
( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R2)))
<=> ( dvd_dvd(int,L,K)
| ( R2 = zero_zero(int) ) ) ) ).
% dvd_mult_sgn_iff
tff(fact_2415_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K: int] :
( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),K))
<=> ( dvd_dvd(int,L,K)
| ( R2 = zero_zero(int) ) ) ) ).
% dvd_sgn_mult_iff
tff(fact_2416_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K: int] :
( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R2)),K)
<=> ( dvd_dvd(int,L,K)
& ( ( R2 = zero_zero(int) )
=> ( K = zero_zero(int) ) ) ) ) ).
% mult_sgn_dvd_iff
tff(fact_2417_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K: int] :
( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),L),K)
<=> ( dvd_dvd(int,L,K)
& ( ( R2 = zero_zero(int) )
=> ( K = zero_zero(int) ) ) ) ) ).
% sgn_mult_dvd_iff
tff(fact_2418_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> ( aa(A,A,sgn_sgn(A),A4) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% sgn_neg
tff(fact_2419_zero__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).
% zero_less_floor
tff(fact_2420_floor__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).
% floor_le_zero
tff(fact_2421_one__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).
% one_le_floor
tff(fact_2422_floor__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).
% floor_less_one
tff(fact_2423_ceiling__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ).
% ceiling_le_one
tff(fact_2424_one__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ).
% one_less_ceiling
tff(fact_2425_ceiling__add__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).
% ceiling_add_one
tff(fact_2426_floor__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archim6421214686448440834_floor(A,minus_minus(A,X,one_one(A))) = minus_minus(int,archim6421214686448440834_floor(A,X),one_one(int)) ) ).
% floor_diff_one
tff(fact_2427_ceiling__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,minus_minus(A,X,one_one(A))) = minus_minus(int,archimedean_ceiling(A,X),one_one(int)) ) ).
% ceiling_diff_one
tff(fact_2428_ceiling__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).
% ceiling_less_zero
tff(fact_2429_zero__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).
% zero_le_ceiling
tff(fact_2430_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).
% numeral_less_floor
tff(fact_2431_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).
% floor_le_numeral
tff(fact_2432_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),minus_minus(A,aa(num,A,numeral_numeral(A),V),one_one(A))) ) ) ).
% ceiling_less_numeral
tff(fact_2433_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(num,A,numeral_numeral(A),V),one_one(A))),X) ) ) ).
% numeral_le_ceiling
tff(fact_2434_sgn__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A4: A,B3: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A4)),aa(A,A,sgn_sgn(A),B3)) ) ).
% sgn_mult
tff(fact_2435_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L3: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L3)),aa(nat,int,semiring_1_of_nat(int),N3)) ).
% int_sgnE
tff(fact_2436_mod__eq__0D,axiom,
! [M: nat,D3: nat] :
( ( modulo_modulo(nat,M,D3) = zero_zero(nat) )
=> ? [Q7: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q7) ) ).
% mod_eq_0D
tff(fact_2437_nat__mod__eq__iff,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( modulo_modulo(nat,X,N2) = modulo_modulo(nat,Y,N2) )
<=> ? [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),N2),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q22)) ) ).
% nat_mod_eq_iff
tff(fact_2438_sgn__minus__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% sgn_minus_1
tff(fact_2439_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: A] : abs_abs(A,K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).
% linordered_idom_class.abs_sgn
tff(fact_2440_abs__mult__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A4)),aa(A,A,sgn_sgn(A),A4)) = A4 ) ).
% abs_mult_sgn
tff(fact_2441_sgn__mult__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A4)),abs_abs(A,A4)) = A4 ) ).
% sgn_mult_abs
tff(fact_2442_mult__sgn__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),abs_abs(A,X)) = X ) ).
% mult_sgn_abs
tff(fact_2443_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( modulo_modulo(nat,X,N2) = modulo_modulo(nat,Y,N2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
=> ? [Q7: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q7)) ) ) ).
% nat_mod_eq_lemma
tff(fact_2444_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N2: nat] :
( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N2,Q3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ~ ! [S6: nat] : N2 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S6)) ) ) ).
% mod_eq_nat2E
tff(fact_2445_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N2: nat] :
( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N2,Q3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M)
=> ~ ! [S6: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S6)) ) ) ).
% mod_eq_nat1E
tff(fact_2446_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( ( aa(A,A,sgn_sgn(A),A4) = one_one(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4) ) ) ).
% sgn_1_pos
tff(fact_2447_div__mod__decomp,axiom,
! [A3: nat,N2: nat] : A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A3,N2)),N2)),modulo_modulo(nat,A3,N2)) ).
% div_mod_decomp
tff(fact_2448_mod__mult2__eq,axiom,
! [M: nat,N2: nat,Q3: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),modulo_modulo(nat,divide_divide(nat,M,N2),Q3))),modulo_modulo(nat,M,N2)) ).
% mod_mult2_eq
tff(fact_2449_modulo__nat__def,axiom,
! [M: nat,N2: nat] : modulo_modulo(nat,M,N2) = minus_minus(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N2)),N2)) ).
% modulo_nat_def
tff(fact_2450_abs__sgn__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
abs_abs(A,aa(A,A,sgn_sgn(A),A4)) = $ite(A4 = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% abs_sgn_eq
tff(fact_2451_one__add__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).
% one_add_floor
tff(fact_2452_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V != zero_zero(int) )
=> ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),abs_abs(int,K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),abs_abs(int,L))) = divide_divide(int,abs_abs(int,K),abs_abs(int,L)) ) ) ).
% div_sgn_abs_cancel
tff(fact_2453_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( dvd_dvd(int,L,K)
=> ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),divide_divide(int,abs_abs(int,K),abs_abs(int,L))) ) ) ).
% div_dvd_sgn_abs
tff(fact_2454_split__mod,axiom,
! [P: fun(nat,$o),M: nat,N2: nat] :
( aa(nat,$o,P,modulo_modulo(nat,M,N2))
<=> ( ( ( N2 = zero_zero(nat) )
=> aa(nat,$o,P,M) )
& ( ( N2 != zero_zero(nat) )
=> ! [I4: nat,J3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),N2)
=> ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),I4)),J3) )
=> aa(nat,$o,P,J3) ) ) ) ) ) ).
% split_mod
tff(fact_2455_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( ( aa(A,A,sgn_sgn(A),A4) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ).
% sgn_1_neg
tff(fact_2456_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
aa(A,A,sgn_sgn(A),X) = $ite(
X = zero_zero(A),
zero_zero(A),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).
% sgn_if
tff(fact_2457_le__mult__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A4)),archim6421214686448440834_floor(A,B3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))) ) ) ) ).
% le_mult_floor
tff(fact_2458_mult__ceiling__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A4)),archimedean_ceiling(A,B3))) ) ) ) ).
% mult_ceiling_le
tff(fact_2459_Suc__times__mod__eq,axiom,
! [M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
=> ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),M) = one_one(nat) ) ) ).
% Suc_times_mod_eq
tff(fact_2460_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q3: int] :
( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_abs(int,R2)),abs_abs(int,L))
=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R2) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_2461_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
=> ( ( ( A22 = zero_zero(int) )
=> ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
=> ( ! [Q7: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),zero_zero(int)) )
=> ( ( A22 != zero_zero(int) )
=> ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q7),A22) ) ) )
=> ~ ! [R: int,Q7: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R) )
=> ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),A22) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_abs(int,R)),abs_abs(int,A22))
=> ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q7),A22)),R) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_2462_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),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(Xa,modulo_modulo(nat,X,Xa)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa)))))) ) ) ).
% bezw.elims
tff(fact_2463_bezw_Osimps,axiom,
! [X: nat,Y: nat] :
bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y)))))) ).
% bezw.simps
tff(fact_2464_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( aa(nat,$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,modulo_modulo(nat,X,Y)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ).
% bezw_non_0
tff(fact_2465_divide__int__unfold,axiom,
! [K: int,M: nat,L: int,N2: nat] :
divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N2))) = $ite(
( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
| ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
| ( N2 = zero_zero(nat) ) ),
zero_zero(int),
$ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,N2)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,M,N2)),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,N2,M)))))) ) ).
% divide_int_unfold
tff(fact_2466_modulo__int__unfold,axiom,
! [K: int,M: nat,L: int,N2: nat] :
modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N2))) = $ite(
( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
| ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
| ( N2 = zero_zero(nat) ) ),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),
$ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,N2,M)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N2))))) ) ).
% modulo_int_unfold
tff(fact_2467_floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).
% floor_add
tff(fact_2468_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).
% of_bool_eq(2)
tff(fact_2469_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: $o] :
( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
<=> (P) ) ) ).
% of_bool_eq_1_iff
tff(fact_2470_of__bool__less__one__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: $o] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
<=> ~ (P) ) ) ).
% of_bool_less_one_iff
tff(fact_2471_of__bool__not__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = minus_minus(A,one_one(A),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).
% of_bool_not_iff
tff(fact_2472_sgn__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A4)),aa(A,A,sgn_sgn(A),A4)) = aa($o,A,zero_neq_one_of_bool(A),A4 != zero_zero(A)) ) ).
% sgn_mult_self_eq
tff(fact_2473_of__bool__conj,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [P: $o,Q: $o] :
aa($o,A,zero_neq_one_of_bool(A),
( (P)
& (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).
% of_bool_conj
tff(fact_2474_of__bool__less__eq__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).
% of_bool_less_eq_one
tff(fact_2475_of__bool__def,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P2: $o] :
aa($o,A,zero_neq_one_of_bool(A),(P2)) = $ite((P2),one_one(A),zero_zero(A)) ) ).
% of_bool_def
tff(fact_2476_split__of__bool,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,$o),P2: $o] :
( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
<=> ( ( (P2)
=> aa(A,$o,P,one_one(A)) )
& ( ~ (P2)
=> aa(A,$o,P,zero_zero(A)) ) ) ) ) ).
% split_of_bool
tff(fact_2477_split__of__bool__asm,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,$o),P2: $o] :
( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
<=> ~ ( ( (P2)
& ~ aa(A,$o,P,one_one(A)) )
| ( ~ (P2)
& ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).
% split_of_bool_asm
tff(fact_2478_frac__lt__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,X)),one_one(A)) ) ).
% frac_lt_1
tff(fact_2479_frac__1__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).
% frac_1_eq
tff(fact_2480_frac__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( archimedean_frac(A,X) = X )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).
% frac_eq
tff(fact_2481_frac__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),one_one(A))) ) ).
% frac_add
tff(fact_2482_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
=> ~ ( ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa)))))) )
=> ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).
% bezw.pelims
tff(fact_2483_frac__unique__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A4: A] :
( ( archimedean_frac(A,X) = A4 )
<=> ( member(A,minus_minus(A,X,A4),ring_1_Ints(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A)) ) ) ) ).
% frac_unique_iff
tff(fact_2484_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : minus_minus(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N2)) ) ).
% diff_numeral_special(3)
tff(fact_2485_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).
% diff_numeral_special(4)
tff(fact_2486_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q3))),one_one(A))),Q3)),P2) ) ) ).
% ceiling_divide_lower
tff(fact_2487_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),one_one(A))),Q3)) ) ) ).
% floor_divide_upper
tff(fact_2488_semiring__norm_I12_J,axiom,
! [N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N2) = N2 ).
% semiring_norm(12)
tff(fact_2489_semiring__norm_I11_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).
% semiring_norm(11)
tff(fact_2490_numeral__eq__one__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: num] :
( ( aa(num,A,numeral_numeral(A),N2) = one_one(A) )
<=> ( N2 = one2 ) ) ) ).
% numeral_eq_one_iff
tff(fact_2491_one__eq__numeral__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: num] :
( ( one_one(A) = aa(num,A,numeral_numeral(A),N2) )
<=> ( one2 = N2 ) ) ) ).
% one_eq_numeral_iff
tff(fact_2492_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( aa(int,A,ring_1_of_int(A),Z2) = one_one(A) )
<=> ( Z2 = one_one(int) ) ) ) ).
% of_int_eq_1_iff
tff(fact_2493_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).
% of_int_1
tff(fact_2494_of__int__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_mult
tff(fact_2495_numeral__le__one__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),N2)),one_one(A))
<=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N2),one2) ) ) ).
% numeral_le_one_iff
tff(fact_2496_one__less__numeral__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N2))
<=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),N2) ) ) ).
% one_less_numeral_iff
tff(fact_2497_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N2: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( N2 = one2 ) ) ) ).
% numeral_eq_neg_one_iff
tff(fact_2498_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N2: num] :
( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2)) )
<=> ( N2 = one2 ) ) ) ).
% neg_one_eq_numeral_iff
tff(fact_2499_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
<=> ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_2500_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
<=> ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_2501_of__int__le__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),one_one(int)) ) ) ).
% of_int_le_1_iff
tff(fact_2502_of__int__1__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2) ) ) ).
% of_int_1_le_iff
tff(fact_2503_numeral__plus__one,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N2)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N2),one2)) ) ).
% numeral_plus_one
tff(fact_2504_one__plus__numeral,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N2)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N2)) ) ).
% one_plus_numeral
tff(fact_2505_of__int__less__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),one_one(int)) ) ) ).
% of_int_less_1_iff
tff(fact_2506_of__int__1__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ) ).
% of_int_1_less_iff
tff(fact_2507_mult__of__int__commute,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).
% mult_of_int_commute
tff(fact_2508_Ints__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A4: A,B3: A] :
( member(A,A4,ring_1_Ints(A))
=> ( member(A,B3,ring_1_Ints(A))
=> member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),ring_1_Ints(A)) ) ) ) ).
% Ints_mult
tff(fact_2509_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> member(A,one_one(A),ring_1_Ints(A)) ) ).
% Ints_1
tff(fact_2510_mult__numeral__1,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A4) = A4 ) ).
% mult_numeral_1
tff(fact_2511_mult__numeral__1__right,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),one2)) = A4 ) ).
% mult_numeral_1_right
tff(fact_2512_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_2513_le__mult__floor__Ints,axiom,
! [B: $tType,A: $tType] :
( ( archim2362893244070406136eiling(A)
& linordered_idom(B) )
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( member(A,A4,ring_1_Ints(A))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A4)),archim6421214686448440834_floor(A,B3)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)))) ) ) ) ).
% le_mult_floor_Ints
tff(fact_2514_mult__ceiling__le__Ints,axiom,
! [B: $tType,A: $tType] :
( ( archim2362893244070406136eiling(A)
& linordered_idom(B) )
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( member(A,A4,ring_1_Ints(A))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A4)),archimedean_ceiling(A,B3)))) ) ) ) ).
% mult_ceiling_le_Ints
tff(fact_2515_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B3) = aa(A,A,uminus_uminus(A),B3) ) ).
% mult_1s_ring_1(1)
tff(fact_2516_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B3) ) ).
% mult_1s_ring_1(2)
tff(fact_2517_uminus__numeral__One,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% uminus_numeral_One
tff(fact_2518_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [A4: A] :
( member(A,A4,ring_1_Ints(A))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4)),A4) != zero_zero(A) ) ) ) ).
% Ints_odd_nonzero
tff(fact_2519_of__int__leD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,aa(int,A,ring_1_of_int(A),N2))),X)
=> ( ( N2 = zero_zero(int) )
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).
% of_int_leD
tff(fact_2520_of__int__lessD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,aa(int,A,ring_1_of_int(A),N2))),X)
=> ( ( N2 = zero_zero(int) )
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).
% of_int_lessD
tff(fact_2521_Ints__odd__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A] :
( member(A,A4,ring_1_Ints(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4)),A4)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A)) ) ) ) ).
% Ints_odd_less_0
tff(fact_2522_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( member(A,X,ring_1_Ints(A))
=> ( ( X != zero_zero(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),abs_abs(A,X)) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_2523_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( member(A,X,ring_1_Ints(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,X)),one_one(A))
=> ( X = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_2524_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( member(A,X,ring_1_Ints(A))
=> ( member(A,Y,ring_1_Ints(A))
=> ( ( X = Y )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,minus_minus(A,X,Y))),one_one(A)) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_2525_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,$o),T5: A] :
( aa(int,$o,P,archim6421214686448440834_floor(A,T5))
<=> ! [I4: int] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),T5),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
=> aa(int,$o,P,I4) ) ) ) ).
% floor_split
tff(fact_2526_floor__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A4: int] :
( ( archim6421214686448440834_floor(A,X) = A4 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A4)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A4)),one_one(A))) ) ) ) ).
% floor_eq_iff
tff(fact_2527_floor__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
=> ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).
% floor_unique
tff(fact_2528_ceiling__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)),one_one(A))),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).
% ceiling_correct
tff(fact_2529_ceiling__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),Z2),one_one(A))),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))
=> ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).
% ceiling_unique
tff(fact_2530_ceiling__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A4: int] :
( ( archimedean_ceiling(A,X) = A4 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),A4),one_one(A))),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A4)) ) ) ) ).
% ceiling_eq_iff
tff(fact_2531_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,$o),T5: A] :
( aa(int,$o,P,archimedean_ceiling(A,T5))
<=> ! [I4: int] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),I4),one_one(A))),T5)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T5),aa(int,A,ring_1_of_int(A),I4)) )
=> aa(int,$o,P,I4) ) ) ) ).
% ceiling_split
tff(fact_2532_less__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X) ) ) ).
% less_floor_iff
tff(fact_2533_floor__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).
% floor_le_iff
tff(fact_2534_ceiling__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),minus_minus(A,aa(int,A,ring_1_of_int(A),Z2),one_one(A))) ) ) ).
% ceiling_less_iff
tff(fact_2535_le__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),Z2),one_one(A))),X) ) ) ).
% le_ceiling_iff
tff(fact_2536_floor__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),Q3)),P2) ) ) ).
% floor_divide_lower
tff(fact_2537_ceiling__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q3)))),Q3)) ) ) ).
% ceiling_divide_upper
tff(fact_2538_frac__neg,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = $ite(member(A,X,ring_1_Ints(A)),zero_zero(A),minus_minus(A,one_one(A),archimedean_frac(A,X))) ) ).
% frac_neg
tff(fact_2539_even__succ__mod__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A4,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2))) ) ) ) ) ).
% even_succ_mod_exp
tff(fact_2540_even__succ__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = divide_divide(A,A4,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) ) ) ) ) ).
% even_succ_div_exp
tff(fact_2541_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),zero_zero(int))
=> ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A4),one_one(int)),B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2),one_one(int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_2542_fact__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N2: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2))),semiring_char_0_fact(A,N2)) ) ).
% fact_double
tff(fact_2543_pochhammer__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))),comm_s3205402744901411588hammer(A,Z2,N2))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N2)) ) ).
% pochhammer_double
tff(fact_2544_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,M: nat,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M)
| ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2) = zero_zero(A) )
| ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
& dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,A4,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N2,M)))) ) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_2545_semiring__norm_I13_J,axiom,
! [M: num,N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(N2)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ).
% semiring_norm(13)
tff(fact_2546_num__double,axiom,
! [N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),N2) = bit0(N2) ).
% num_double
tff(fact_2547_one__add__one,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).
% one_add_one
tff(fact_2548_even__mult__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
<=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
| dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),B3) ) ) ) ).
% even_mult_iff
tff(fact_2549_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).
% bits_one_mod_two_eq_one
tff(fact_2550_one__mod__two__eq__one,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).
% one_mod_two_eq_one
tff(fact_2551_zmod__numeral__Bit0,axiom,
! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit0(V)),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).
% zmod_numeral_Bit0
tff(fact_2552_one__div__two__eq__zero,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).
% one_div_two_eq_zero
tff(fact_2553_bits__1__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).
% bits_1_div_2
tff(fact_2554_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).
% add_neg_numeral_special(9)
tff(fact_2555_even__plus__one__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)))
<=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4) ) ) ).
% even_plus_one_iff
tff(fact_2556_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( minus_minus(A,one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).
% diff_numeral_special(11)
tff(fact_2557_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A)),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).
% diff_numeral_special(10)
tff(fact_2558_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
<=> ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_2559_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
<=> ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_2560_minus__1__div__2__eq,axiom,
! [A: $tType] :
( euclid8789492081693882211th_nat(A)
=> ( divide_divide(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% minus_1_div_2_eq
tff(fact_2561_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).
% bits_minus_1_mod_2_eq
tff(fact_2562_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( euclid8789492081693882211th_nat(A)
=> ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).
% minus_1_mod_2_eq
tff(fact_2563_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A4: A,B3: A] :
( ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
<=> ( A4 != B3 ) ) ).
% card_doubleton_eq_2_iff
tff(fact_2564_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ) ).
% Power.ring_1_class.power_minus_even
tff(fact_2565_odd__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).
% odd_succ_div_two
tff(fact_2566_even__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_two
tff(fact_2567_even__succ__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_2
tff(fact_2568_power__minus1__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) = one_one(A) ) ).
% power_minus1_even
tff(fact_2569_neg__one__even__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] :
( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2) = one_one(A) ) ) ) ).
% neg_one_even_power
tff(fact_2570_neg__one__odd__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] :
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% neg_one_odd_power
tff(fact_2571_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A4 ) ) ) ).
% odd_two_times_div_two_succ
tff(fact_2572_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2),one_one(A)))
<=> ( N2 = zero_zero(nat) ) ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_2573_one__div__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa($o,A,zero_neq_one_of_bool(A),N2 = zero_zero(nat)) ) ).
% one_div_2_pow_eq
tff(fact_2574_bits__1__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N2: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa($o,A,zero_neq_one_of_bool(A),N2 = zero_zero(nat)) ) ).
% bits_1_div_exp
tff(fact_2575_odd__two__times__div__two__nat,axiom,
! [N2: nat] :
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = minus_minus(nat,N2,one_one(nat)) ) ) ).
% odd_two_times_div_two_nat
tff(fact_2576_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)) ) ).
% one_mod_2_pow_eq
tff(fact_2577_left__add__twice,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),B3) ) ).
% left_add_twice
tff(fact_2578_mult__2__right,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2_right
tff(fact_2579_mult__2,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2
tff(fact_2580_evenE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ~ ! [B2: A] : A4 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ).
% evenE
tff(fact_2581_odd__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),one_one(A)) ) ).
% odd_one
tff(fact_2582_power4__eq__xxxx,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).
% power4_eq_xxxx
tff(fact_2583_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4) ) ).
% power2_eq_square
tff(fact_2584_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).
% one_power2
tff(fact_2585_Suc__double__not__eq__double,axiom,
! [M: nat,N2: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) ).
% Suc_double_not_eq_double
tff(fact_2586_double__not__eq__Suc__double,axiom,
! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ).
% double_not_eq_Suc_double
tff(fact_2587_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).
% power_even_eq
tff(fact_2588_even__two__times__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))) = A4 ) ) ) ).
% even_two_times_div_two
tff(fact_2589_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
<=> ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_2590_power2__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A4: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
<=> ( ( A4 = one_one(A) )
| ( A4 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% power2_eq_1_iff
tff(fact_2591_abs__square__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
<=> ( abs_abs(A,X) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_2592_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set(A)] :
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),finite_card(A,A3))
=> ( A3 != bot_bot(set(A)) ) ) ).
% odd_card_imp_not_empty
tff(fact_2593_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
=> ( modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)) = modulo_modulo(A,A4,B3) ) ) ) ) ).
% divmod_digit_0(2)
tff(fact_2594_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ~ ! [B2: A] : A4 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),one_one(A)) ) ) ).
% oddE
tff(fact_2595_parity__cases,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
( ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
=> ~ ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).
% parity_cases
tff(fact_2596_mod2__eq__if,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A4: A] :
modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4),zero_zero(A),one_one(A)) ) ).
% mod2_eq_if
tff(fact_2597_power2__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).
% power2_sum
tff(fact_2598_square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).
% square_le_1
tff(fact_2599_zero__le__even__power_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) ) ).
% zero_le_even_power'
tff(fact_2600_abs__square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,X)),one_one(A)) ) ) ).
% abs_square_le_1
tff(fact_2601_abs__square__less__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,X)),one_one(A)) ) ) ).
% abs_square_less_1
tff(fact_2602_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,$o),A4: A] :
( ! [A5: A] :
( ( divide_divide(A,A5,aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
=> aa(A,$o,P,A5) )
=> ( ! [A5: A,B2: $o] :
( aa(A,$o,P,A5)
=> ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5)),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5))) ) )
=> aa(A,$o,P,A4) ) ) ) ).
% bits_induct
tff(fact_2603_minus__power__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A4: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A4)),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A4)),N2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ) ).
% minus_power_mult_self
tff(fact_2604_power__odd__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),N2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).
% power_odd_eq
tff(fact_2605_nat__bit__induct,axiom,
! [P: fun(nat,$o),N2: nat] :
( aa(nat,$o,P,zero_zero(nat))
=> ( ! [N3: nat] :
( aa(nat,$o,P,N3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N3)
=> aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)) ) )
=> ( ! [N3: nat] :
( aa(nat,$o,P,N3)
=> aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))) )
=> aa(nat,$o,P,N2) ) ) ) ).
% nat_bit_induct
tff(fact_2606_minus__one__power__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N2) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% minus_one_power_iff
tff(fact_2607_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))) = divide_divide(A,A4,B3) ) ) ) ) ).
% divmod_digit_0(1)
tff(fact_2608_power2__diff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),minus_minus(A,X,Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).
% power2_diff
tff(fact_2609_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4) ) ) ).
% odd_0_le_power_imp_0_le
tff(fact_2610_odd__power__less__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A4: A,N2: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))),zero_zero(A)) ) ) ).
% odd_power_less_zero
tff(fact_2611_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N2: nat,A4: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A4,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N2,M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_2612_power__minus1__odd,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% power_minus1_odd
tff(fact_2613_exp__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N2: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).
% exp_mod_exp
tff(fact_2614_int__bit__induct,axiom,
! [P: fun(int,$o),K: int] :
( aa(int,$o,P,zero_zero(int))
=> ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
=> ( ! [K3: int] :
( aa(int,$o,P,K3)
=> ( ( K3 != zero_zero(int) )
=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
=> ( ! [K3: int] :
( aa(int,$o,P,K3)
=> ( ( K3 != aa(int,int,uminus_uminus(int),one_one(int)) )
=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
=> aa(int,$o,P,K) ) ) ) ) ).
% int_bit_induct
tff(fact_2615_mod__double__modulus,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,X,M) )
| ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).
% mod_double_modulus
tff(fact_2616_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
=> ( minus_minus(A,modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),B3) = modulo_modulo(A,A4,B3) ) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_2617_even__mask__div__iff_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ).
% even_mask_div_iff'
tff(fact_2618_pos__zdiv__mult__2,axiom,
! [A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A4)
=> ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)) = divide_divide(int,B3,A4) ) ) ).
% pos_zdiv_mult_2
tff(fact_2619_neg__zdiv__mult__2,axiom,
! [A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),zero_zero(int))
=> ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int)),A4) ) ) ).
% neg_zdiv_mult_2
tff(fact_2620_pos__zmod__mult__2,axiom,
! [A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A4)
=> ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B3,A4))) ) ) ).
% pos_zmod_mult_2
tff(fact_2621_even__mask__div__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2) = zero_zero(A) )
| aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2) ) ) ) ).
% even_mask_div_iff
tff(fact_2622_exp__div__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N2: nat] :
divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,
aa(A,fun(A,A),times_times(A),
aa($o,A,zero_neq_one_of_bool(A),
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M) ))),
aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,M,N2))) ) ).
% exp_div_exp_eq
tff(fact_2623_neg__zmod__mult__2,axiom,
! [A4: int,B3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),zero_zero(int))
=> ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)) = minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int)),A4)),one_one(int)) ) ) ).
% neg_zmod_mult_2
tff(fact_2624_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B3)
=> ( eucl_rel_int(A4,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A4)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_2625_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))),one_one(A)) = divide_divide(A,A4,B3) ) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_2626_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Q3: A,R2: A] :
unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),one_one(A))),minus_minus(A,R2,aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),R2)) ) ).
% divmod_step_eq
tff(fact_2627_flip__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% flip_bit_0
tff(fact_2628_card__2__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
<=> ? [X2: A,Y3: A] :
( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))) )
& ( X2 != Y3 ) ) ) ).
% card_2_iff
tff(fact_2629_set__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% set_bit_0
tff(fact_2630_unset__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% unset_bit_0
tff(fact_2631_unset__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,N2,divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% unset_bit_Suc
tff(fact_2632_set__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,N2,divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% set_bit_Suc
tff(fact_2633_flip__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,N2,divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% flip_bit_Suc
tff(fact_2634_signed__take__bit__rec,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat,A4: A] :
bit_ri4674362597316999326ke_bit(A,N2,A4) = $ite(N2 = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,minus_minus(nat,N2,one_one(nat)),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).
% signed_take_bit_rec
tff(fact_2635_round__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,X,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
=> ( archimedean_round(A,X) = Y ) ) ) ) ).
% round_unique
tff(fact_2636_dbl__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).
% dbl_simps(4)
tff(fact_2637_round__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
archimedean_round(A,X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).
% round_altdef
tff(fact_2638_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),abs_abs(A,minus_minus(A,X,aa(int,A,ring_1_of_int(A),N2)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))
=> ( archimedean_round(A,X) = N2 ) ) ) ).
% round_unique'
tff(fact_2639_of__int__round__abs__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),abs_abs(A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X)),X))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% of_int_round_abs_le
tff(fact_2640_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] : bit_ri4674362597316999326ke_bit(A,N2,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% signed_take_bit_of_minus_1
tff(fact_2641_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N2),one_one(A)) = one_one(A) ) ).
% signed_take_bit_Suc_1
tff(fact_2642_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [K: num] : bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ).
% signed_take_bit_numeral_of_1
tff(fact_2643_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_2644_signed__take__bit__Suc__bit0,axiom,
! [N2: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N2),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N2,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_Suc_bit0
tff(fact_2645_dbl__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).
% dbl_simps(3)
tff(fact_2646_signed__take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_Suc_minus_bit0
tff(fact_2647_signed__take__bit__mult,axiom,
! [N2: nat,K: int,L: int] : bit_ri4674362597316999326ke_bit(int,N2,aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N2,K)),bit_ri4674362597316999326ke_bit(int,N2,L))) = bit_ri4674362597316999326ke_bit(int,N2,aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).
% signed_take_bit_mult
tff(fact_2648_round__def,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% round_def
tff(fact_2649_signed__take__bit__Suc,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat,A4: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,N2,divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% signed_take_bit_Suc
tff(fact_2650_of__int__round__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% of_int_round_le
tff(fact_2651_of__int__round__ge,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),minus_minus(A,X,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).
% of_int_round_ge
tff(fact_2652_of__int__round__gt,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),minus_minus(A,X,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).
% of_int_round_gt
tff(fact_2653_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( Y = $ite(
( member(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).
% and_int.elims
tff(fact_2654_and__int_Osimps,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).
% and_int.simps
tff(fact_2655_signed__take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N2,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_2656_concat__bit__Suc,axiom,
! [N2: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N2),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_concat_bit(N2,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))),L))) ).
% concat_bit_Suc
tff(fact_2657_fact__code,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N2: nat] : semiring_char_0_fact(A,N2) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N2,one_one(nat))) ) ).
% fact_code
tff(fact_2658_take__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] :
bit_se2584673776208193580ke_bit(A,N2,A4) = $ite(N2 = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,minus_minus(nat,N2,one_one(nat)),divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% take_bit_rec
tff(fact_2659_take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),one_one(A)) = one_one(A) ) ).
% take_bit_Suc_1
tff(fact_2660_and_Oleft__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A4) = A4 ) ).
% and.left_neutral
tff(fact_2661_and_Oright__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),aa(A,A,uminus_uminus(A),one_one(A))) = A4 ) ).
% and.right_neutral
tff(fact_2662_bit_Oconj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).
% bit.conj_one_right
tff(fact_2663_take__bit__numeral__1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [L: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ).
% take_bit_numeral_1
tff(fact_2664_semiring__norm_I14_J,axiom,
! [M: num,N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit1(N2)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),bit1(N2))) ).
% semiring_norm(14)
tff(fact_2665_semiring__norm_I15_J,axiom,
! [M: num,N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),bit0(N2)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),N2)) ).
% semiring_norm(15)
tff(fact_2666_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit(A,N2,one_one(A)) = zero_zero(A) )
<=> ( N2 = zero_zero(nat) ) ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_2667_and__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = one_one(A) ) ).
% and_numerals(2)
tff(fact_2668_and__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = one_one(A) ) ).
% and_numerals(8)
tff(fact_2669_semiring__norm_I16_J,axiom,
! [M: num,N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),bit1(N2)) = bit1(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)))) ).
% semiring_norm(16)
tff(fact_2670_and__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).
% and_numerals(5)
tff(fact_2671_and__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = zero_zero(A) ) ).
% and_numerals(1)
tff(fact_2672_and__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(3)
tff(fact_2673_take__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : bit_se2584673776208193580ke_bit(A,N2,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)) ) ).
% take_bit_of_1
tff(fact_2674_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit1(one2)) ) ) ).
% dbl_inc_simps(3)
tff(fact_2675_and__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(4)
tff(fact_2676_and__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(6)
tff(fact_2677_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit1(one2))) ) ) ).
% dbl_dec_simps(4)
tff(fact_2678_and__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% and_numerals(7)
tff(fact_2679_take__bit__of__exp,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: nat,N2: nat] : bit_se2584673776208193580ke_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) ) ).
% take_bit_of_exp
tff(fact_2680_take__bit__of__2,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: nat] : bit_se2584673776208193580ke_bit(A,N2,aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_of_2
tff(fact_2681_zmod__numeral__Bit1,axiom,
! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit1(V)),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).
% zmod_numeral_Bit1
tff(fact_2682_signed__take__bit__Suc__bit1,axiom,
! [N2: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N2),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N2,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_bit1
tff(fact_2683_take__bit__mult,axiom,
! [N2: nat,K: int,L: int] : bit_se2584673776208193580ke_bit(int,N2,aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N2,K)),bit_se2584673776208193580ke_bit(int,N2,L))) = bit_se2584673776208193580ke_bit(int,N2,aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).
% take_bit_mult
tff(fact_2684_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( A4 = aa(A,A,uminus_uminus(A),one_one(A)) )
& ( B3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_2685_numeral__Bit1,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N2: num] : aa(num,A,numeral_numeral(A),bit1(N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N2)),aa(num,A,numeral_numeral(A),N2))),one_one(A)) ) ).
% numeral_Bit1
tff(fact_2686_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N2,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).
% take_bit_Suc_bit1
tff(fact_2687_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A4: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(num,nat,numeral_numeral(nat),bit1(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),A4)),A4) ) ).
% power3_eq_cube
tff(fact_2688_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N2,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_Suc_bit0
tff(fact_2689_and__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),one_one(A)) = modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% and_one_eq
tff(fact_2690_one__and__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A4) = modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% one_and_eq
tff(fact_2691_card__3__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
<=> ? [X2: A,Y3: A,Z3: A] :
( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z3),bot_bot(set(A))))) )
& ( X2 != Y3 )
& ( Y3 != Z3 )
& ( X2 != Z3 ) ) ) ).
% card_3_iff
tff(fact_2692_take__bit__Suc__minus__bit0,axiom,
! [N2: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% take_bit_Suc_minus_bit0
tff(fact_2693_and__int__rec,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% and_int_rec
tff(fact_2694_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),aa(A,A,uminus_uminus(A),one_one(A))) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,N2)),one_one(A)) ) ).
% take_bit_Suc_minus_1_eq
tff(fact_2695_take__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N2,divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A4,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% take_bit_Suc
tff(fact_2696_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K),aa(A,A,uminus_uminus(A),one_one(A))) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K)),one_one(A)) ) ).
% take_bit_numeral_minus_1_eq
tff(fact_2697_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A,N2: nat] :
( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
=> ( bit_se2584673776208193580ke_bit(A,N2,A4) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4),zero_zero(A),minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2),one_one(A))) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_2698_and__int__unfold,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
( ( K = zero_zero(int) )
| ( L = zero_zero(int) ) ),
zero_zero(int),
$ite(
K = aa(int,int,uminus_uminus(int),one_one(int)),
L,
$ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).
% and_int_unfold
tff(fact_2699_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_minus_bit1
tff(fact_2700_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] :
unique8689654367752047608divmod(A,bit1(M),bit1(N2)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),N2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit1(M))),unique1321980374590559556d_step(A,bit1(N2),unique8689654367752047608divmod(A,bit1(M),bit0(bit1(N2))))) ) ).
% divmod_algorithm_code(8)
tff(fact_2701_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] :
unique8689654367752047608divmod(A,bit0(M),bit1(N2)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),N2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,bit1(N2),unique8689654367752047608divmod(A,bit0(M),bit0(bit1(N2))))) ) ).
% divmod_algorithm_code(7)
tff(fact_2702_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_bit1
tff(fact_2703_and__int_Opsimps,axiom,
! [K: int,L: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).
% and_int.psimps
tff(fact_2704_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
=> ~ ( ( Y = $ite(
( member(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
=> ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).
% and_int.pelims
tff(fact_2705_numeral__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).
% numeral_mod_numeral
tff(fact_2706_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).
% divmod_algorithm_code(2)
tff(fact_2707_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N2: num] : unique8689654367752047608divmod(A,one2,bit0(N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(3)
tff(fact_2708_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N2: num] : unique8689654367752047608divmod(A,one2,bit1(N2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(4)
tff(fact_2709_one__div__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N2: num] : divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),N2)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N2)) ) ).
% one_div_numeral
tff(fact_2710_one__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N2: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N2)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N2)) ) ).
% one_mod_numeral
tff(fact_2711_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_numeral_bit0
tff(fact_2712_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_numeral_minus_bit0
tff(fact_2713_snd__divmod,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,N2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2)) ) ).
% snd_divmod
tff(fact_2714_fact__numeral,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).
% fact_numeral
tff(fact_2715_divmod__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] : unique8689654367752047608divmod(A,M,N2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N2))) ) ).
% divmod_def
tff(fact_2716_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_numeral_bit0
tff(fact_2717_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% take_bit_numeral_minus_bit0
tff(fact_2718_and__nat__unfold,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N2) = $ite(
( ( M = zero_zero(nat) )
| ( N2 = zero_zero(nat) ) ),
zero_zero(nat),
aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).
% and_nat_unfold
tff(fact_2719_and__nat__rec,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N2) = aa(nat,nat,
aa(nat,fun(nat,nat),plus_plus(nat),
aa($o,nat,zero_neq_one_of_bool(nat),
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),M)
& ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2) ))),
aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% and_nat_rec
tff(fact_2720_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
=> ( ! [K3: int,L3: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K3),L3))
=> ( ( ~ ( member(int,K3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
=> aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K3,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),bit0(one2)))) )
=> aa(int,$o,aa(int,fun(int,$o),P,K3),L3) ) )
=> aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).
% and_int.pinduct
tff(fact_2721_divmod__divmod__step,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] :
unique8689654367752047608divmod(A,M,N2) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),N2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,N2,unique8689654367752047608divmod(A,M,bit0(N2)))) ) ).
% divmod_divmod_step
tff(fact_2722_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).
% take_bit_numeral_bit1
tff(fact_2723_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% take_bit_numeral_minus_bit1
tff(fact_2724_take__bit__Suc__minus__bit1,axiom,
! [N2: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% take_bit_Suc_minus_bit1
tff(fact_2725_binomial__code,axiom,
! [N2: nat,K: nat] :
aa(nat,nat,binomial(N2),K) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),K),
zero_zero(nat),
$ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(N2),minus_minus(nat,N2,K)),divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),minus_minus(nat,N2,K)),one_one(nat)),N2,one_one(nat)),semiring_char_0_fact(nat,K))) ) ).
% binomial_code
tff(fact_2726_modulo__int__def,axiom,
! [K: int,L: int] :
modulo_modulo(int,K,L) = $ite(
L = zero_zero(int),
K,
$ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,L)),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(abs_abs(int,K)),nat2(abs_abs(int,L))))))) ) ).
% modulo_int_def
tff(fact_2727_mask__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N2)))) ) ).
% mask_numeral
tff(fact_2728_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),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).
% or_int_unfold
tff(fact_2729_bit_Odisj__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_left
tff(fact_2730_bit_Odisj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_right
tff(fact_2731_or__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).
% or_numerals(8)
tff(fact_2732_or__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).
% or_numerals(2)
tff(fact_2733_mask__Suc__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).
% mask_Suc_0
tff(fact_2734_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] : bit_se2584673776208193580ke_bit(A,N2,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N2) ) ).
% take_bit_minus_one_eq_mask
tff(fact_2735_or__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% or_numerals(3)
tff(fact_2736_or__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).
% or_numerals(5)
tff(fact_2737_or__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).
% or_numerals(1)
tff(fact_2738_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N2))) ) ).
% add_neg_numeral_special(5)
tff(fact_2739_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).
% add_neg_numeral_special(6)
tff(fact_2740_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),N2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N2))) ) ).
% diff_numeral_special(5)
tff(fact_2741_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : minus_minus(A,aa(num,A,numeral_numeral(A),M),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).
% diff_numeral_special(6)
tff(fact_2742_or__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(4)
tff(fact_2743_or__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(6)
tff(fact_2744_or__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(7)
tff(fact_2745_Suc__times__binomial__eq,axiom,
! [N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N2)),aa(nat,nat,binomial(N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N2)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).
% Suc_times_binomial_eq
tff(fact_2746_Suc__times__binomial,axiom,
! [K: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(aa(nat,nat,suc,N2)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N2)),aa(nat,nat,binomial(N2),K)) ).
% Suc_times_binomial
tff(fact_2747_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R2)),M)) ).
% choose_mult_lemma
tff(fact_2748_Suc__times__binomial__add,axiom,
! [A4: nat,B3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A4)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))),aa(nat,nat,suc,A4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))),A4)) ).
% Suc_times_binomial_add
tff(fact_2749_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,minus_minus(nat,N2,K)))),aa(nat,nat,binomial(N2),K)) = semiring_char_0_fact(nat,N2) ) ) ).
% binomial_fact_lemma
tff(fact_2750_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N2),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N2),K)),aa(nat,nat,binomial(minus_minus(nat,N2,K)),minus_minus(nat,M,K))) ) ) ) ).
% choose_mult
tff(fact_2751_binomial__absorb__comp,axiom,
! [N2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,N2,K)),aa(nat,nat,binomial(N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,binomial(minus_minus(nat,N2,one_one(nat))),K)) ).
% binomial_absorb_comp
tff(fact_2752_binomial__Suc__Suc__eq__times,axiom,
! [N2: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N2)),aa(nat,nat,suc,K)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N2)),aa(nat,nat,binomial(N2),K)),aa(nat,nat,suc,K)) ).
% binomial_Suc_Suc_eq_times
tff(fact_2753_nat__abs__mult__distrib,axiom,
! [W: int,Z2: int] : nat2(abs_abs(int,aa(int,int,aa(int,fun(int,int),times_times(int),W),Z2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(abs_abs(int,W))),nat2(abs_abs(int,Z2))) ).
% nat_abs_mult_distrib
tff(fact_2754_mask__Suc__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N2))) ) ).
% mask_Suc_double
tff(fact_2755_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_2756_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),X) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A4),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A4),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
tff(fact_2757_numeral__inc,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).
% numeral_inc
tff(fact_2758_binomial__absorption,axiom,
! [K: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N2),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,binomial(minus_minus(nat,N2,one_one(nat))),K)) ).
% binomial_absorption
tff(fact_2759_nat__mult__distrib,axiom,
! [Z2: int,Z7: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z2)
=> ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z2)),nat2(Z7)) ) ) ).
% nat_mult_distrib
tff(fact_2760_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( aa(nat,nat,binomial(N2),K) = divide_divide(nat,semiring_char_0_fact(nat,N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,minus_minus(nat,N2,K)))) ) ) ).
% binomial_altdef_nat
tff(fact_2761_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N2),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,binomial(minus_minus(nat,N2,one_one(nat))),minus_minus(nat,K,one_one(nat)))) ) ) ).
% times_binomial_minus1_eq
tff(fact_2762_nat__mult__distrib__neg,axiom,
! [Z2: int,Z7: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),zero_zero(int))
=> ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,uminus_uminus(int),Z2))),nat2(aa(int,int,uminus_uminus(int),Z7))) ) ) ).
% nat_mult_distrib_neg
tff(fact_2763_fact__binomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K))) = divide_divide(A,semiring_char_0_fact(A,N2),semiring_char_0_fact(A,minus_minus(nat,N2,K))) ) ) ) ).
% fact_binomial
tff(fact_2764_binomial__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N2),K)) = divide_divide(A,semiring_char_0_fact(A,N2),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,minus_minus(nat,N2,K)))) ) ) ) ).
% binomial_fact
tff(fact_2765_or__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A4),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))) ) ).
% or_one_eq
tff(fact_2766_one__or__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))) ) ).
% one_or_eq
tff(fact_2767_choose__two,axiom,
! [N2: nat] : aa(nat,nat,binomial(N2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),minus_minus(nat,N2,one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% choose_two
tff(fact_2768_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : bit_se2239418461657761734s_mask(A,N2) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2),one_one(A)) ) ).
% mask_eq_exp_minus_1
tff(fact_2769_or__int__rec,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
| ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% or_int_rec
tff(fact_2770_signed__take__bit__eq__take__bit__minus,axiom,
! [N2: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N2,K) = minus_minus(int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N2),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N2))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2)))) ).
% signed_take_bit_eq_take_bit_minus
tff(fact_2771_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),N2))) ) ).
% push_bit_numeral_minus_1
tff(fact_2772_or__nat__unfold,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N2) = $ite(
M = zero_zero(nat),
N2,
$ite(N2 = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).
% or_nat_unfold
tff(fact_2773_xor__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(4)
tff(fact_2774_xor__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(6)
tff(fact_2775_one__xor__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A4) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))) ) ).
% one_xor_eq
tff(fact_2776_bit__0__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).
% bit_0_eq
tff(fact_2777_xor__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% xor_numerals(3)
tff(fact_2778_xor__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).
% xor_numerals(1)
tff(fact_2779_xor__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).
% xor_numerals(2)
tff(fact_2780_xor__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).
% xor_numerals(5)
tff(fact_2781_xor__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).
% xor_numerals(8)
tff(fact_2782_push__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N2),A4) = bit_se4730199178511100633sh_bit(A,N2,aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% push_bit_Suc
tff(fact_2783_xor__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% xor_numerals(7)
tff(fact_2784_push__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : bit_se4730199178511100633sh_bit(A,N2,one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2) ) ).
% push_bit_of_1
tff(fact_2785_flip__bit__eq__xor,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se8732182000553998342ip_bit(A,N2,A4) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A4),bit_se4730199178511100633sh_bit(A,N2,one_one(A))) ) ).
% flip_bit_eq_xor
tff(fact_2786_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A,N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),N2)
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),bit_se4730199178511100633sh_bit(A,N2,one_one(A))) != zero_zero(A) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_2787_bit__1__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),N2)
<=> ( N2 = zero_zero(nat) ) ) ) ).
% bit_1_iff
tff(fact_2788_not__bit__1__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N2)) ) ).
% not_bit_1_Suc
tff(fact_2789_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N2: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N2)) ) ).
% bit_numeral_simps(1)
tff(fact_2790_set__bit__eq__or,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se5668285175392031749et_bit(A,N2,A4) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A4),bit_se4730199178511100633sh_bit(A,N2,one_one(A))) ) ).
% set_bit_eq_or
tff(fact_2791_push__bit__int__def,axiom,
! [N2: nat,K: int] : bit_se4730199178511100633sh_bit(int,N2,K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N2)) ).
% push_bit_int_def
tff(fact_2792_push__bit__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se4730199178511100633sh_bit(A,N2,aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N2,A4)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% push_bit_double
tff(fact_2793_push__bit__nat__def,axiom,
! [N2: nat,M: nat] : bit_se4730199178511100633sh_bit(nat,N2,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ).
% push_bit_nat_def
tff(fact_2794_push__bit__eq__mult,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat,A4: A] : bit_se4730199178511100633sh_bit(A,N2,A4) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) ) ).
% push_bit_eq_mult
tff(fact_2795_set__bit__eq,axiom,
! [N2: nat,K: int] : bit_se5668285175392031749et_bit(int,N2,K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N2))) ).
% set_bit_eq
tff(fact_2796_unset__bit__eq,axiom,
! [N2: nat,K: int] : bit_se2638667681897837118et_bit(int,N2,K) = minus_minus(int,K,aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N2))) ).
% unset_bit_eq
tff(fact_2797_even__bit__succ__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,N2: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A4)),N2)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),N2)
| ( N2 = zero_zero(nat) ) ) ) ) ) ).
% even_bit_succ_iff
tff(fact_2798_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,N2: nat] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4)
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),N2)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,minus_minus(A,A4,one_one(A))),N2)
| ( N2 = zero_zero(nat) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_2799_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A,B3: A,N2: nat] :
( ! [J4: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),aa(nat,nat,suc,J4))
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),N2)
<=> $ite(N2 = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),N2)) ) ) ) ).
% bit_sum_mult_2_cases
tff(fact_2800_or__nat__rec,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N2) = aa(nat,nat,
aa(nat,fun(nat,nat),plus_plus(nat),
aa($o,nat,zero_neq_one_of_bool(nat),
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),M)
| ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2) ))),
aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% or_nat_rec
tff(fact_2801_xor__nat__unfold,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N2) = $ite(
M = zero_zero(nat),
N2,
$ite(N2 = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).
% xor_nat_unfold
tff(fact_2802_take__bit__Suc__from__most,axiom,
! [N2: nat,K: int] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N2),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N2)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2)))),bit_se2584673776208193580ke_bit(int,N2,K)) ).
% take_bit_Suc_from_most
tff(fact_2803_xor__int__rec,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K) != ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% xor_int_rec
tff(fact_2804_xor__nat__rec,axiom,
! [M: nat,N2: nat] :
aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N2) = aa(nat,nat,
aa(nat,fun(nat,nat),plus_plus(nat),
aa($o,nat,zero_neq_one_of_bool(nat),
~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),M) != ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N2))),
aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% xor_nat_rec
tff(fact_2805_xor__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A4),one_one(A)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A4))) ) ).
% xor_one_eq
tff(fact_2806_xor__int__unfold,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
K = aa(int,int,uminus_uminus(int),one_one(int)),
aa(int,int,bit_ri4277139882892585799ns_not(int),L),
$ite(
L = aa(int,int,uminus_uminus(int),one_one(int)),
aa(int,int,bit_ri4277139882892585799ns_not(int),K),
$ite(
K = zero_zero(int),
L,
$ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).
% xor_int_unfold
tff(fact_2807_not__int__rec,axiom,
! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% not_int_rec
tff(fact_2808_card__UNION,axiom,
! [A: $tType,A3: set(set(A))] :
( aa(set(set(A)),$o,finite_finite2(set(A)),A3)
=> ( ! [X3: set(A)] :
( member(set(A),X3,A3)
=> aa(set(A),$o,finite_finite2(A),X3) )
=> ( finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = nat2(aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_aa(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_ab(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).
% card_UNION
tff(fact_2809_or__not__numerals_I8_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2))))) ).
% or_not_numerals(8)
tff(fact_2810_or__not__numerals_I9_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2))))) ).
% or_not_numerals(9)
tff(fact_2811_and__not__numerals_I8_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2))))) ).
% and_not_numerals(8)
tff(fact_2812_image__ident,axiom,
! [A: $tType,Y4: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ac(A,A)),Y4) = Y4 ).
% image_ident
tff(fact_2813_vimage__Collect__eq,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),P: fun(B,$o)] : vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_ad(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P)) ).
% vimage_Collect_eq
tff(fact_2814_vimage__ident,axiom,
! [A: $tType,Y4: set(A)] : vimage(A,A,aTP_Lamp_ac(A,A),Y4) = Y4 ).
% vimage_ident
tff(fact_2815_map__prod__ident,axiom,
! [B: $tType,A: $tType,X4: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_ac(A,A),aTP_Lamp_ae(B,B)),X4) = X4 ).
% map_prod_ident
tff(fact_2816_bool__assn__proper_I4_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_af(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).
% bool_assn_proper(4)
tff(fact_2817_bool__assn__proper_I3_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ag(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).
% bool_assn_proper(3)
tff(fact_2818_bool__assn__proper_I2_J,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
% bool_assn_proper(2)
tff(fact_2819_singleton__conv2,axiom,
! [A: $tType,A4: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) ).
% singleton_conv2
tff(fact_2820_singleton__conv,axiom,
! [A: $tType,A4: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(A,fun(A,$o),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) ).
% singleton_conv
tff(fact_2821_Collect__const,axiom,
! [A: $tType,P: $o] :
aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aj($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).
% Collect_const
tff(fact_2822_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ak(B,A)),A3) = one_one(A) ) ).
% prod.neutral_const
tff(fact_2823_Field__square,axiom,
! [A: $tType,X: set(A)] : aa(set(product_prod(A,A)),set(A),field2(A),product_Sigma(A,A,X,aTP_Lamp_al(set(A),fun(A,set(A)),X))) = X ).
% Field_square
tff(fact_2824_bool__assn__proper_I5_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_am(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ) ).
% bool_assn_proper(5)
tff(fact_2825_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_an(B,A)),A3)) = bot_bot(A) ) ).
% SUP_bot
tff(fact_2826_SUP__bot__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: fun(B,A),A3: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A3)) = bot_bot(A) )
<=> ! [X2: B] :
( member(B,X2,A3)
=> ( aa(B,A,B5,X2) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(1)
tff(fact_2827_SUP__bot__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: fun(B,A),A3: set(B)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A3)) )
<=> ! [X2: B] :
( member(B,X2,A3)
=> ( aa(B,A,B5,X2) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(2)
tff(fact_2828_cSUP__const,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),C2: B] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ao(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).
% cSUP_const
tff(fact_2829_SUP__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [A3: set(A),F2: B] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ap(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).
% SUP_const
tff(fact_2830_INF__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [A3: set(A),F2: B] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ap(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).
% INF_const
tff(fact_2831_cINF__const,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),C2: B] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ao(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).
% cINF_const
tff(fact_2832_prod_Odelta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A4: A,B3: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_aq(A,fun(fun(A,B),fun(A,B)),A4),B3)),S) = $ite(member(A,A4,S),aa(A,B,B3,A4),one_one(B)) ) ) ) ).
% prod.delta
tff(fact_2833_prod_Odelta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A4: A,B3: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ar(A,fun(fun(A,B),fun(A,B)),A4),B3)),S) = $ite(member(A,A4,S),aa(A,B,B3,A4),one_one(B)) ) ) ) ).
% prod.delta'
tff(fact_2834_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_as(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image2(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_at(fun(B,$o),fun(B,$o),P))))) ).
% if_image_distrib
tff(fact_2835_UN__constant,axiom,
! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_au(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),C2) ).
% UN_constant
tff(fact_2836_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A3: set(B)] :
vimage(A,B,aTP_Lamp_av(B,fun(A,B),C2),A3) = $ite(member(B,C2,A3),top_top(set(A)),bot_bot(set(A))) ).
% vimage_const
tff(fact_2837_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)) = bot_bot(set(product_prod(A,B))) )
<=> ( ( A3 = bot_bot(set(A)) )
| ( B5 = bot_bot(set(B)) ) ) ) ).
% Times_empty
tff(fact_2838_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set(A)] : product_Sigma(A,B,A3,aTP_Lamp_ax(A,set(B))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty2
tff(fact_2839_Compl__Times__UNIV1,axiom,
! [A: $tType,B: $tType,A3: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aw(set(B),fun(A,set(B)),A3))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ay(set(B),fun(A,set(B)),A3)) ).
% Compl_Times_UNIV1
tff(fact_2840_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A3: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_az(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A3),aTP_Lamp_az(A,set(B))) ).
% Compl_Times_UNIV2
tff(fact_2841_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_az(A,set(B))) = top_top(set(product_prod(A,B))) ).
% UNIV_Times_UNIV
tff(fact_2842_bit_Ocompl__one,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% bit.compl_one
tff(fact_2843_bit_Ocompl__zero,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% bit.compl_zero
tff(fact_2844_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_left
tff(fact_2845_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_right
tff(fact_2846_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_right
tff(fact_2847_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_left
tff(fact_2848_bit_Oxor__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).
% bit.xor_one_right
tff(fact_2849_bit_Oxor__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).
% bit.xor_one_left
tff(fact_2850_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F2: fun(B,A),A3: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = bot_bot(A) )
<=> ! [X2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
=> ? [Xa2: B] :
( member(B,Xa2,A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa2)),X2) ) ) ) ) ).
% INF_eq_bot_iff
tff(fact_2851_range__constant,axiom,
! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_ba(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).
% range_constant
tff(fact_2852_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,A4: A,B5: fun(B,set(A)),C6: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bb(A,fun(fun(B,set(A)),fun(B,set(A))),A4),B5)),C6)) = $ite(C6 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6)))) ).
% UN_simps(1)
tff(fact_2853_UN__singleton,axiom,
! [A: $tType,A3: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_bc(A,set(A))),A3)) = A3 ).
% UN_singleton
tff(fact_2854_sum__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bd(A,fun(B,A),Y)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A3))),Y) ) ).
% sum_constant
tff(fact_2855_UN__simps_I2_J,axiom,
! [B: $tType,A: $tType,A3: fun(B,set(A)),B5: set(A),C6: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_be(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6))),B5)) ).
% UN_simps(2)
tff(fact_2856_UN__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bf(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6)))) ).
% UN_simps(3)
tff(fact_2857_UN__insert,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A4: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B5,A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3))) ).
% UN_insert
tff(fact_2858_Max__const,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [A3: set(A),C2: B] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_bg(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).
% Max_const
tff(fact_2859_Min__const,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [A3: set(A),C2: B] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_bg(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).
% Min_const
tff(fact_2860_INT__constant,axiom,
! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_au(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(set(A)),C2) ).
% INT_constant
tff(fact_2861_INT__insert,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A4: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B5,A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3))) ).
% INT_insert
tff(fact_2862_fst__image__times,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) = $ite(B5 = bot_bot(set(B)),bot_bot(set(A)),A3) ).
% fst_image_times
tff(fact_2863_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set(B),B5: set(A)] :
aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),B5))) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),B5) ).
% snd_image_times
tff(fact_2864_vimage__if,axiom,
! [B: $tType,A: $tType,B5: set(A),C2: B,D3: B,A3: set(B)] :
vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_bh(set(A),fun(B,fun(B,fun(A,B))),B5),C2),D3),A3) = $ite(
member(B,C2,A3),
$ite(member(B,D3,A3),top_top(set(A)),B5),
$ite(member(B,D3,A3),aa(set(A),set(A),uminus_uminus(set(A)),B5),bot_bot(set(A))) ) ).
% vimage_if
tff(fact_2865_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A3: set(A),X5: set(B)] : minus_minus(set(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),X5)),product_Sigma(A,B,A3,aTP_Lamp_az(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_UNIV_cancel
tff(fact_2866_pairself__image__cart,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B5: set(B)] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F2)),product_Sigma(B,B,A3,aTP_Lamp_bi(set(B),fun(B,set(B)),B5))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F2),A3),aa(set(B),fun(A,set(A)),aTP_Lamp_bj(fun(B,A),fun(set(B),fun(A,set(A))),F2),B5)) ).
% pairself_image_cart
tff(fact_2867_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] : bit_se4730199178511100633sh_bit(A,N2,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N2)) ) ).
% push_bit_minus_one_eq_not_mask
tff(fact_2868_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C6: set(A),A3: set(B),B5: set(B)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,C6,aTP_Lamp_aw(set(B),fun(A,set(B)),A3))),product_Sigma(A,B,C6,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
<=> ( ( C6 = bot_bot(set(A)) )
| aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),A3),B5) ) ) ).
% disjnt_Times1_iff
tff(fact_2869_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A3: set(A),C6: set(B),B5: set(A)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),C6)))
<=> ( ( C6 = bot_bot(set(B)) )
| aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A3),B5) ) ) ).
% disjnt_Times2_iff
tff(fact_2870_sum__zero__power,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: fun(nat,A),A3: set(nat)] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
( aa(set(nat),$o,finite_finite2(nat),A3)
& member(nat,zero_zero(nat),A3) ),
aa(nat,A,C2,zero_zero(nat)),
zero_zero(A) ) ) ).
% sum_zero_power
tff(fact_2871_sum__of__bool__mult__eq,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [A3: set(A),P: fun(A,$o),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_bl(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).
% sum_of_bool_mult_eq
tff(fact_2872_sum__mult__of__bool__eq,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [A3: set(A),F2: fun(A,B),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_bm(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).
% sum_mult_of_bool_eq
tff(fact_2873_INT__simps_I1_J,axiom,
! [B: $tType,A: $tType,A3: fun(B,set(A)),B5: set(A),C6: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bn(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6))),B5)) ).
% INT_simps(1)
tff(fact_2874_INT__simps_I2_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bo(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6)))) ).
% INT_simps(2)
tff(fact_2875_INT__simps_I3_J,axiom,
! [B: $tType,A: $tType,A3: fun(B,set(A)),B5: set(A),C6: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bp(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),top_top(set(A)),minus_minus(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6)),B5)) ).
% INT_simps(3)
tff(fact_2876_insert__Times__insert,axiom,
! [A: $tType,B: $tType,A4: A,A3: set(A),B3: B,B5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3),aa(set(B),fun(A,set(B)),aTP_Lamp_bq(B,fun(set(B),fun(A,set(B))),B3),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A3,aa(set(B),fun(A,set(B)),aTP_Lamp_bq(B,fun(set(B),fun(A,set(B))),B3),B5))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3),aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))) ).
% insert_Times_insert
tff(fact_2877_not__one__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).
% not_one_eq
tff(fact_2878_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(A,C),A3: set(A)] :
( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F2),product_Sigma(A,B,A3,aTP_Lamp_az(A,set(B))))
<=> inj_on(A,C,F2,A3) ) ).
% inj_on_apfst
tff(fact_2879_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,C),A3: set(B)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aw(set(B),fun(A,set(B)),A3)))
<=> inj_on(B,C,F2,A3) ) ).
% inj_on_apsnd
tff(fact_2880_sum__zero__power_H,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: fun(nat,A),D3: fun(nat,A),A3: set(nat)] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A3) = $ite(
( aa(set(nat),$o,finite_finite2(nat),A3)
& member(nat,zero_zero(nat),A3) ),
divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))),
zero_zero(A) ) ) ).
% sum_zero_power'
tff(fact_2881_INT__simps_I4_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bs(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6)) = $ite(C6 = bot_bot(set(B)),top_top(set(A)),minus_minus(set(A),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6)))) ).
% INT_simps(4)
tff(fact_2882_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),S))
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R3),S) ) ).
% pred_subset_eq2
tff(fact_2883_Pow__def,axiom,
! [A: $tType,A3: set(A)] : pow2(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_bu(set(A),fun(set(A),$o),A3)) ).
% Pow_def
tff(fact_2884_less__set__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B5)
<=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B5)) ) ).
% less_set_def
tff(fact_2885_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B5: set(B)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
=> member(B,aa(A,B,F2,X3),B5) )
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(fun(A,$o),set(A),collect(A),P))),B5) ) ).
% image_Collect_subsetI
tff(fact_2886_Collect__subset,axiom,
! [A: $tType,A3: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).
% Collect_subset
tff(fact_2887_less__eq__set__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
<=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B5)) ) ).
% less_eq_set_def
tff(fact_2888_pred__subset__eq,axiom,
! [A: $tType,R3: set(A),S: set(A)] :
( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S))
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R3),S) ) ).
% pred_subset_eq
tff(fact_2889_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set(A),X5: set(A),P: fun(A,$o)] :
( member(A,X,Z5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
=> aa(A,$o,P,X) ) ) ).
% prop_restrict
tff(fact_2890_Collect__restrict,axiom,
! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).
% Collect_restrict
tff(fact_2891_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),S2: set(B)] :
( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F2),S2))
<=> pairwise(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_bw(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F2),S2) ) ).
% pairwise_image
tff(fact_2892_inv__imagep__def,axiom,
! [B: $tType,A: $tType,R2: fun(B,fun(B,$o)),F2: fun(A,B),X4: A,Xa3: A] :
( inv_imagep(B,A,R2,F2,X4,Xa3)
<=> aa(B,$o,aa(B,fun(B,$o),R2,aa(A,B,F2,X4)),aa(A,B,F2,Xa3)) ) ).
% inv_imagep_def
tff(fact_2893_uminus__set__def,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3))) ).
% uminus_set_def
tff(fact_2894_Collect__neg__eq,axiom,
! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P)) ).
% Collect_neg_eq
tff(fact_2895_Compl__eq,axiom,
! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_by(set(A),fun(A,$o),A3)) ).
% Compl_eq
tff(fact_2896_minus__set__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : minus_minus(set(A),A3,B5) = aa(fun(A,$o),set(A),collect(A),minus_minus(fun(A,$o),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B5))) ).
% minus_set_def
tff(fact_2897_set__diff__eq,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : minus_minus(set(A),A3,B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bz(set(A),fun(set(A),fun(A,$o)),A3),B5)) ).
% set_diff_eq
tff(fact_2898_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F2: fun(B,A)] :
( member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
=> ~ ! [X3: B] : B3 != aa(B,A,F2,X3) ) ).
% rangeE
tff(fact_2899_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ca(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F2),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).
% range_composition
tff(fact_2900_vimage__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),B5: set(B)] : vimage(A,B,F2,B5) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_cb(fun(A,B),fun(set(B),fun(A,$o)),F2),B5)) ).
% vimage_def
tff(fact_2901_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set(A),A3: set(B),B5: set(B)] :
( member(A,X,C6)
=> ( ( product_Sigma(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),C6)) = product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),C6)) )
<=> ( A3 = B5 ) ) ) ).
% Times_eq_cancel2
tff(fact_2902_Sigma__cong,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(A),C6: fun(A,set(B)),D4: fun(A,set(B))] :
( ( A3 = B5 )
=> ( ! [X3: A] :
( member(A,X3,B5)
=> ( aa(A,set(B),C6,X3) = aa(A,set(B),D4,X3) ) )
=> ( product_Sigma(A,B,A3,C6) = product_Sigma(A,B,B5,D4) ) ) ) ).
% Sigma_cong
tff(fact_2903_Collect__conj__eq,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).
% Collect_conj_eq
tff(fact_2904_Int__Collect,axiom,
! [A: $tType,X: A,A3: set(A),P: fun(A,$o)] :
( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))
<=> ( member(A,X,A3)
& aa(A,$o,P,X) ) ) ).
% Int_Collect
tff(fact_2905_Int__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_cd(set(A),fun(set(A),fun(A,$o)),A3),B5)) ).
% Int_def
tff(fact_2906_inf__set__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B5))) ).
% inf_set_def
tff(fact_2907_inf__Int__eq,axiom,
! [A: $tType,R3: set(A),S: set(A),X4: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
<=> member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R3),S)) ) ).
% inf_Int_eq
tff(fact_2908_pairwise__trivial,axiom,
! [A: $tType,I: set(A)] : pairwise(A,aTP_Lamp_ce(A,fun(A,$o)),I) ).
% pairwise_trivial
tff(fact_2909_curry__K,axiom,
! [B: $tType,A: $tType,C: $tType,C2: C,X4: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aTP_Lamp_cf(C,fun(product_prod(A,B),C),C2)),X4),Xa3) = C2 ).
% curry_K
tff(fact_2910_Collect__disj__eq,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).
% Collect_disj_eq
tff(fact_2911_Un__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ch(set(A),fun(set(A),fun(A,$o)),A3),B5)) ).
% Un_def
tff(fact_2912_sup__set__def,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B5))) ).
% sup_set_def
tff(fact_2913_sup__Un__eq,axiom,
! [A: $tType,R3: set(A),S: set(A),X4: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
<=> member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R3),S)) ) ).
% sup_Un_eq
tff(fact_2914_UNIV__def,axiom,
! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ci(A,$o)) ).
% UNIV_def
tff(fact_2915_Compr__image__eq,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A3),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A3),P))) ).
% Compr_image_eq
tff(fact_2916_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image2(B,A,F2),aa(set(C),set(B),image2(C,B,G),A3)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cl(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).
% image_image
tff(fact_2917_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F2: fun(B,A),A3: set(B)] :
( member(A,B3,aa(set(B),set(A),image2(B,A,F2),A3))
=> ~ ! [X3: B] :
( ( B3 = aa(B,A,F2,X3) )
=> ~ member(B,X3,A3) ) ) ).
% imageE
tff(fact_2918_Collect__imp__eq,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).
% Collect_imp_eq
tff(fact_2919_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Ha: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cn(fun(B,A),fun(fun(B,A),fun(B,A)),G),Ha)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Ha),A3)) ) ).
% prod.distrib
tff(fact_2920_sum__product,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),B5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B5)),A3) ) ).
% sum_product
tff(fact_2921_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),A3: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F2),R2)),A3) ) ).
% sum_distrib_right
tff(fact_2922_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [R2: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cr(A,fun(fun(B,A),fun(B,A)),R2),F2)),A3) ) ).
% sum_distrib_left
tff(fact_2923_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_cs(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_2924_lambda__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ( aTP_Lamp_ct(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).
% lambda_zero
tff(fact_2925_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R3),S)) ) ).
% inf_Int_eq2
tff(fact_2926_underS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] : order_underS(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_cu(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A4)) ).
% underS_def
tff(fact_2927_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) ) ).
% SUP_Sup_eq2
tff(fact_2928_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( ! [X2: A,Xa2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2),R3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2),S) )
<=> ( R3 = S ) ) ).
% pred_equals_eq2
tff(fact_2929_under__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] : order_under(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_cv(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A4)) ).
% under_def
tff(fact_2930_top__empty__eq2,axiom,
! [B: $tType,A: $tType,X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),top_top(set(product_prod(A,B)))) ) ).
% top_empty_eq2
tff(fact_2931_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),bot_bot(set(product_prod(A,B)))) ) ).
% bot_empty_eq2
tff(fact_2932_insert__Collect,axiom,
! [A: $tType,A4: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cw(A,fun(fun(A,$o),fun(A,$o)),A4),P)) ).
% insert_Collect
tff(fact_2933_insert__compr,axiom,
! [A: $tType,A4: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_cx(A,fun(set(A),fun(A,$o)),A4),B5)) ).
% insert_compr
tff(fact_2934_Set_Oempty__def,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cy(A,$o)) ).
% Set.empty_def
tff(fact_2935_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B),C6: set(A),D4: set(B)] :
( ( product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)) = product_Sigma(A,B,C6,aTP_Lamp_aw(set(B),fun(A,set(B)),D4)) )
<=> ( ( ( A3 = C6 )
& ( B5 = D4 ) )
| ( ( ( A3 = bot_bot(set(A)) )
| ( B5 = bot_bot(set(B)) ) )
& ( ( C6 = bot_bot(set(A)) )
| ( D4 = bot_bot(set(B)) ) ) ) ) ) ).
% times_eq_iff
tff(fact_2936_insert__def,axiom,
! [A: $tType,A4: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ai(A,fun(A,$o),A4))),B5) ).
% insert_def
tff(fact_2937_inj__singleton,axiom,
! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_bc(A,set(A)),A3) ).
% inj_singleton
tff(fact_2938_Collect__conv__if2,axiom,
! [A: $tType,A4: A,P: fun(A,$o)] :
aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cz(A,fun(fun(A,$o),fun(A,$o)),A4),P)) = $ite(aa(A,$o,P,A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))),bot_bot(set(A))) ).
% Collect_conv_if2
tff(fact_2939_Collect__conv__if,axiom,
! [A: $tType,A4: A,P: fun(A,$o)] :
aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_da(A,fun(fun(A,$o),fun(A,$o)),A4),P)) = $ite(aa(A,$o,P,A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))),bot_bot(set(A))) ).
% Collect_conv_if
tff(fact_2940_INF__filter__not__bot,axiom,
! [A: $tType,B: $tType,B5: set(A),F4: fun(A,filter(B))] :
( ! [X7: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5)
=> ( aa(set(A),$o,finite_finite2(A),X7)
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),X7)) != bot_bot(filter(B)) ) ) )
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B5)) != bot_bot(filter(B)) ) ) ).
% INF_filter_not_bot
tff(fact_2941_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_db(fun(A,B),fun(A,product_prod(A,B)),F2),X5) ).
% inj_on_convol_ident
tff(fact_2942_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_db(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).
% inj_Pair(1)
tff(fact_2943_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_dc(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).
% inj_Pair(2)
tff(fact_2944_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X4: fun(product_prod(A,B),C),Xa3: A,Xb2: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),X4),Xa3),Xb2) = aa(product_prod(A,B),C,X4,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa3),Xb2)) ).
% curry_def
tff(fact_2945_SUP__Sup__eq,axiom,
! [A: $tType,S: set(set(A)),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ).
% SUP_Sup_eq
tff(fact_2946_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S)) ) ).
% INF_Int_eq2
tff(fact_2947_INF__Int__eq,axiom,
! [A: $tType,S: set(set(A)),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ).
% INF_Int_eq
tff(fact_2948_bot__assn__def,axiom,
bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
% bot_assn_def
tff(fact_2949_prod_Ofinite__Collect__op,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I: set(A),X: fun(A,B),Y: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),I),X)))
=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),I),Y)))
=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_de(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I),X),Y))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_2950_image__constant__conv,axiom,
! [B: $tType,A: $tType,C2: A,A3: set(B)] :
aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_ba(A,fun(B,A)),C2)),A3) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A)))) ).
% image_constant_conv
tff(fact_2951_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A3: set(A),C2: B] :
( member(A,X,A3)
=> ( aa(set(A),set(B),image2(A,B,aTP_Lamp_av(B,fun(A,B),C2)),A3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C2),bot_bot(set(B))) ) ) ).
% image_constant
tff(fact_2952_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,M: nat,I: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),X),M)),I) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),I)) ) ).
% sum_power_add
tff(fact_2953_prod_Ointer__filter,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_dg(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).
% prod.inter_filter
tff(fact_2954_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(B,A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( inj_on(B,A,F2,top_top(set(B)))
=> aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_dh(set(A),fun(fun(B,A),fun(B,$o)),S),F2))) ) ) ).
% finite_Collect
tff(fact_2955_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_di(B,set(A))),A3)) = bot_bot(set(A)) ).
% UN_empty2
tff(fact_2956_UN__empty,axiom,
! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),bot_bot(set(B)))) = bot_bot(set(A)) ).
% UN_empty
tff(fact_2957_UNION__empty__conv_I1_J,axiom,
! [B: $tType,A: $tType,B5: fun(B,set(A)),A3: set(B)] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3)) )
<=> ! [X2: B] :
( member(B,X2,A3)
=> ( aa(B,set(A),B5,X2) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(1)
tff(fact_2958_UNION__empty__conv_I2_J,axiom,
! [B: $tType,A: $tType,B5: fun(B,set(A)),A3: set(B)] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3)) = bot_bot(set(A)) )
<=> ! [X2: B] :
( member(B,X2,A3)
=> ( aa(B,set(A),B5,X2) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(2)
tff(fact_2959_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set(A),A4: B,B5: fun(A,set(B))] :
( member(A,U,A3)
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_dj(B,fun(fun(A,set(B)),fun(A,set(B))),A4),B5)),A3)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A3))) ) ) ).
% UN_insert_distrib
tff(fact_2960_vimage__fst,axiom,
! [B: $tType,A: $tType,A3: set(A)] : vimage(product_prod(A,B),A,product_fst(A,B),A3) = product_Sigma(A,B,A3,aTP_Lamp_az(A,set(B))) ).
% vimage_fst
tff(fact_2961_vimage__snd,axiom,
! [A: $tType,B: $tType,A3: set(B)] : vimage(product_prod(A,B),B,product_snd(A,B),A3) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aw(set(B),fun(A,set(B)),A3)) ).
% vimage_snd
tff(fact_2962_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set(A),A3: set(B),B5: set(B)] :
( member(A,X,C6)
=> ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),C6))),product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),C6)))
<=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),B5) ) ) ).
% Times_subset_cancel2
tff(fact_2963_in__prod__fst__sndI,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A3: set(A),B5: set(B)] :
( member(A,aa(product_prod(A,B),A,product_fst(A,B),X),A3)
=> ( member(B,aa(product_prod(A,B),B,product_snd(A,B),X),B5)
=> member(product_prod(A,B),X,product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) ) ) ).
% in_prod_fst_sndI
tff(fact_2964_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A3: set(A),B5: set(B)] :
( member(product_prod(A,B),X,product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
<=> ( member(A,aa(product_prod(A,B),A,product_fst(A,B),X),A3)
& member(B,aa(product_prod(A,B),B,product_snd(A,B),X),B5) ) ) ).
% mem_Times_iff
tff(fact_2965_R__subset__Field,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R3),aTP_Lamp_dk(set(product_prod(A,A)),fun(A,set(A)),R3))) ).
% R_subset_Field
tff(fact_2966_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set(A),A4: B,B5: fun(A,set(B))] :
( member(A,U,A3)
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_dj(B,fun(fun(A,set(B)),fun(A,set(B))),A4),B5)),A3)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A3))) ) ) ).
% INT_insert_distrib
tff(fact_2967_INT__extend__simps_I5_J,axiom,
! [A: $tType,B: $tType,A4: A,B5: fun(B,set(A)),C6: set(B)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bb(A,fun(fun(B,set(A)),fun(B,set(A))),A4),B5)),C6)) ).
% INT_extend_simps(5)
tff(fact_2968_Sigma__empty__iff,axiom,
! [A: $tType,B: $tType,I: set(A),X5: fun(A,set(B))] :
( ( product_Sigma(A,B,I,X5) = bot_bot(set(product_prod(A,B))) )
<=> ! [X2: A] :
( member(A,X2,I)
=> ( aa(A,set(B),X5,X2) = bot_bot(set(B)) ) ) ) ).
% Sigma_empty_iff
tff(fact_2969_Image__UN,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B5: fun(C,set(B)),A3: set(C)] : image(B,A,R2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dl(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B5)),A3)) ).
% Image_UN
tff(fact_2970_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ~ ! [F: fun(A,nat)] :
( ? [N3: nat] : aa(set(A),set(nat),image2(A,nat,F),A3) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_dm(nat,fun(nat,$o),N3))
=> ~ inj_on(A,nat,F,A3) ) ) ).
% finite_imp_inj_to_nat_seg'
tff(fact_2971_Times__Int__Times,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B),C6: set(A),D4: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))),product_Sigma(A,B,C6,aTP_Lamp_aw(set(B),fun(A,set(B)),D4))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6),aa(set(B),fun(A,set(B)),aTP_Lamp_dn(set(B),fun(set(B),fun(A,set(B))),B5),D4)) ).
% Times_Int_Times
tff(fact_2972_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I: set(A),A3: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_do(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I,A3)),product_Sigma(A,B,I,B5)) ).
% Sigma_Int_distrib2
tff(fact_2973_Times__Int__distrib1,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(A),C6: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5),aTP_Lamp_aw(set(B),fun(A,set(B)),C6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))) ).
% Times_Int_distrib1
tff(fact_2974_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I: set(A),A3: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_dp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I,A3)),product_Sigma(A,B,I,B5)) ).
% Sigma_Un_distrib2
tff(fact_2975_Times__Un__distrib1,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(A),C6: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5),aTP_Lamp_aw(set(B),fun(A,set(B)),C6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))) ).
% Times_Un_distrib1
tff(fact_2976_Id__fstsnd__eq,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_dq(product_prod(A,A),$o)) ).
% Id_fstsnd_eq
tff(fact_2977_relcomp__subset__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,B)),A3: set(A),B5: set(B),S2: set(product_prod(B,C)),C6: set(C)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
=> ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S2),product_Sigma(B,C,B5,aTP_Lamp_dr(set(C),fun(B,set(C)),C6)))
=> aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R2,S2)),product_Sigma(A,C,A3,aTP_Lamp_ds(set(C),fun(A,set(C)),C6))) ) ) ).
% relcomp_subset_Sigma
tff(fact_2978_Sigma__Union,axiom,
! [B: $tType,A: $tType,X5: set(set(A)),B5: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X5),B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_dt(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B5)),X5)) ).
% Sigma_Union
tff(fact_2979_Times__Diff__distrib1,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(A),C6: set(B)] : product_Sigma(A,B,minus_minus(set(A),A3,B5),aTP_Lamp_aw(set(B),fun(A,set(B)),C6)) = minus_minus(set(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),C6)),product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))) ).
% Times_Diff_distrib1
tff(fact_2980_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I: set(A),A3: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_du(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B5)) = minus_minus(set(product_prod(A,B)),product_Sigma(A,B,I,A3),product_Sigma(A,B,I,B5)) ).
% Sigma_Diff_distrib2
tff(fact_2981_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R3),S)) ) ).
% sup_Un_eq2
tff(fact_2982_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set(A),S3: set(A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_dv(set(A),fun(set(A),fun(set(A),$o)),S),S3))) ) ).
% finite_if_eq_beyond_finite
tff(fact_2983_SUP__UN__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_dw(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).
% SUP_UN_eq2
tff(fact_2984_Id__on__subset__Times,axiom,
! [A: $tType,A3: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id_on(A,A3)),product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),A3))) ).
% Id_on_subset_Times
tff(fact_2985_SUP__UN__eq,axiom,
! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_dx(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).
% SUP_UN_eq
tff(fact_2986_INF__INT__eq,axiom,
! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X4: A] :
( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_dx(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X4)
<=> member(A,X4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).
% INF_INT_eq
tff(fact_2987_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_dy(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).
% converse_UNION
tff(fact_2988_relcomp__UNION__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,S2: set(product_prod(A,C)),R2: fun(D,set(product_prod(C,B))),I: set(D)] : relcomp(A,C,B,S2,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R2),I))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_dz(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S2),R2)),I)) ).
% relcomp_UNION_distrib
tff(fact_2989_relcomp__UNION__distrib2,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R2: fun(D,set(product_prod(A,C))),I: set(D),S2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image2(D,set(product_prod(A,C)),R2),I)),S2) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ea(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R2),S2)),I)) ).
% relcomp_UNION_distrib2
tff(fact_2990_INF__INT__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_dw(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).
% INF_INT_eq2
tff(fact_2991_product__swap,axiom,
! [A: $tType,B: $tType,A3: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),A3)) ).
% product_swap
tff(fact_2992_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_dy(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).
% converse_INTER
tff(fact_2993_numeral__code_I3_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N2: num] :
aa(num,A,numeral_numeral(A),bit1(N2)) = $let(
m: A,
m:= aa(num,A,numeral_numeral(A),N2),
aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).
% numeral_code(3)
tff(fact_2994_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [C2: A,A3: set(B)] :
aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_eb(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(A),C2) ) ).
% SUP_constant
tff(fact_2995_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).
% SUP_empty
tff(fact_2996_INF__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).
% INF_empty
tff(fact_2997_INF__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [C2: A,A3: set(B)] :
aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_eb(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(A),C2) ) ).
% INF_constant
tff(fact_2998_INF__inf__const1,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),X: B,F2: fun(A,B)] :
( ( I != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ec(B,fun(fun(A,B),fun(A,B)),X),F2)),I)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I))) ) ) ) ).
% INF_inf_const1
tff(fact_2999_INF__inf__const2,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I: set(A),F2: fun(A,B),X: B] :
( ( I != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_ed(fun(A,B),fun(B,fun(A,B)),F2),X)),I)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I))),X) ) ) ) ).
% INF_inf_const2
tff(fact_3000_power__numeral__even,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W: num] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),bit0(W))) = $let(
w: A,
w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W)),
aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).
% power_numeral_even
tff(fact_3001_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),A4: B,A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),A3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A4)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))) ) ).
% SUP_insert
tff(fact_3002_INF__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),A4: B,A3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),A3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A4)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))) ) ).
% INF_insert
tff(fact_3003_sup__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ee(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).
% sup_assn_def
tff(fact_3004_inf__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ef(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).
% inf_assn_def
tff(fact_3005_power__numeral__odd,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W: num] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),bit1(W))) = $let(
w: A,
w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W)),
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),w)),w) ) ) ).
% power_numeral_odd
tff(fact_3006_prod_Ointer__restrict,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_eg(fun(A,B),fun(set(A),fun(A,B)),G),B5)),A3) ) ) ) ).
% prod.inter_restrict
tff(fact_3007_Image__singleton,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A4: B] : image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A4)) ).
% Image_singleton
tff(fact_3008_prod_Osetdiff__irrelevant,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),minus_minus(set(A),A3,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ei(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).
% prod.setdiff_irrelevant
tff(fact_3009_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,A4: A,B5: fun(B,set(A)),C6: set(B)] :
aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6))) = $ite(C6 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bb(A,fun(fun(B,set(A)),fun(B,set(A))),A4),B5)),C6))) ).
% UN_extend_simps(1)
tff(fact_3010_UN__extend__simps_I2_J,axiom,
! [A: $tType,B: $tType,A3: fun(B,set(A)),C6: set(B),B5: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6))),B5) = $ite(C6 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_be(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6))) ).
% UN_extend_simps(2)
tff(fact_3011_UN__extend__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6))) = $ite(C6 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bf(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6))) ).
% UN_extend_simps(3)
tff(fact_3012_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set(A),C6: set(B),B5: set(A),D4: set(B)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),D4)))
<=> ( ( A3 = bot_bot(set(A)) )
| ( C6 = bot_bot(set(B)) )
| ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C6),D4) ) ) ) ).
% times_subset_iff
tff(fact_3013_INT__empty,axiom,
! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),bot_bot(set(B)))) = top_top(set(A)) ).
% INT_empty
tff(fact_3014_INT__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,A3: fun(B,set(A)),C6: set(B),B5: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6))),B5) = $ite(C6 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bn(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6))) ).
% INT_extend_simps(1)
tff(fact_3015_INT__extend__simps_I2_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6))) = $ite(C6 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bo(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6))) ).
% INT_extend_simps(2)
tff(fact_3016_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),A3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_rtrancl(A,R2))
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),A3)))
=> ( ( A4 = B3 )
| member(A,A4,A3) ) ) ) ).
% trancl_subset_Sigma_aux
tff(fact_3017_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I: set(A),F2: fun(B,C),A3: fun(A,set(B))] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> inj_on(B,C,F2,aa(A,set(B),A3,I3)) )
=> inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I))) ) ) ).
% inj_on_INTER
tff(fact_3018_wfI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B5: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),B5)))
=> ( ! [X3: A,P3: fun(A,$o)] :
( ! [Xa3: A] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R2)
=> aa(A,$o,P3,Y2) )
=> aa(A,$o,P3,Xa3) )
=> ( member(A,X3,A3)
=> ( member(A,X3,B5)
=> aa(A,$o,P3,X3) ) ) )
=> wf(A,R2) ) ) ).
% wfI
tff(fact_3019_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
<=> ( ( A3 = bot_bot(set(A)) )
| ( B5 = bot_bot(set(B)) )
| ( aa(set(A),$o,finite_finite2(A),A3)
& aa(set(B),$o,finite_finite2(B),B5) ) ) ) ).
% finite_cartesian_product_iff
tff(fact_3020_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
=> ( ( A3 != bot_bot(set(A)) )
=> aa(set(B),$o,finite_finite2(B),B5) ) ) ).
% finite_cartesian_productD2
tff(fact_3021_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
=> ( ( B5 != bot_bot(set(B)) )
=> aa(set(A),$o,finite_finite2(A),A3) ) ) ).
% finite_cartesian_productD1
tff(fact_3022_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: fun(A,set(B))] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ej(set(A),fun(fun(A,set(B)),fun(A,$o)),A3),B5)))
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,A5)) )
=> aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,B5)) ) ) ).
% finite_SigmaI2
tff(fact_3023_quotient__diff1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),A4: A] :
( inj_on(A,set(set(A)),aTP_Lamp_ek(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
=> ( member(A,A4,A3)
=> ( equiv_quotient(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))),R2) = minus_minus(set(set(A)),equiv_quotient(A,A3,R2),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))),R2)) ) ) ) ).
% quotient_diff1
tff(fact_3024_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B)),A3: set(A),B5: set(B),C6: set(A)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R2,C6)),B5) ) ).
% Image_subset
tff(fact_3025_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E3: set(product_prod(A,A)),U2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E3),product_Sigma(A,A,U2,aTP_Lamp_al(set(A),fun(A,set(A)),U2)))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_rtrancl(A,E3)) ) ).
% Restr_rtrancl_mono
tff(fact_3026_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set(A),T3: set(B),R3: fun(A,fun(B,$o)),K: nat] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(B),$o,finite_finite2(B),T3)
=> ( ! [X3: B] :
( member(B,X3,T3)
=> ( finite_card(A,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_el(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R3),X3))) = K ) )
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_en(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R3)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(B,T3)) ) ) ) ) ).
% sum_multicount
tff(fact_3027_Image__INT__subset,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B5: fun(C,set(B)),A3: set(C)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A3)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dl(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B5)),A3))) ).
% Image_INT_subset
tff(fact_3028_nat__times__as__int,axiom,
! [X4: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa3) = 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),Xa3))) ).
% nat_times_as_int
tff(fact_3029_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,B5)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ej(set(A),fun(fun(A,set(B)),fun(A,$o)),A3),B5)) ).
% fst_image_Sigma
tff(fact_3030_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E3: set(product_prod(A,A)),U2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E3),product_Sigma(A,A,U2,aTP_Lamp_al(set(A),fun(A,set(A)),U2)))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,E3)) ) ).
% Restr_trancl_mono
tff(fact_3031_refl__on__def,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,A3,R2)
<=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),A3)))
& ! [X2: A] :
( member(A,X2,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2) ) ) ) ).
% refl_on_def
tff(fact_3032_refl__onI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),A3)))
=> ( ! [X3: A] :
( member(A,X3,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2) )
=> refl_on(A,A3,R2) ) ) ).
% refl_onI
tff(fact_3033_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_bot(A)
=> ordering_top(A,aTP_Lamp_eo(A,fun(A,$o)),aTP_Lamp_ep(A,fun(A,$o)),bot_bot(A)) ) ).
% bot.ordering_top_axioms
tff(fact_3034_homo__rel__restrict__mono,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B5: set(A),A3: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_al(set(A),fun(A,set(A)),B5)))
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R3,A3)),product_Sigma(A,A,minus_minus(set(A),B5,A3),aa(set(A),fun(A,set(A)),aTP_Lamp_eq(set(A),fun(set(A),fun(A,set(A))),B5),A3))) ) ).
% homo_rel_restrict_mono
tff(fact_3035_reflcl__set__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X4),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),id2(A))) ) ).
% reflcl_set_eq
tff(fact_3036_map__prod__surj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F2: fun(B,A),A3: set(B),A9: set(A),G: fun(D,C),B5: set(D),B12: set(C)] :
( ( aa(set(B),set(A),image2(B,A,F2),A3) = A9 )
=> ( ( aa(set(D),set(C),image2(D,C,G),B5) = B12 )
=> ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F2,G)),product_Sigma(B,D,A3,aTP_Lamp_es(set(D),fun(B,set(D)),B5))) = product_Sigma(A,C,A9,aTP_Lamp_ds(set(C),fun(A,set(C)),B12)) ) ) ) ).
% map_prod_surj_on
tff(fact_3037_rel__restrict__alt__def,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : rel_restrict(A,R3,A3) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(A),set(A),uminus_uminus(set(A)),A3),aTP_Lamp_et(set(A),fun(A,set(A)),A3))) ).
% rel_restrict_alt_def
tff(fact_3038_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] : finite_card(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,A3)),finite_card(B,B5)) ).
% card_cartesian_product
tff(fact_3039_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_eu(C,set(B))),F2) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G)),aTP_Lamp_ev(A,set(D))) ).
% empty_natural
tff(fact_3040_Int__Inter__eq_I1_J,axiom,
! [A: $tType,A3: set(A),B9: set(set(A))] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)) = $ite(B9 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B9))) ).
% Int_Inter_eq(1)
tff(fact_3041_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B9: set(set(A)),A3: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)),A3) = $ite(B9 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_ew(set(A),fun(set(A),set(A)),A3)),B9))) ).
% Int_Inter_eq(2)
tff(fact_3042_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F2: fun(A,B),A3: set(A),G: fun(C,D),B5: set(C)] :
( inj_on(A,B,F2,A3)
=> ( inj_on(C,D,G,B5)
=> inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F2,G),product_Sigma(A,C,A3,aTP_Lamp_ds(set(C),fun(A,set(C)),B5))) ) ) ).
% map_prod_inj_on
tff(fact_3043_fst__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ex(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% fst_diag_fst
tff(fact_3044_snd__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ey(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% snd_diag_snd
tff(fact_3045_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ez(fun(B,A),fun(B,set(A)),F2)),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).
% UNION_singleton_eq_range
tff(fact_3046_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),G: fun(A,B),B5: set(A)] :
( inj_on(A,B,F2,A3)
=> ( inj_on(A,B,G,B5)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,G),B5)) = bot_bot(set(B)) )
=> inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_fa(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ) ) ) ).
% inj_on_disjoint_Un
tff(fact_3047_Max__add__commute,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [S: set(A),F2: fun(A,B),K: B] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_fb(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F2),S))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_3048_Min__add__commute,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [S: set(A),F2: fun(A,B),K: B] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_fb(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F2),S))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_3049_less__cSUP__iff,axiom,
! [B: $tType,A: $tType] :
( condit6923001295902523014norder(B)
=> ! [A3: set(A),F2: fun(A,B),A4: B] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A4),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)))
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),A4),aa(A,B,F2,X2)) ) ) ) ) ) ).
% less_cSUP_iff
tff(fact_3050_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( condit6923001295902523014norder(B)
=> ! [A3: set(A),F2: fun(A,B),A4: B] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),A4)
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),A4) ) ) ) ) ) ).
% cINF_less_iff
tff(fact_3051_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),A3))
=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A3))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3052_cINF__inf__distrib,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),A3))
=> ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A3))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).
% cINF_inf_distrib
tff(fact_3053_Image__eq__UN,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B5: set(B)] : image(B,A,R2,B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fe(set(product_prod(B,A)),fun(B,set(A)),R2)),B5)) ).
% Image_eq_UN
tff(fact_3054_INT__extend__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: fun(B,set(A)),C6: set(B),B5: set(A)] :
minus_minus(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),C6)),B5) = $ite(C6 = bot_bot(set(B)),minus_minus(set(A),top_top(set(A)),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bp(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B5)),C6))) ).
% INT_extend_simps(3)
tff(fact_3055_prod_OIf__cases,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: set(A),P: fun(A,$o),Ha: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ff(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),Ha),G)),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Ha),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).
% prod.If_cases
tff(fact_3056_vimage__eq__UN,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),B5: set(B)] : vimage(A,B,F2,B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fg(fun(A,B),fun(B,set(A)),F2)),B5)) ).
% vimage_eq_UN
tff(fact_3057_INT__extend__simps_I4_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: fun(B,set(A)),C6: set(B)] :
minus_minus(set(A),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C6))) = $ite(C6 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bs(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C6))) ).
% INT_extend_simps(4)
tff(fact_3058_Sigma__Image,axiom,
! [A: $tType,B: $tType,A3: set(B),B5: fun(B,set(A)),X5: set(B)] : image(B,A,product_Sigma(B,A,A3,B5),X5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X5),A3))) ).
% Sigma_Image
tff(fact_3059_Domain__Union,axiom,
! [B: $tType,A: $tType,S: set(set(product_prod(A,B)))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,B))),set(set(A)),image2(set(product_prod(A,B)),set(A),domain(A,B)),S)) ).
% Domain_Union
tff(fact_3060_rtrancl__last__visit_H,axiom,
! [A: $tType,Q3: A,Q5: A,R3: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q5),transitive_rtrancl(A,R3))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q5),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),R3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),S)))))
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),R3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_visit'
tff(fact_3061_rtrancl__last__touch,axiom,
! [A: $tType,Q3: A,Q5: A,R3: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q5),transitive_rtrancl(A,R3))
=> ( member(A,Q3,S)
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),R3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_touch
tff(fact_3062_Range__Union,axiom,
! [A: $tType,B: $tType,S: set(set(product_prod(B,A)))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(B,A))),set(set(A)),image2(set(product_prod(B,A)),set(A),range2(B,A)),S)) ).
% Range_Union
tff(fact_3063_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),set(B),image2(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_fh(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).
% image_fold_insert
tff(fact_3064_product__fold,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(B),$o,finite_finite2(B),B5)
=> ( product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_fj(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B5),bot_bot(set(product_prod(A,B))),A3) ) ) ) ).
% product_fold
tff(fact_3065_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A3: set(B),B5: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A3)) ).
% snd_image_Sigma
tff(fact_3066_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A3),aTP_Lamp_fk(set(product_prod(A,B)),fun(A,set(B)),A3))) ).
% subset_fst_snd
tff(fact_3067_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A3: set(A),R3: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A3,aTP_Lamp_al(set(A),fun(A,set(A)),A3))),R3)),transitive_trancl(A,product_Sigma(A,A,minus_minus(set(A),A3,R3),aa(set(A),fun(A,set(A)),aTP_Lamp_eq(set(A),fun(set(A),fun(A,set(A))),A3),R3)))) ).
% rel_restrict_Sigma_sub
tff(fact_3068_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A3: set(B)] : finite_card(product_prod(A,B),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),aTP_Lamp_aw(set(B),fun(A,set(B)),A3))) = finite_card(B,A3) ).
% card_cartesian_product_singleton
tff(fact_3069_Id__on__fold,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( id_on(A,A3) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_fl(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A3) ) ) ).
% Id_on_fold
tff(fact_3070_UN__Image,axiom,
! [B: $tType,A: $tType,C: $tType,X5: fun(C,set(product_prod(B,A))),I: set(C),S: set(B)] : image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),X5),I)),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_fm(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X5),S)),I)) ).
% UN_Image
tff(fact_3071_divide__nat__def,axiom,
! [M: nat,N2: nat] :
divide_divide(nat,M,N2) = $ite(N2 = zero_zero(nat),zero_zero(nat),lattic643756798349783984er_Max(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_fn(nat,fun(nat,fun(nat,$o)),M),N2)))) ).
% divide_nat_def
tff(fact_3072_Chains__alt__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R2)
=> ( chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))) ) ) ).
% Chains_alt_def
tff(fact_3073_Chains__subset,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).
% Chains_subset
tff(fact_3074_uminus__assn__def,axiom,
! [P: assn] : aa(assn,assn,uminus_uminus(assn),P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_fo(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ).
% uminus_assn_def
tff(fact_3075_snd__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ex(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% snd_diag_fst
tff(fact_3076_fst__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ey(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% fst_diag_snd
tff(fact_3077_sum_Odelta__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [S: set(A),A4: A,B3: fun(A,B),C2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A4),B3),C2)),S) = $ite(member(A,A4,S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B3,A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ) ).
% sum.delta_remove
tff(fact_3078_prod_Odelta__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A4: A,B3: fun(A,B),C2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fq(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A4),B3),C2)),S) = $ite(member(A,A4,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ) ).
% prod.delta_remove
tff(fact_3079_sum_OUNION__disjoint,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_add(C)
=> ! [I: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ! [X3: A] :
( member(A,X3,I)
=> aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I)
=> ! [Xa4: A] :
( member(A,Xa4,I)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_fr(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I) ) ) ) ) ) ).
% sum.UNION_disjoint
tff(fact_3080_prod_OUNION__disjoint,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [I: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ! [X3: A] :
( member(A,X3,I)
=> aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I)
=> ! [Xa4: A] :
( member(A,Xa4,I)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_fs(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I) ) ) ) ) ) ).
% prod.UNION_disjoint
tff(fact_3081_cSUP__UNION,axiom,
! [B: $tType,C: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [A3: set(A),B5: fun(A,set(B)),F2: fun(B,C)] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(A,set(B),B5,X3) != bot_bot(set(B)) ) )
=> ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ft(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B5),F2)),A3)))
=> ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A3)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fu(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B5),F2)),A3)) ) ) ) ) ) ).
% cSUP_UNION
tff(fact_3082_cINF__UNION,axiom,
! [B: $tType,C: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [A3: set(A),B5: fun(A,set(B)),F2: fun(B,C)] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(A,set(B),B5,X3) != bot_bot(set(B)) ) )
=> ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ft(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B5),F2)),A3)))
=> ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A3)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fv(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B5),F2)),A3)) ) ) ) ) ) ).
% cINF_UNION
tff(fact_3083_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A13: set(A),R1: set(product_prod(A,A)),A23: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A1: A,A22: B] :
( equiv_equiv(A,A13,R1)
=> ( equiv_equiv(B,A23,R22)
=> ( equiv_congruent2(A,B,set(C),R1,R22,F2)
=> ( member(A,A1,A13)
=> ( member(B,A22,A23)
=> ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_fw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A22)),image(A,A,R1,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F2,A1),A22) ) ) ) ) ) ) ).
% UN_equiv_class2
tff(fact_3084_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,E3))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E3,S)),S)
=> ( member(A,U,S)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E3),product_Sigma(A,A,S,aTP_Lamp_al(set(A),fun(A,set(A)),S))))) ) ) ) ).
% trancl_restrict_reachable
tff(fact_3085_rtrancl__last__visit,axiom,
! [A: $tType,Q3: A,Q5: A,R3: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q5),transitive_rtrancl(A,R3))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q5),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),R3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),S)))))
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_trancl(A,R3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q5),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),R3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_visit
tff(fact_3086_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E3: set(product_prod(A,A)),R3: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,minus_minus(set(product_prod(A,A)),E3,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_al(set(A),fun(A,set(A)),R3)))))
=> ( ~ member(A,U,R3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E3,R3))) ) ) ).
% rtrancl_restrictI
tff(fact_3087_Image__INT__eq,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),A3: set(C),B5: fun(C,set(B))] :
( single_valued(A,B,converse(B,A,R2))
=> ( ( A3 != bot_bot(set(C)) )
=> ( image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dl(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B5)),A3)) ) ) ) ).
% Image_INT_eq
tff(fact_3088_card__quotient__disjoint,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( inj_on(A,set(set(A)),aTP_Lamp_ek(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
=> ( finite_card(set(A),equiv_quotient(A,A3,R2)) = finite_card(A,A3) ) ) ) ).
% card_quotient_disjoint
tff(fact_3089_Sigma__def,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: fun(A,set(B))] : product_Sigma(A,B,A3,B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_fy(fun(A,set(B)),fun(A,set(product_prod(A,B))),B5)),A3)) ).
% Sigma_def
tff(fact_3090_quotient__def,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A3,R2) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_fz(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A3)) ).
% quotient_def
tff(fact_3091_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A] : aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A4)),one_one(A)) ) ).
% minus_eq_not_plus_1
tff(fact_3092_Chains__subset_H,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R2)
=> aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).
% Chains_subset'
tff(fact_3093_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A] : aa(A,A,uminus_uminus(A),A4) = aa(A,A,bit_ri4277139882892585799ns_not(A),minus_minus(A,A4,one_one(A))) ) ).
% minus_eq_not_minus_1
tff(fact_3094_not__eq__complement,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A4) = minus_minus(A,aa(A,A,uminus_uminus(A),A4),one_one(A)) ) ).
% not_eq_complement
tff(fact_3095_Id__on__def,axiom,
! [A: $tType,A3: set(A)] : id_on(A,A3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_ga(A,set(product_prod(A,A)))),A3)) ).
% Id_on_def
tff(fact_3096_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I: set(A),X: fun(A,B),A4: fun(A,B),B3: B,Delta: B] :
( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3)) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I) = one_one(B) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),abs_abs(B,minus_minus(B,aa(A,B,A4,I3),B3))),Delta) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),abs_abs(B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gb(fun(A,B),fun(fun(A,B),fun(A,B)),X),A4)),I),B3))),Delta) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_3097_mask__eq__sum__exp,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N2: nat] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_dm(nat,fun(nat,$o),N2))) ) ).
% mask_eq_sum_exp
tff(fact_3098_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S: set(A),R3: set(B),G: fun(A,B),F2: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(B),$o,finite_finite2(B),R3)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),R3)
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_gc(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),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(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ge(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R3) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_3099_prod__gen__delta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A4: A,B3: fun(A,B),C2: B] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_gf(A,fun(fun(A,B),fun(B,fun(A,B))),A4),B3),C2)),S) = $ite(member(A,A4,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A4)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),minus_minus(nat,finite_card(A,S),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),finite_card(A,S))) ) ) ) ).
% prod_gen_delta
tff(fact_3100_Gcd__eq__Max,axiom,
! [M2: set(nat)] :
( aa(set(nat),$o,finite_finite2(nat),M2)
=> ( ( M2 != bot_bot(set(nat)) )
=> ( ~ member(nat,zero_zero(nat),M2)
=> ( gcd_Gcd(nat,M2) = 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_gh(nat,set(nat))),M2))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_3101_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I: set(A),A3: fun(A,set(B))] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ! [X3: A] :
( member(A,X3,I)
=> aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I)
=> ! [Xa4: A] :
( member(A,Xa4,I)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( finite_card(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_gi(fun(A,set(B)),fun(A,nat),A3)),I) ) ) ) ) ).
% card_UN_disjoint
tff(fact_3102_trancl__multi__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),M: A,X5: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),M),bot_bot(set(A))),aTP_Lamp_al(set(A),fun(A,set(A)),X5)))))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),M),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B3),transitive_rtrancl(A,R2)) ) ) ) ) ).
% trancl_multi_insert2
tff(fact_3103_trancl__multi__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),X5: set(A),M: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),product_Sigma(A,A,X5,aTP_Lamp_gj(A,fun(A,set(A)),M)))))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,R2))
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),X3),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),B3),transitive_rtrancl(A,R2)) ) ) ) ) ).
% trancl_multi_insert
tff(fact_3104_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set(A),X: B,A3: set(product_prod(B,A))] :
( aa(set(A),$o,finite_finite2(A),B5)
=> ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_gk(B,fun(A,set(product_prod(B,A))),X)),B5))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_gl(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A3,B5) ) ) ).
% fold_union_pair
tff(fact_3105_wf__UN,axiom,
! [B: $tType,A: $tType,I: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [I3: A] :
( member(A,I3,I)
=> wf(B,aa(A,set(product_prod(B,B)),R2,I3)) )
=> ( ! [I3: A,J4: A] :
( member(A,I3,I)
=> ( member(A,J4,I)
=> ( ( aa(A,set(product_prod(B,B)),R2,I3) != aa(A,set(product_prod(B,B)),R2,J4) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(product_prod(B,B)),set(B),domain(B,B),aa(A,set(product_prod(B,B)),R2,I3))),aa(set(product_prod(B,B)),set(B),range2(B,B),aa(A,set(product_prod(B,B)),R2,J4))) = bot_bot(set(B)) ) ) ) )
=> wf(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),I))) ) ) ).
% wf_UN
tff(fact_3106_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(A,product_prod(B,C)),A3: set(B),B5: set(C)] : vimage(A,product_prod(B,C),F2,product_Sigma(B,C,A3,aTP_Lamp_dr(set(C),fun(B,set(C)),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F2),A3)),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F2),B5)) ).
% vimage_Times
tff(fact_3107_rtrancl__last__visit__node,axiom,
! [A: $tType,S2: A,S3: A,R3: set(product_prod(A,A)),Sh: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3),transitive_rtrancl(A,R3))
=> ( ( ( S2 != Sh )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_gm(A,fun(A,set(A)),Sh))))) )
| ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),Sh),transitive_rtrancl(A,R3))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_gm(A,fun(A,set(A)),Sh))))) ) ) ) ).
% rtrancl_last_visit_node
tff(fact_3108_signed__take__bit__code,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat,A4: A] :
bit_ri4674362597316999326ke_bit(A,N2,A4) = $let(
l: A,
l:= bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N2),A4),
$ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),N2),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N2),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).
% signed_take_bit_code
tff(fact_3109_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A3: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gn(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),A3) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A3),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_go(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A3)) ).
% image_split_eq_Sigma
tff(fact_3110_trans__wf__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> ( wf(A,R2)
<=> ! [A7: A] : wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,image(A,A,converse(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_gp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A7)))) ) ) ).
% trans_wf_iff
tff(fact_3111_range__prod,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(C,product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F2),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,aa(fun(C,product_prod(A,B)),fun(C,A),comp(product_prod(A,B),A,C,product_fst(A,B)),F2)),top_top(set(C))),aTP_Lamp_gq(fun(C,product_prod(A,B)),fun(A,set(B)),F2))) ).
% range_prod
tff(fact_3112_Pow__fold,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( pow2(A,A3) = finite_fold(A,set(set(A)),aTP_Lamp_gr(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A3) ) ) ).
% Pow_fold
tff(fact_3113_pochhammer__code,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A,N2: nat] :
comm_s3205402744901411588hammer(A,A4,N2) = $ite(N2 = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_gs(A,fun(nat,fun(A,A)),A4),zero_zero(nat),minus_minus(nat,N2,one_one(nat)),one_one(A))) ) ).
% pochhammer_code
tff(fact_3114_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat,A4: A] : bit_se2638667681897837118et_bit(A,N2,A4) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A4),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se4730199178511100633sh_bit(A,N2,one_one(A)))) ) ).
% unset_bit_eq_and_not
tff(fact_3115_and__not__numerals_I5_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2)))) ).
% and_not_numerals(5)
tff(fact_3116_bit_Ocompl__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).
% bit.compl_unique
tff(fact_3117_gbinomial__code,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] :
aa(nat,A,gbinomial(A,A4),K) = $ite(K = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_gt(A,fun(nat,fun(A,A)),A4),zero_zero(nat),minus_minus(nat,K,one_one(nat)),one_one(A)),semiring_char_0_fact(A,K))) ) ).
% gbinomial_code
tff(fact_3118_and__not__numerals_I6_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2)))) ).
% and_not_numerals(6)
tff(fact_3119_and__not__numerals_I9_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2)))) ).
% and_not_numerals(9)
tff(fact_3120_or__not__numerals_I6_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2)))) ).
% or_not_numerals(6)
tff(fact_3121_or__not__numerals_I5_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N2))))) ).
% or_not_numerals(5)
tff(fact_3122_signed__take__bit__def,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat,A4: A] : bit_ri4674362597316999326ke_bit(A,N2,A4) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se2584673776208193580ke_bit(A,N2,A4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),N2))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N2)))) ) ).
% signed_take_bit_def
tff(fact_3123_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N2))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,N2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gu(A,fun(nat,A),Z2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)),one_one(nat)))) ) ).
% pochhammer_times_pochhammer_half
tff(fact_3124_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),R3: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P,R3)) = same_fst(A,B,P,aTP_Lamp_gv(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R3)) ).
% same_fst_trancl
tff(fact_3125_subset__mset_OcSUP__const,axiom,
! [A: $tType,B: $tType,A3: set(A),C2: multiset(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_gw(multiset(B),fun(A,multiset(B)),C2)),A3)) = C2 ) ) ).
% subset_mset.cSUP_const
tff(fact_3126_subset__mset_OcINF__const,axiom,
! [A: $tType,B: $tType,A3: set(A),C2: multiset(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_gw(multiset(B),fun(A,multiset(B)),C2)),A3)) = C2 ) ) ).
% subset_mset.cINF_const
tff(fact_3127_choose__even__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gx(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2) ) ) ) ).
% choose_even_sum
tff(fact_3128_choose__odd__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gy(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2) ) ) ) ).
% choose_odd_sum
tff(fact_3129_predicate2I,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),P,X3),Y2)
=> aa(B,$o,aa(A,fun(B,$o),Q,X3),Y2) )
=> aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).
% predicate2I
tff(fact_3130_subset__mset_OcInf__singleton,axiom,
! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.cInf_singleton
tff(fact_3131_subset__mset_OcSup__singleton,axiom,
! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.cSup_singleton
tff(fact_3132_predicate1I,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).
% predicate1I
tff(fact_3133_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Y) ).
% top2I
tff(fact_3134_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A4,B3) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% atLeastatMost_empty_iff2
tff(fact_3135_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( ( set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ).
% atLeastatMost_empty_iff
tff(fact_3136_atLeastatMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)) ) ) ) ).
% atLeastatMost_empty
tff(fact_3137_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A] : set_or1337092689740270186AtMost(A,A4,A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) ) ).
% atLeastAtMost_singleton
tff(fact_3138_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A,C2: A] :
( ( set_or1337092689740270186AtMost(A,A4,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A))) )
<=> ( ( A4 = B3 )
& ( B3 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_3139_Inf__atMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),X)) = bot_bot(A) ) ).
% Inf_atMost
tff(fact_3140_prod_OatMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N2))),aa(nat,A,G,aa(nat,nat,suc,N2))) ) ).
% prod.atMost_Suc
tff(fact_3141_atMost__0,axiom,
aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ).
% atMost_0
tff(fact_3142_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D3: A,A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A4,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A4),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B3)) ) ) ) ).
% image_mult_atLeastAtMost
tff(fact_3143_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N2)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G,aa(nat,nat,suc,N2)))) ) ).
% prod.cl_ivl_Suc
tff(fact_3144_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Y) ).
% bot2E
tff(fact_3145_Inf__multiset__empty,axiom,
! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).
% Inf_multiset_empty
tff(fact_3146_Sup__multiset__empty,axiom,
! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).
% Sup_multiset_empty
tff(fact_3147_Inf__filter__not__bot,axiom,
! [A: $tType,B5: set(filter(A))] :
( ! [X7: set(filter(A))] :
( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B5)
=> ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X7)
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X7) != bot_bot(filter(A)) ) ) )
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5) != bot_bot(filter(A)) ) ) ).
% Inf_filter_not_bot
tff(fact_3148_INF__filter__bot__base,axiom,
! [A: $tType,B: $tType,I: set(A),F4: fun(A,filter(B))] :
( ! [I3: A] :
( member(A,I3,I)
=> ! [J4: A] :
( member(A,J4,I)
=> ? [X4: A] :
( member(A,X4,I)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),aa(A,filter(B),F4,J4))) ) ) )
=> ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I)) = bot_bot(filter(B)) )
<=> ? [X2: A] :
( member(A,X2,I)
& ( aa(A,filter(B),F4,X2) = bot_bot(filter(B)) ) ) ) ) ).
% INF_filter_bot_base
tff(fact_3149_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,Q: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
=> aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).
% rev_predicate2D
tff(fact_3150_rev__predicate1D,axiom,
! [A: $tType,P: fun(A,$o),X: A,Q: fun(A,$o)] :
( aa(A,$o,P,X)
=> ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
=> aa(A,$o,Q,X) ) ) ).
% rev_predicate1D
tff(fact_3151_predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),X: A,Y: B] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
=> ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).
% predicate2D
tff(fact_3152_predicate1D,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),X: A] :
( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
=> ( aa(A,$o,P,X)
=> aa(A,$o,Q,X) ) ) ).
% predicate1D
tff(fact_3153_refl__ge__eq,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( ! [X3: A] : aa(A,$o,aa(A,fun(A,$o),R3,X3),X3)
=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R3) ) ).
% refl_ge_eq
tff(fact_3154_ge__eq__refl,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),X: A] :
( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R3)
=> aa(A,$o,aa(A,fun(A,$o),R3,X),X) ) ).
% ge_eq_refl
tff(fact_3155_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Ha: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),Ha) ) ).
% not_empty_eq_Iic_eq_empty
tff(fact_3156_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
=> ( set_or1337092689740270186AtMost(A,A4,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))) ) ) ) ).
% atLeastAtMost_singleton'
tff(fact_3157_sum__power__shift,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N2: nat,X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),minus_minus(nat,N2,M)))) ) ) ) ).
% sum_power_shift
tff(fact_3158_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2) = minus_minus(set(nat),aa(nat,set(nat),set_ord_atMost(nat),N2),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_atMost_eq_remove0
tff(fact_3159_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( bounded_lattice(A)
=> ! [X: A,Y: A] :
( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
<=> ( ( X = bot_bot(A) )
& ( Y = top_top(A) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_3160_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) ) ).
% prod.atLeast0_atMost_Suc_shift
tff(fact_3161_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))),aa(nat,A,G,aa(nat,nat,suc,N2))) ) ).
% prod.atLeast0_atMost_Suc
tff(fact_3162_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,N2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_3163_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_3164_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gz(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N2))) ) ).
% prod.atMost_Suc_shift
tff(fact_3165_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2))),aa(nat,A,G,aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gz(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N2))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_3166_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [F2: fun(nat,A),A4: nat,B3: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A4,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ha(fun(nat,A),fun(nat,fun(A,A)),F2),A4,B3,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_3167_prod_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A),P2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),P2)))) ) ) ) ).
% prod.ub_add_nat
tff(fact_3168_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hb(A,fun(nat,A),A4)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),aa(nat,A,semiring_1_of_nat(A),N2))),one_one(A))),N2) ) ).
% gbinomial_parallel_sum
tff(fact_3169_vandermonde,axiom,
! [M: nat,N2: nat,R2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hc(nat,fun(nat,fun(nat,fun(nat,nat))),M),N2),R2)),aa(nat,set(nat),set_ord_atMost(nat),R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2)),R2) ).
% vandermonde
tff(fact_3170_fact__eq__fact__times,axiom,
! [N2: nat,M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M)
=> ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N2)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_hd(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N2),M))) ) ) ).
% fact_eq_fact_times
tff(fact_3171_sum__gp__basic,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,one_one(A),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N2))) = minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))) ) ).
% sum_gp_basic
tff(fact_3172_binomial,axiom,
! [A4: nat,B3: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)),N2) = 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_he(nat,fun(nat,fun(nat,fun(nat,nat))),A4),B3),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ).
% binomial
tff(fact_3173_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)),bot_bot(set(A))) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_3174_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
aa(set(A),set(A),image2(A,A,aTP_Lamp_hf(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2))),
bot_bot(set(A)) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_3175_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M: A,C2: A,A4: A,B3: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A4,B3)) = $ite(
set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A4)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A4)),C2))) ) ) ).
% image_affinity_atLeastAtMost
tff(fact_3176_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M: A,C2: A,A4: A,B3: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A4,B3)) = $ite(
set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A4),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3),C2)),set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A4),C2))) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_3177_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M: A,C2: A,A4: A,B3: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A4,B3)) = $ite(
set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A4,M)),C2))) ) ) ).
% image_affinity_atLeastAtMost_div
tff(fact_3178_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M: A,C2: A,A4: A,B3: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hj(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A4,B3)) = $ite(
set_or1337092689740270186AtMost(A,A4,B3) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,minus_minus(A,divide_divide(A,A4,M),C2),minus_minus(A,divide_divide(A,B3,M),C2)),set_or1337092689740270186AtMost(A,minus_minus(A,divide_divide(A,B3,M),C2),minus_minus(A,divide_divide(A,A4,M),C2))) ) ) ).
% image_affinity_atLeastAtMost_div_diff
tff(fact_3179_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hk(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).
% sum.in_pairs_0
tff(fact_3180_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hl(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).
% prod.in_pairs_0
tff(fact_3181_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hm(A,fun(nat,A),A4)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,minus_minus(A,A4,one_one(A))),M)) ) ).
% gbinomial_sum_lower_neg
tff(fact_3182_binomial__ring,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A,B3: A,N2: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),N2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hn(A,fun(A,fun(nat,fun(nat,A))),A4),B3),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).
% binomial_ring
tff(fact_3183_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N2: nat,X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,one_one(A),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2))) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))) ) ) ) ).
% sum_gp_multiplied
tff(fact_3184_sum_Oin__pairs,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hk(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).
% sum.in_pairs
tff(fact_3185_prod_Oin__pairs,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hl(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N2)) ) ).
% prod.in_pairs
tff(fact_3186_choose__square__sum,axiom,
! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ho(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)),N2) ).
% choose_square_sum
tff(fact_3187_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A4: A,B3: A,N2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),N2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),A4),B3),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ).
% pochhammer_binomial_sum
tff(fact_3188_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P2: nat,K: nat,G: fun(nat,A),Ha: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),Ha)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hr(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),Ha)),aa(nat,set(nat),set_ord_atMost(nat),minus_minus(nat,P2,aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% prod.zero_middle
tff(fact_3189_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A4: A,X: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hs(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A4),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ht(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A4),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).
% gbinomial_partial_sum_poly
tff(fact_3190_gauss__sum__nat,axiom,
! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hd(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,N2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% gauss_sum_nat
tff(fact_3191_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hu(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).
% gbinomial_sum_up_index
tff(fact_3192_sum__gp0,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))),divide_divide(A,minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))),minus_minus(A,one_one(A),X))) ) ).
% sum_gp0
tff(fact_3193_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A4: A,X: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hs(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A4),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hv(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A4),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).
% gbinomial_partial_sum_poly_xpos
tff(fact_3194_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat] :
( ( N2 != one_one(nat) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hw(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_3195_double__gauss__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))) ) ).
% double_gauss_sum
tff(fact_3196_double__arith__series,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A4: A,D3: A,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_hx(A,fun(A,fun(nat,A)),A4),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),D3))) ) ).
% double_arith_series
tff(fact_3197_binomial__r__part__sum,axiom,
! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).
% binomial_r_part_sum
tff(fact_3198_choose__linear__sum,axiom,
! [N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hy(nat,fun(nat,nat),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),minus_minus(nat,N2,one_one(nat)))) ).
% choose_linear_sum
tff(fact_3199_arith__series__nat,axiom,
! [A4: nat,D3: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_hz(nat,fun(nat,fun(nat,nat)),A4),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),D3))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% arith_series_nat
tff(fact_3200_Sum__Icc__nat,axiom,
! [M: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hd(nat,nat)),set_or1337092689740270186AtMost(nat,M,N2)) = divide_divide(nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),minus_minus(nat,M,one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% Sum_Icc_nat
tff(fact_3201_choose__alternating__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ia(nat,fun(nat,A),N2)),aa(nat,set(nat),set_ord_atMost(nat),N2)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_3202_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))) ) ).
% double_gauss_sum_from_Suc_0
tff(fact_3203_gauss__sum,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% gauss_sum
tff(fact_3204_arith__series,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A,D3: A,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ib(A,fun(A,fun(nat,A)),A4),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),D3))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% arith_series
tff(fact_3205_sum__gp__offset,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M: nat,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N2))) = $ite(X = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2)))),minus_minus(A,one_one(A),X))) ) ).
% sum_gp_offset
tff(fact_3206_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N2)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% gauss_sum_from_Suc_0
tff(fact_3207_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [R2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ic(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).
% gchoose_row_sum_weighted
tff(fact_3208_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).
% gbinomial_r_part_sum
tff(fact_3209_sum__gp,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M: nat,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N2)) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M),
zero_zero(A),
$ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)),M)),divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2))),minus_minus(A,one_one(A),X))) ) ) ).
% sum_gp
tff(fact_3210_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ic(A,fun(nat,A),A4)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).
% gbinomial_partial_row_sum
tff(fact_3211_same__fstI,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),X: A,Y7: B,Y: B,R3: fun(A,set(product_prod(B,B)))] :
( aa(A,$o,P,X)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y),aa(A,set(product_prod(B,B)),R3,X))
=> member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),same_fst(A,B,P,R3)) ) ) ).
% same_fstI
tff(fact_3212_congruent2__implies__congruent__UN,axiom,
! [A: $tType,C: $tType,B: $tType,A13: set(A),R1: set(product_prod(A,A)),A23: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A4: B] :
( equiv_equiv(A,A13,R1)
=> ( equiv_equiv(B,A23,R22)
=> ( equiv_congruent2(A,B,set(C),R1,R22,F2)
=> ( member(B,A4,A23)
=> equiv_congruent(A,set(C),R1,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_fw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A4)) ) ) ) ) ).
% congruent2_implies_congruent_UN
tff(fact_3213_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),A4: A] :
( equiv_equiv(A,A3,R2)
=> ( equiv_congruent(A,set(B),R2,F2)
=> ( member(A,A4,A3)
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) = aa(A,set(B),F2,A4) ) ) ) ) ).
% UN_equiv_class
tff(fact_3214_of__nat__code,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat] : aa(nat,A,semiring_1_of_nat(A),N2) = semiri8178284476397505188at_aux(A,aTP_Lamp_id(A,A),N2,zero_zero(A)) ) ).
% of_nat_code
tff(fact_3215_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] : unique8689654367752047608divmod(A,bit1(M),bit0(N2)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ie(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N2)) ) ).
% divmod_algorithm_code(6)
tff(fact_3216_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A4: B,B3: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3)) = aa(C,A,aa(B,fun(C,A),F2,A4),B3) ).
% case_prod_conv
tff(fact_3217_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),F2)) = F2 ).
% case_prod_curry
tff(fact_3218_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C))] : aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)) = F2 ).
% curry_case_prod
tff(fact_3219_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(C,fun(B,A)),P2: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_if(fun(C,fun(B,A)),fun(B,fun(C,A)),F2)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P2)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F2),P2) ).
% case_swap
tff(fact_3220_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A4: A,B3: B,A3: set(product_prod(A,B)),F2: fun(A,fun(B,C))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),A3)
=> member(C,aa(B,C,aa(A,fun(B,C),F2,A4),B3),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)),A3)) ) ).
% pair_imageI
tff(fact_3221_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N2: num] : unique8689654367752047608divmod(A,bit0(M),bit0(N2)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ig(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N2)) ) ).
% divmod_algorithm_code(5)
tff(fact_3222_union__diff__assoc,axiom,
! [A: $tType,C6: multiset(A),B5: multiset(A),A3: multiset(A)] :
( ( minus_minus(multiset(A),C6,B5) = zero_zero(multiset(A)) )
=> ( minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5),C6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),minus_minus(multiset(A),B5,C6)) ) ) ).
% union_diff_assoc
tff(fact_3223_less__filter__def,axiom,
! [A: $tType,F4: filter(A),F7: filter(A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F4),F7)
<=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
& ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F7),F4) ) ) ).
% less_filter_def
tff(fact_3224_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P2: product_prod(A,B)] :
( ! [X3: A,Y2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) = Q3 )
=> ( aa(B,C,aa(A,fun(B,C),F2,X3),Y2) = aa(B,C,aa(A,fun(B,C),G,X3),Y2) ) )
=> ( ( P2 = Q3 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q3) ) ) ) ).
% split_cong
tff(fact_3225_old_Oprod_Ocase,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),X1: B,X22: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X22)) = aa(C,A,aa(B,fun(C,A),F2,X1),X22) ).
% old.prod.case
tff(fact_3226_nested__case__prod__simp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F2),X),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_ih(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),X) ).
% nested_case_prod_simp
tff(fact_3227_prod_Ocase__distrib,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Ha: fun(B,A),F2: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,Ha,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F2),Prod)) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_ii(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Ha),F2)),Prod) ).
% prod.case_distrib
tff(fact_3228_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P2) = P2 ).
% case_prod_Pair_iden
tff(fact_3229_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2))
=> ~ ! [X3: B,Y2: C] :
( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X3),Y2)) ) ) ).
% case_prodE2
tff(fact_3230_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ij(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).
% case_prod_eta
tff(fact_3231_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
( ! [X3: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F2,X3),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) = G ) ) ).
% cond_case_prod_eta
tff(fact_3232_fst__def,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_ba(A,fun(B,A))),Prod) ).
% fst_def
tff(fact_3233_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(A,C)] : aTP_Lamp_ik(fun(A,C),fun(product_prod(A,B),C),F2) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_il(fun(A,C),fun(A,fun(B,C)),F2)) ).
% fn_fst_conv
tff(fact_3234_fn__snd__conv,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,C)] : aTP_Lamp_im(fun(B,C),fun(product_prod(A,B),C),F2) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_in(fun(B,C),fun(A,fun(B,C)),F2)) ).
% fn_snd_conv
tff(fact_3235_snd__def,axiom,
! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_io(B,fun(A,A))),Prod) ).
% snd_def
tff(fact_3236_split__beta,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod) = aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) ).
% split_beta
tff(fact_3237_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,A)),P2: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),P2) = aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),P2)),aa(product_prod(B,C),C,product_snd(B,C),P2)) ).
% case_prod_beta
tff(fact_3238_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C))] : uncurry(A,B,C,F2) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) ).
% uncurry_def
tff(fact_3239_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc5280177257484947105e_prod(A,B,C) = product_case_prod(A,B,C) ).
% internal_case_prod_def
tff(fact_3240_map__prod__def,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F2,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ip(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F2),G)) ).
% map_prod_def
tff(fact_3241_congruentI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B)] :
( ! [Y2: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> ( aa(A,B,F2,Y2) = aa(A,B,F2,Z4) ) )
=> equiv_congruent(A,B,R2,F2) ) ).
% congruentI
tff(fact_3242_congruentD,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B),Y: A,Z2: A] :
( equiv_congruent(A,B,R2,F2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> ( aa(A,B,F2,Y) = aa(A,B,F2,Z2) ) ) ) ).
% congruentD
tff(fact_3243_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A),Q: fun(C,$o),F2: fun(B,fun(A,C))] :
( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
=> ( ! [X3: B,Y2: A] :
( aa(B,$o,aa(A,fun(B,$o),P,Y2),X3)
=> aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F2,X3),Y2)) )
=> aa(C,$o,Q,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F2),P2)) ) ) ).
% exE_realizer
tff(fact_3244_split__comp__eq,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F2: fun(D,fun(B,C)),G: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_iq(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F2),G) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_ir(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F2),G)) ).
% split_comp_eq
tff(fact_3245_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),X4: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),X4) = aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),X4)),aa(product_prod(A,B),B,product_snd(A,B),X4)) ).
% case_prod_beta'
tff(fact_3246_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X4),Xa3) = aa(B,C,aa(A,fun(B,C),X4,aa(product_prod(A,B),A,product_fst(A,B),Xa3)),aa(product_prod(A,B),B,product_snd(A,B),Xa3)) ).
% case_prod_unfold
tff(fact_3247_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_is(A,fun(B,product_prod(B,A)))),A3) ).
% swap_inj_on
tff(fact_3248_prod_Osplit__sel__asm,axiom,
! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
<=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
& ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).
% prod.split_sel_asm
tff(fact_3249_prod_Osplit__sel,axiom,
! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
<=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
=> aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).
% prod.split_sel
tff(fact_3250_swap__product,axiom,
! [A: $tType,B: $tType,A3: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_aw(set(B),fun(A,set(B)),A3)) ).
% swap_product
tff(fact_3251_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(D,fun(C,A)),G: fun(B,D),X: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F2),G)),X) = aa(C,A,aa(D,fun(C,A),F2,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).
% case_prod_comp
tff(fact_3252_image__paired__Times,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A3: set(C),B5: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_iu(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),product_Sigma(C,D,A3,aTP_Lamp_iv(set(D),fun(C,set(D)),B5))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A3),aa(set(D),fun(A,set(B)),aTP_Lamp_iw(fun(D,B),fun(set(D),fun(A,set(B))),G),B5)) ).
% image_paired_Times
tff(fact_3253_fst__snd__flip,axiom,
! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_is(A,fun(B,product_prod(B,A))))),Xy) ).
% fst_snd_flip
tff(fact_3254_snd__fst__flip,axiom,
! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))))),Xy) ).
% snd_fst_flip
tff(fact_3255_periodic__finite__ex,axiom,
! [D3: int,P: fun(int,$o)] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
=> ( ! [X3: int,K3: int] :
( aa(int,$o,P,X3)
<=> aa(int,$o,P,minus_minus(int,X3,aa(int,int,aa(int,fun(int,int),times_times(int),K3),D3))) )
=> ( ? [X_12: int] : aa(int,$o,P,X_12)
<=> ? [X2: int] :
( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D3))
& aa(int,$o,P,X2) ) ) ) ) ).
% periodic_finite_ex
tff(fact_3256_simp__from__to,axiom,
! [I2: int,J2: int] :
set_or1337092689740270186AtMost(int,I2,J2) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J2),I2),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2))) ).
% simp_from_to
tff(fact_3257_cpmi,axiom,
! [D4: int,P: fun(int,$o),P5: fun(int,$o),B5: set(int)] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
=> ( ? [Z6: int] :
! [X3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z6)
=> ( aa(int,$o,P,X3)
<=> aa(int,$o,P5,X3) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
=> ! [Xb2: int] :
( member(int,Xb2,B5)
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( aa(int,$o,P,X3)
=> aa(int,$o,P,minus_minus(int,X3,D4)) ) )
=> ( ! [X3: int,K3: int] :
( aa(int,$o,P5,X3)
<=> aa(int,$o,P5,minus_minus(int,X3,aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4))) )
=> ( ? [X_12: int] : aa(int,$o,P,X_12)
<=> ( ? [X2: int] :
( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D4))
& aa(int,$o,P5,X2) )
| ? [X2: int] :
( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D4))
& ? [Xa2: int] :
( member(int,Xa2,B5)
& aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X2)) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_3258_cppi,axiom,
! [D4: int,P: fun(int,$o),P5: fun(int,$o),A3: set(int)] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
=> ( ? [Z6: int] :
! [X3: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z6),X3)
=> ( aa(int,$o,P,X3)
<=> aa(int,$o,P5,X3) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
=> ! [Xb2: int] :
( member(int,Xb2,A3)
=> ( X3 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( aa(int,$o,P,X3)
=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)) ) )
=> ( ! [X3: int,K3: int] :
( aa(int,$o,P5,X3)
<=> aa(int,$o,P5,minus_minus(int,X3,aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4))) )
=> ( ? [X_12: int] : aa(int,$o,P,X_12)
<=> ( ? [X2: int] :
( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D4))
& aa(int,$o,P5,X2) )
| ? [X2: int] :
( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D4))
& ? [Xa2: int] :
( member(int,Xa2,A3)
& aa(int,$o,P,minus_minus(int,Xa2,X2)) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_3259_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ix(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_3260_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_iy(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_3261_Sum__Icc__int,axiom,
! [M: int,N2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),N2)
=> ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_iz(int,int)),set_or1337092689740270186AtMost(int,M,N2)) = divide_divide(int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(int,int,aa(int,fun(int,int),plus_plus(int),N2),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),M),minus_minus(int,M,one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).
% Sum_Icc_int
tff(fact_3262_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),X5: set(A),Y4: set(A)] :
( equiv_equiv(A,A3,R2)
=> ( equiv_congruent(A,set(B),R2,F2)
=> ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X5)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),Y4)) )
=> ( member(set(A),X5,equiv_quotient(A,A3,R2))
=> ( member(set(A),Y4,equiv_quotient(A,A3,R2))
=> ( ! [X3: A,Y2: A] :
( member(A,X3,A3)
=> ( member(A,Y2,A3)
=> ( ( aa(A,set(B),F2,X3) = aa(A,set(B),F2,Y2) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2) ) ) )
=> ( X5 = Y4 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
tff(fact_3263_divmod__step__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ja(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_3264_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> boolea3799213064322606851m_diff(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_3265_Set__filter__fold,axiom,
! [A: $tType,A3: set(A),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_jb(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).
% Set_filter_fold
tff(fact_3266_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),X5: set(product_prod(C,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
=> ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),X),bot_bot(set(product_prod(C,A)))),S)),X5) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_jc(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X5,S) ) ) ).
% insert_relcomp_union_fold
tff(fact_3267_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [F4: set(A),I: set(A),F2: fun(A,B),I2: A] :
( aa(set(A),$o,finite_finite2(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jd(set(A),fun(fun(A,B),fun(A,$o)),I),F2))),F4)
=> ( groups1027152243600224163dd_sum(A,B,F2,minus_minus(set(A),I,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I2),bot_bot(set(A))))) = $ite(member(A,I2,I),minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I),aa(A,B,F2,I2)),groups1027152243600224163dd_sum(A,B,F2,I)) ) ) ) ) ).
% sum_diff1'_aux
tff(fact_3268_Sum__Ico__nat,axiom,
! [M: nat,N2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hd(nat,nat)),set_or7035219750837199246ssThan(nat,M,N2)) = divide_divide(nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),minus_minus(nat,N2,one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),minus_minus(nat,M,one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% Sum_Ico_nat
tff(fact_3269_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
( ! [A5: A,B2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) )
=> aa(B,$o,aa(A,fun(B,$o),C2,A5),B2) )
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2) ) ).
% case_prodI2
tff(fact_3270_case__prodI,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),F2,A4),B3)
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)) ) ).
% case_prodI
tff(fact_3271_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
( ! [A5: A,B2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) )
=> member(C,Z2,aa(B,set(C),aa(A,fun(B,set(C)),C2,A5),B2)) )
=> member(C,Z2,aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P2)) ) ).
% mem_case_prodI2
tff(fact_3272_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A4: B,B3: C] :
( member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,A4),B3))
=> member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B3))) ) ).
% mem_case_prodI
tff(fact_3273_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),X: C] :
( ! [A5: A,B2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) = P2 )
=> aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A5),B2),X) )
=> aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),X) ) ).
% case_prodI2'
tff(fact_3274_Collect__case__prod,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),Q: fun(B,$o)] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_je(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_jf(fun(B,$o),fun(A,set(B)),Q)) ).
% Collect_case_prod
tff(fact_3275_member__filter,axiom,
! [A: $tType,X: A,P: fun(A,$o),A3: set(A)] :
( member(A,X,filter3(A,P,A3))
<=> ( member(A,X,A3)
& aa(A,$o,P,X) ) ) ).
% member_filter
tff(fact_3276_pair__set__inverse,axiom,
! [A: $tType,B: $tType,P: fun(B,fun(A,$o))] : converse(B,A,aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P))) ).
% pair_set_inverse
tff(fact_3277_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_jh($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).
% Collect_const_case_prod
tff(fact_3278_atLeastLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
=> ( set_or7035219750837199246ssThan(A,A4,B3) = bot_bot(set(A)) ) ) ) ).
% atLeastLessThan_empty
tff(fact_3279_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A4,B3) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% atLeastLessThan_empty_iff2
tff(fact_3280_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: A,B3: A] :
( ( set_or7035219750837199246ssThan(A,A4,B3) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% atLeastLessThan_empty_iff
tff(fact_3281_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty'
tff(fact_3282_atLeastLessThan__singleton,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),bot_bot(set(nat))) ).
% atLeastLessThan_singleton
tff(fact_3283_sum_Oinsert_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [I: set(A),P2: fun(A,B),I2: A] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ji(set(A),fun(fun(A,B),fun(A,$o)),I),P2)))
=> ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I2),I)) = $ite(member(A,I2,I),groups1027152243600224163dd_sum(A,B,P2,I),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I2)),groups1027152243600224163dd_sum(A,B,P2,I))) ) ) ) ).
% sum.insert'
tff(fact_3284_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G,N2))) ) ).
% prod.op_ivl_Suc
tff(fact_3285_same__fst__def,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),R3: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R3) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_jk(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R3)))) ).
% same_fst_def
tff(fact_3286_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
( member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P2))
=> ~ ! [X3: B,Y2: C] :
( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,X3),Y2)) ) ) ).
% mem_case_prodE
tff(fact_3287_lex__prod__def,axiom,
! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_jm(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).
% lex_prod_def
tff(fact_3288_Set_Ofilter__def,axiom,
! [A: $tType,P: fun(A,$o),A3: set(A)] : filter3(A,P,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jn(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).
% Set.filter_def
tff(fact_3289_mset__distrib,axiom,
! [A: $tType,A3: multiset(A),B5: multiset(A),M2: multiset(A),N: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N) )
=> ~ ! [Am: multiset(A),An: multiset(A)] :
( ( A3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
=> ! [Bm: multiset(A),Bn: multiset(A)] :
( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
=> ( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
=> ( N != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).
% mset_distrib
tff(fact_3290_case__prodE,axiom,
! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2)
=> ~ ! [X3: A,Y2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ aa(B,$o,aa(A,fun(B,$o),C2,X3),Y2) ) ) ).
% case_prodE
tff(fact_3291_case__prodD,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> aa(B,$o,aa(A,fun(B,$o),F2,A4),B3) ) ).
% case_prodD
tff(fact_3292_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(A,fun(B,$o))] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_jo(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_jp(fun(A,fun(B,$o)),fun(A,set(B)),Q)) ).
% Collect_case_prod_Sigma
tff(fact_3293_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P2: product_prod(A,B),Z2: C] :
( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),Z2)
=> ~ ! [X3: A,Y2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X3),Y2),Z2) ) ) ).
% case_prodE'
tff(fact_3294_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: fun(A,fun(B,fun(C,$o))),A4: A,B3: B,C2: C] :
( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),C2)
=> aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R3,A4),B3),C2) ) ).
% case_prodD'
tff(fact_3295_Id__on__def_H,axiom,
! [A: $tType,A3: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A3)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_jq(fun(A,$o),fun(A,fun(A,$o)),A3))) ).
% Id_on_def'
tff(fact_3296_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),A3: fun(A,fun(B,$o))] :
( member(product_prod(A,B),X,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A3)))
=> aa(B,$o,aa(A,fun(B,$o),A3,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X)) ) ).
% Product_Type.Collect_case_prodD
tff(fact_3297_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(3)
tff(fact_3298_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A3),B5)
=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A3))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B5))) ) ).
% Collect_case_prod_mono
tff(fact_3299_atLeastLessThan0,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).
% atLeastLessThan0
tff(fact_3300_converse__unfold,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : converse(B,A,R2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_jr(set(product_prod(B,A)),fun(A,fun(B,$o)),R2))) ).
% converse_unfold
tff(fact_3301_rel__restrict__def,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] : rel_restrict(A,R3,A3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_js(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),R3),A3))) ).
% rel_restrict_def
tff(fact_3302_inv__image__def,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F2: fun(A,B)] : inv_image(B,A,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jt(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F2))) ).
% inv_image_def
tff(fact_3303_bsqr__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_jv(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).
% bsqr_def
tff(fact_3304_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(7)
tff(fact_3305_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,P2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),P2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N2,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_3306_wf__bounded__supset,axiom,
! [A: $tType,S: set(A)] :
( aa(set(A),$o,finite_finite2(A),S)
=> wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_jw(set(A),fun(set(A),fun(set(A),$o)),S)))) ) ).
% wf_bounded_supset
tff(fact_3307_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).
% Sup_SUP_eq2
tff(fact_3308_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).
% Inf_INT_eq2
tff(fact_3309_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X5: set(A),A3: set(product_prod(A,B)),Y4: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
( ( X5 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A3) )
=> ( ( Y4 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A3) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ! [Xa4: B] :
( member(B,Xa4,Y4)
=> ( aa(B,$o,aa(A,fun(B,$o),P,X3),Xa4)
=> aa(B,$o,aa(A,fun(B,$o),Q,X3),Xa4) ) ) )
=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))
=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q))) ) ) ) ) ).
% Collect_split_mono_strong
tff(fact_3310_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))),aa(nat,A,G,N2)) ) ).
% prod.atLeast0_lessThan_Suc
tff(fact_3311_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] : set_or7035219750837199246ssThan(A,A4,B3) = minus_minus(set(A),set_or1337092689740270186AtMost(A,A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3312_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N2))) ) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_3313_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: nat,B3: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A4),B3)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A4,aa(nat,nat,suc,B3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A4,B3))),aa(nat,A,G,B3)) ) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_3314_prod_Olast__plus,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N2))) ) ) ) ).
% prod.last_plus
tff(fact_3315_atLeastLessThanSuc,axiom,
! [M: nat,N2: nat] :
set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),N2),set_or7035219750837199246ssThan(nat,M,N2)),bot_bot(set(nat))) ).
% atLeastLessThanSuc
tff(fact_3316_rtrancl__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A4),B3),R2)))) ).
% rtrancl_insert
tff(fact_3317_trancl__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A4),B3),R2)))) ).
% trancl_insert2
tff(fact_3318_Image__fold,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(A)] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R3)
=> ( image(A,B,R3,S) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_jz(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R3) ) ) ).
% Image_fold
tff(fact_3319_UN__Times__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,E3: fun(C,set(A)),F4: fun(D,set(B)),A3: set(C),B5: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_kb(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E3),F4))),product_Sigma(C,D,A3,aTP_Lamp_iv(set(D),fun(C,set(D)),B5)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E3),A3)),aa(set(D),fun(A,set(B)),aTP_Lamp_kc(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B5)) ).
% UN_Times_distrib
tff(fact_3320_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(6)
tff(fact_3321_prod_Ohead__if,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N2))),aa(nat,A,G,N2))) ) ).
% prod.head_if
tff(fact_3322_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) ) ).
% prod.atLeast0_lessThan_Suc_shift
tff(fact_3323_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),X),R2)))) ).
% trancl_insert
tff(fact_3324_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),pred_numeral(K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).
% atLeastLessThan_nat_numeral
tff(fact_3325_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,X: nat,Y: nat] :
aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_kd(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,minus_minus(nat,X,C2),minus_minus(nat,Y,C2)),
$ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_3326_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R3)
=> ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
=> ( relcomp(A,B,C,R3,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_kf(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R3) ) ) ) ).
% relcomp_fold
tff(fact_3327_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),R3: set(product_prod(C,A))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
=> ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),X),R3),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_jc(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R3,S),S) ) ) ).
% insert_relcomp_fold
tff(fact_3328_fact__split,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: nat,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2)
=> ( semiring_char_0_fact(A,N2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,minus_minus(nat,N2,K),N2)))),semiring_char_0_fact(A,minus_minus(nat,N2,K))) ) ) ) ).
% fact_split
tff(fact_3329_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [I: set(A),F2: fun(A,B),I2: A] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jd(set(A),fun(fun(A,B),fun(A,$o)),I),F2)))
=> ( groups1027152243600224163dd_sum(A,B,F2,minus_minus(set(A),I,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I2),bot_bot(set(A))))) = $ite(member(A,I2,I),minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I),aa(A,B,F2,I2)),groups1027152243600224163dd_sum(A,B,F2,I)) ) ) ) ).
% sum_diff1'
tff(fact_3330_gbinomial__mult__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A4),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_kg(A,fun(nat,A),A4)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact
tff(fact_3331_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A4),K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_kg(A,fun(nat,A),A4)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact'
tff(fact_3332_comp__fun__commute__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
=> finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ki(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).
% comp_fun_commute_relcomp_fold
tff(fact_3333_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_kj(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_3334_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,A4),aa(A,fun(A,set(A)),aTP_Lamp_kk(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A4)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ).
% underS_Restr_ordLess
tff(fact_3335_triangle__def,axiom,
! [N2: nat] : nat_triangle(N2) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(nat,nat,suc,N2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% triangle_def
tff(fact_3336_max__extp__max__ext__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X4: set(A),Xa3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R3)),X4),Xa3)
<=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X4),Xa3),max_ext(A,R3)) ) ).
% max_extp_max_ext_eq
tff(fact_3337_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X4: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kl($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X4)
<=> ( (P)
& aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X4) ) ) ).
% split_part
tff(fact_3338_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_km(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),minus_minus(code_integer,U,L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).
% image_add_integer_atLeastLessThan
tff(fact_3339_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_3340_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_kn(A,fun(B,$o))),Prod) ).
% prod.disc_eq_case
tff(fact_3341_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_3342_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_3343_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_jb(fun(A,$o),fun(A,fun(set(A),set(A))),P)) ).
% comp_fun_commute_filter_fold
tff(fact_3344_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_jz(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).
% comp_fun_commute_Image_fold
tff(fact_3345_max__ext__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : max_ext(A,X4) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),X4)))) ).
% max_ext_def
tff(fact_3346_comp__fun__commute__product__fold,axiom,
! [B: $tType,A: $tType,B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),B5)
=> finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_ko(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).
% comp_fun_commute_product_fold
tff(fact_3347_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),bit0(one2))),aa(int,code_integer,code_integer_of_int,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
$ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).
% integer_of_int_code
tff(fact_3348_mlex__eq,axiom,
! [A: $tType,F2: fun(A,nat),R3: set(product_prod(A,A))] : mlex_prod(A,F2,R3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_kp(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R3))) ).
% mlex_eq
tff(fact_3349_enumerate__Suc,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N2: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N2)) = infini527867602293511546merate(A,minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_kq(set(A),fun(A,$o),S))),bot_bot(set(A)))),N2) ) ).
% enumerate_Suc
tff(fact_3350_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_kr(A,fun(A,$o)),aTP_Lamp_ks(A,fun(A,$o))) ) ).
% sup_bot.semilattice_neutr_order_axioms
tff(fact_3351_fun_Oin__rel,axiom,
! [B: $tType,C: $tType,A: $tType,R3: fun(B,fun(C,$o)),A4: fun(A,B),B3: fun(A,C)] :
( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R3),A4),B3)
<=> ? [Z3: fun(A,product_prod(B,C))] :
( member(fun(A,product_prod(B,C)),Z3,aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_kt(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R3)))
& ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),Z3) = A4 )
& ( aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),Z3) = B3 ) ) ) ).
% fun.in_rel
tff(fact_3352_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D)] :
( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),A3,X3),Y2)
=> aa(D,$o,aa(C,fun(D,$o),B5,aa(A,C,F2,X3)),aa(B,D,G,Y2)) )
=> aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A3,B5),F2),G) ) ).
% rel_funI
tff(fact_3353_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U)) ).
% finite_atLeastLessThan_integer
tff(fact_3354_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_3355_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& semiring_numeral(B)
& monoid_add(A)
& semiring_numeral(A) )
=> ! [R3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
=> aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R3),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).
% transfer_rule_numeral
tff(fact_3356_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_3357_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( power(B)
& power(A) )
=> ! [R3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),times_times(A)),times_times(B))
=> aa(fun(B,fun(nat,B)),$o,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),$o),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R3,bNF_rel_fun(nat,nat,A,B,fequal(nat),R3)),power_power(A)),power_power(B)) ) ) ) ).
% power_transfer
tff(fact_3358_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_3359_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_3360_LeastI,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),K: A] :
( aa(A,$o,P,K)
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% LeastI
tff(fact_3361_LeastI2__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),Q: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_ex
tff(fact_3362_LeastI__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% LeastI_ex
tff(fact_3363_LeastI2,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A4: A,Q: fun(A,$o)] :
( aa(A,$o,P,A4)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2
tff(fact_3364_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: fun(A,fun(B,$o)),X5: fun(A,fun(B,$o)),A3: fun(C,fun(D,$o)),B5: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D)] :
( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),Y4,X3),Y2)
=> aa(B,$o,aa(A,fun(B,$o),X5,X3),Y2) )
=> ( ! [X3: C,Y2: D] :
( aa(D,$o,aa(C,fun(D,$o),A3,X3),Y2)
=> aa(D,$o,aa(C,fun(D,$o),B5,X3),Y2) )
=> ( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,X5,A3),F2),G)
=> aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,Y4,B5),F2),G) ) ) ) ).
% rel_fun_mono'
tff(fact_3365_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X5: fun(A,fun(B,$o)),A3: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D),Y4: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o))] :
( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,X5,A3),F2),G)
=> ( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),Y4,X3),Y2)
=> aa(B,$o,aa(A,fun(B,$o),X5,X3),Y2) )
=> ( ! [X3: C,Y2: D] :
( aa(D,$o,aa(C,fun(D,$o),A3,X3),Y2)
=> aa(D,$o,aa(C,fun(D,$o),B5,X3),Y2) )
=> aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,Y4,B5),F2),G) ) ) ) ).
% rel_fun_mono
tff(fact_3366_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D),X: A,Y: B] :
( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A3,B5),F2),G)
=> ( aa(B,$o,aa(A,fun(B,$o),A3,X),Y)
=> aa(D,$o,aa(C,fun(D,$o),B5,aa(A,C,F2,X)),aa(B,D,G,Y)) ) ) ).
% rel_funD
tff(fact_3367_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ring_1(B)
& ring_1(A) )
=> ! [R3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
=> ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R3,R3),uminus_uminus(A)),uminus_uminus(B))
=> aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R3),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).
% transfer_rule_of_int
tff(fact_3368_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),Q: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> ( ! [A5: A] :
( aa(A,$o,P,A5)
=> ( ! [B13: A] :
( aa(A,$o,P,B13)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B13) )
=> aa(A,$o,Q,A5) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_wellorder_ex
tff(fact_3369_LeastI2__wellorder,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A4: A,Q: fun(A,$o)] :
( aa(A,$o,P,A4)
=> ( ! [A5: A] :
( aa(A,$o,P,A5)
=> ( ! [B13: A] :
( aa(A,$o,P,B13)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B13) )
=> aa(A,$o,Q,A5) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_wellorder
tff(fact_3370_Least__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
=> ( ord_Least(A,P) = X ) ) ) ) ).
% Least_equality
tff(fact_3371_LeastI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> ( ! [Y5: A] :
( aa(A,$o,P,Y5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y5) )
=> aa(A,$o,Q,X3) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).
% LeastI2_order
tff(fact_3372_Least1__le,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),Z2: A] :
( ? [X4: A] :
( aa(A,$o,P,X4)
& ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
& ! [Y2: A] :
( ( aa(A,$o,P,Y2)
& ! [Ya: A] :
( aa(A,$o,P,Ya)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya) ) )
=> ( Y2 = X4 ) ) )
=> ( aa(A,$o,P,Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z2) ) ) ) ).
% Least1_le
tff(fact_3373_Least1I,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o)] :
( ? [X4: A] :
( aa(A,$o,P,X4)
& ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y2) )
& ! [Y2: A] :
( ( aa(A,$o,P,Y2)
& ! [Ya: A] :
( aa(A,$o,P,Ya)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya) ) )
=> ( Y2 = X4 ) ) )
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% Least1I
tff(fact_3374_Least__le,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),K: A] :
( aa(A,$o,P,K)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).
% Least_le
tff(fact_3375_not__less__Least,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [K: A,P: fun(A,$o)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
=> ~ aa(A,$o,P,K) ) ) ).
% not_less_Least
tff(fact_3376_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),P: fun(A,fun(C,$o)),Q: fun(B,fun(D,$o)),A4: product_prod(A,B),A3: set(product_prod(A,B)),B3: product_prod(C,D),B5: set(product_prod(C,D))] :
( aa(fun(B,fun(D,$o)),$o,aa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),$o),bNF_rel_fun(A,B,fun(C,$o),fun(D,$o),R12,bNF_rel_fun(C,D,$o,$o,R23,fequal($o))),P),Q)
=> ( member(product_prod(A,B),A4,A3)
=> ( member(product_prod(C,D),B3,B5)
=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R12)))
=> ( aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),B5),aa(fun(product_prod(C,D),$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,$o),R23)))
=> ( aa(C,$o,aa(A,fun(C,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A4)),aa(product_prod(C,D),C,product_fst(C,D),B3))
<=> aa(D,$o,aa(B,fun(D,$o),Q,aa(product_prod(A,B),B,product_snd(A,B),A4)),aa(product_prod(C,D),D,product_snd(C,D),B3)) ) ) ) ) ) ) ).
% predicate2_transferD
tff(fact_3377_times__integer_Oabs__eq,axiom,
! [Xa: 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,Xa)),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),Xa),X)) ).
% times_integer.abs_eq
tff(fact_3378_mlex__less,axiom,
! [A: $tType,F2: fun(A,nat),X: A,Y: A,R3: set(product_prod(A,A))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F2,R3)) ) ).
% mlex_less
tff(fact_3379_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F2: fun(A,nat),R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F2,R3))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
| ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3) ) ) ) ).
% mlex_iff
tff(fact_3380_mlex__leq,axiom,
! [A: $tType,F2: fun(A,nat),X: A,Y: A,R3: set(product_prod(A,A))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F2,R3)) ) ) ).
% mlex_leq
tff(fact_3381_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D),X5: set(product_prod(A,B)),X: product_prod(A,B)] :
( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A3,B5),F2),G)
=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),X5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A3)))
=> ( member(product_prod(A,B),X,X5)
=> aa(D,$o,aa(C,fun(D,$o),B5,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F2),product_fst(A,B)),X)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),comp(B,D,product_prod(A,B),G),product_snd(A,B)),X)) ) ) ) ).
% rel_fun_Collect_case_prodD
tff(fact_3382_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_gr(A,fun(set(set(A)),set(set(A))))) ).
% comp_fun_commute_Pow_fold
tff(fact_3383_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& semiring_1(A) )
=> ! [R3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
=> aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R3),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).
% transfer_rule_of_nat
tff(fact_3384_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( zero_neq_one(B)
& zero_neq_one(A) )
=> ! [R3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
=> aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R3),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).
% transfer_rule_of_bool
tff(fact_3385_typedef__rep__transfer,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),T3: fun(B,fun(A,$o))] :
( type_definition(A,B,Rep,Abs,A3)
=> ( ! [X3: B,Xa4: A] :
( aa(A,$o,aa(B,fun(A,$o),T3,X3),Xa4)
<=> ( X3 = aa(A,B,Rep,Xa4) ) )
=> aa(fun(A,B),$o,aa(fun(B,B),fun(fun(A,B),$o),bNF_rel_fun(B,A,B,B,T3,fequal(B)),aTP_Lamp_ae(B,B)),Rep) ) ) ).
% typedef_rep_transfer
tff(fact_3386_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))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),
$ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ku(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).
% divmod_integer_eq_cases
tff(fact_3387_plus__rat_Otransfer,axiom,
aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_kv(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).
% plus_rat.transfer
tff(fact_3388_times__rat_Otransfer,axiom,
aa(fun(rat,fun(rat,rat)),$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_kw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat)) ).
% times_rat.transfer
tff(fact_3389_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))),
$ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_kx(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).
% int_of_integer_code
tff(fact_3390_positive_Otransfer,axiom,
aa(fun(rat,$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($o)),aTP_Lamp_ky(product_prod(int,int),$o)),positive) ).
% positive.transfer
tff(fact_3391_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_la(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).
% times_int.transfer
tff(fact_3392_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> boolea2506097494486148201lgebra(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% bit.abstract_boolean_algebra_axioms
tff(fact_3393_times__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_la(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).
% times_int.abs_eq
tff(fact_3394_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% times_integer.rep_eq
tff(fact_3395_positive__mult,axiom,
! [X: rat,Y: rat] :
( aa(rat,$o,positive,X)
=> ( aa(rat,$o,positive,Y)
=> aa(rat,$o,positive,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),X),Y)) ) ) ).
% positive_mult
tff(fact_3396_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_3397_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_3398_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_3399_positive_Orep__eq,axiom,
! [X: rat] :
( aa(rat,$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_3400_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A4: B] :
( inj_on(A,B,F2,A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B))))),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_lb(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),A3),A4))),bot_bot(set(A)))) ) ).
% inj_on_vimage_singleton
tff(fact_3401_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: B] :
( inj_on(A,B,F2,top_top(set(A)))
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aTP_Lamp_lc(fun(A,B),fun(B,fun(A,$o)),F2),A4))),bot_bot(set(A)))) ) ).
% inj_vimage_singleton
tff(fact_3402_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).
% diff_numeral_special(8)
tff(fact_3403_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N2))) = neg_numeral_sub(A,N2,one2) ) ).
% diff_numeral_special(7)
tff(fact_3404_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
% The_split_eq
tff(fact_3405_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : minus_minus(A,one_one(A),aa(num,A,numeral_numeral(A),N2)) = neg_numeral_sub(A,one2,N2) ) ).
% diff_numeral_special(1)
tff(fact_3406_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : minus_minus(A,aa(num,A,numeral_numeral(A),M),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).
% diff_numeral_special(2)
tff(fact_3407_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N2)) = neg_numeral_sub(A,N2,one2) ) ).
% add_neg_numeral_special(4)
tff(fact_3408_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).
% add_neg_numeral_special(3)
tff(fact_3409_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).
% add_neg_numeral_special(2)
tff(fact_3410_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) = neg_numeral_sub(A,one2,M) ) ).
% add_neg_numeral_special(1)
tff(fact_3411_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num] : minus_minus(A,aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2)),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).
% minus_sub_one_diff_one
tff(fact_3412_The__case__prod,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = the(product_prod(A,B),aTP_Lamp_le(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).
% The_case_prod
tff(fact_3413_the__elem__def,axiom,
! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_lf(set(A),fun(A,$o),X5)) ).
% the_elem_def
tff(fact_3414_old_Orec__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : product_rec_prod(A,B,C,X4,Xa3) = the(C,product_rec_set_prod(A,B,C,X4,Xa3)) ).
% old.rec_prod_def
tff(fact_3415_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_3416_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).
% eq_numeral_iff_iszero(8)
tff(fact_3417_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( aa(code_integer,$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,$o),ord_less(nat),code_nat_of_integer(X)),code_nat_of_integer(Y))
<=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Y) ) ) ) ).
% nat_of_integer_less_iff
tff(fact_3418_prod_Oinsert_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I: set(A),P2: fun(A,B),I2: A] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),I),P2)))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I2),I)) = $ite(member(A,I2,I),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),I),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P2,I2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P2),I))) ) ) ) ).
% prod.insert'
tff(fact_3419_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P2: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty'
tff(fact_3420_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_3421_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_lg(fun(B,A),fun(set(B),fun(B,$o)),G),I))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I) ) ).
% prod.non_neutral'
tff(fact_3422_not__iszero__neg__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% not_iszero_neg_1
tff(fact_3423_prod_Odistrib__triv_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_lh(fun(A,B),fun(fun(A,B),fun(A,B)),G),Ha)),I) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),Ha),I)) ) ) ) ).
% prod.distrib_triv'
tff(fact_3424_prod_Omono__neutral__left_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T3: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T3) ) ) ) ) ).
% prod.mono_neutral_left'
tff(fact_3425_prod_Omono__neutral__right_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T3: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) ) ) ) ) ).
% prod.mono_neutral_right'
tff(fact_3426_prod_Omono__neutral__cong__left_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T3: set(A),Ha: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [I3: A] :
( member(A,I3,minus_minus(set(A),T3,S))
=> ( aa(A,B,Ha,I3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,Ha,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),Ha),T3) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_3427_prod_Omono__neutral__cong__right_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T3: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
=> ( ! [X3: A] :
( member(A,X3,minus_minus(set(A),T3,S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,Ha,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),Ha),S) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_3428_prod_Odistrib_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I: set(A),G: fun(A,B),Ha: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),I),G)))
=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),I),Ha)))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_lh(fun(A,B),fun(fun(A,B),fun(A,B)),G),Ha)),I) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),Ha),I)) ) ) ) ) ).
% prod.distrib'
tff(fact_3429_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P2: fun(B,A),I: set(B)] :
aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_lg(fun(B,A),fun(set(B),fun(B,$o)),P2),I))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_lg(fun(B,A),fun(set(B),fun(B,$o)),P2),I))),one_one(A)) ) ).
% prod.G_def
tff(fact_3430_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
<=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_3431_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
<=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).
% eq_numeral_iff_iszero(6)
tff(fact_3432_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] : minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,one_one(A),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_li(A,fun(nat,fun(nat,A)),X),N2)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% one_diff_power_eq'
tff(fact_3433_times__int_Orsp,axiom,
aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$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_la(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_la(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int.rsp
tff(fact_3434_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X5: set(A),F2: fun(A,set(B))] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_lj(fun(A,set(B)),fun(A,filter(B)),F2)),X5)) = aa(set(B),filter(B),principal(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X5))) ) ) ).
% INF_principal_finite
tff(fact_3435_mask__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat,M: nat] : modulo_modulo(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N2)),one_one(A)) ) ).
% mask_mod_exp
tff(fact_3436_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N2) = minus_minus(set(nat),aa(nat,set(nat),set_ord_lessThan(nat),N2),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_lessThan_eq_remove0
tff(fact_3437_principal__inject,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( ( aa(set(A),filter(A),principal(A),A3) = aa(set(A),filter(A),principal(A),B5) )
<=> ( A3 = B5 ) ) ).
% principal_inject
tff(fact_3438_min__arg__le_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [M: A,N2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M ) ) ) ).
% min_arg_le(2)
tff(fact_3439_min__arg__le_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [N2: A,M: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 ) ) ) ).
% min_arg_le(1)
tff(fact_3440_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N2: A] :
( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),M) ) ) ).
% min_eq_arg(2)
tff(fact_3441_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N2: A] :
( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),N2) ) ) ).
% min_eq_arg(1)
tff(fact_3442_min__simps_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A4),B3) = B3 ) ) ) ).
% min_simps(2)
tff(fact_3443_min__simps_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A4),B3) = A4 ) ) ) ).
% min_simps(1)
tff(fact_3444_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A4),B3)),B3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ).
% min_less_self_conv(2)
tff(fact_3445_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A4),B3)),A4)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) ) ) ).
% min_less_self_conv(1)
tff(fact_3446_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N2: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)),N2)
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = N2 ) ) ) ).
% min_arg_not_ge(2)
tff(fact_3447_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N2: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2)),M)
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N2) = M ) ) ) ).
% min_arg_not_ge(1)
tff(fact_3448_min__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).
% min_bot
tff(fact_3449_min__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).
% min_bot2
tff(fact_3450_min__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).
% min_top
tff(fact_3451_min__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).
% min_top2
tff(fact_3452_max__min__same_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).
% max_min_same(4)
tff(fact_3453_max__min__same_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).
% max_min_same(3)
tff(fact_3454_max__min__same_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).
% max_min_same(2)
tff(fact_3455_max__min__same_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).
% max_min_same(1)
tff(fact_3456_sup__principal,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),aa(set(A),filter(A),principal(A),A3)),aa(set(A),filter(A),principal(A),B5)) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) ).
% sup_principal
tff(fact_3457_min__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).
% min_0_1(1)
tff(fact_3458_min__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).
% min_0_1(2)
tff(fact_3459_min__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).
% min_0_1(6)
tff(fact_3460_min__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).
% min_0_1(5)
tff(fact_3461_min__Suc__gt_I1_J,axiom,
! [A4: nat,B3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B3)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A4)),B3) = aa(nat,nat,suc,A4) ) ) ).
% min_Suc_gt(1)
tff(fact_3462_min__Suc__gt_I2_J,axiom,
! [A4: nat,B3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B3)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B3),aa(nat,nat,suc,A4)) = aa(nat,nat,suc,A4) ) ) ).
% min_Suc_gt(2)
tff(fact_3463_lessThan__0,axiom,
aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).
% lessThan_0
tff(fact_3464_principal__le__iff,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(A),filter(A),principal(A),A3)),aa(set(A),filter(A),principal(A),B5))
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).
% principal_le_iff
tff(fact_3465_inf__principal,axiom,
! [A: $tType,A3: set(A),B5: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(A),filter(A),principal(A),A3)),aa(set(A),filter(A),principal(A),B5)) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5)) ).
% inf_principal
tff(fact_3466_single__Diff__lessThan,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A] : minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A))) ) ).
% single_Diff_lessThan
tff(fact_3467_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N2))),aa(nat,A,G,N2)) ) ).
% prod.lessThan_Suc
tff(fact_3468_SUP__principal,axiom,
! [A: $tType,B: $tType,A3: fun(B,set(A)),I: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aTP_Lamp_lk(fun(B,set(A)),fun(B,filter(A)),A3)),I)) = aa(set(A),filter(A),principal(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I))) ).
% SUP_principal
tff(fact_3469_Inf__atMostLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),X)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).
% Inf_atMostLessThan
tff(fact_3470_Min__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ).
% Min_insert
tff(fact_3471_nat__mult__min__right,axiom,
! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N2),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).
% nat_mult_min_right
tff(fact_3472_nat__mult__min__left,axiom,
! [M: nat,N2: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N2)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),Q3)) ).
% nat_mult_min_left
tff(fact_3473_min__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A4: A,B3: A] :
aa(A,A,aa(A,fun(A,A),ord_min(A),A4),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3),A4,B3) ) ).
% min_def
tff(fact_3474_min__absorb1,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).
% min_absorb1
tff(fact_3475_min__absorb2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).
% min_absorb2
tff(fact_3476_min__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),minus_minus(A,X,Z2)),minus_minus(A,Y,Z2)) ) ).
% min_diff_distrib_left
tff(fact_3477_min__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% min_add_distrib_left
tff(fact_3478_min__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).
% min_add_distrib_right
tff(fact_3479_lessThan__non__empty,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] : aa(A,set(A),set_ord_lessThan(A),X) != bot_bot(set(A)) ) ).
% lessThan_non_empty
tff(fact_3480_top__eq__principal__UNIV,axiom,
! [A: $tType] : top_top(filter(A)) = aa(set(A),filter(A),principal(A),top_top(set(A))) ).
% top_eq_principal_UNIV
tff(fact_3481_minus__max__eq__min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_max_eq_min
tff(fact_3482_minus__min__eq__max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_min_eq_max
tff(fact_3483_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( aa(nat,set(nat),set_ord_lessThan(nat),N2) = bot_bot(set(nat)) )
<=> ( N2 = zero_zero(nat) ) ) ).
% lessThan_empty_iff
tff(fact_3484_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( linorder(A)
& order_bot(A) )
=> ! [N2: A] :
( ( aa(A,set(A),set_ord_lessThan(A),N2) = bot_bot(set(A)) )
<=> ( N2 = bot_bot(A) ) ) ) ).
% Iio_eq_empty_iff
tff(fact_3485_bot__eq__principal__empty,axiom,
! [A: $tType] : bot_bot(filter(A)) = aa(set(A),filter(A),principal(A),bot_bot(set(A))) ).
% bot_eq_principal_empty
tff(fact_3486_principal__eq__bot__iff,axiom,
! [A: $tType,X5: set(A)] :
( ( aa(set(A),filter(A),principal(A),X5) = bot_bot(filter(A)) )
<=> ( X5 = bot_bot(set(A)) ) ) ).
% principal_eq_bot_iff
tff(fact_3487_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(4)
tff(fact_3488_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(2)
tff(fact_3489_max__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P2: A,X: A,Y: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).
% max_mult_distrib_left
tff(fact_3490_min__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P2: A,X: A,Y: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).
% min_mult_distrib_left
tff(fact_3491_max__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A,P2: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).
% max_mult_distrib_right
tff(fact_3492_min__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A,P2: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).
% min_mult_distrib_right
tff(fact_3493_Inf__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S))) ) ) ) ).
% Inf_insert_finite
tff(fact_3494_hom__Min__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ha: fun(A,A),N: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,Ha,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,Ha,X3)),aa(A,A,Ha,Y2))
=> ( aa(set(A),$o,finite_finite2(A),N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,Ha,lattic643756798350308766er_Min(A,N)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,Ha),N)) ) ) ) ) ) ).
% hom_Min_commute
tff(fact_3495_Min_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,B5)),lattic643756798350308766er_Min(A,A3)) = lattic643756798350308766er_Min(A,A3) ) ) ) ) ) ).
% Min.subset
tff(fact_3496_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,A3)) ) ) ) ) ) ).
% Min.insert_not_elem
tff(fact_3497_Min_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic643756798350308766er_Min(A,A3),A3) ) ) ) ) ).
% Min.closed
tff(fact_3498_Min_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,A3)),lattic643756798350308766er_Min(A,B5)) ) ) ) ) ) ) ).
% Min.union
tff(fact_3499_Min_Oeq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,ord_min(A),X,A3) ) ) ) ).
% Min.eq_fold
tff(fact_3500_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gz(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% prod.lessThan_Suc_shift
tff(fact_3501_sum_Onat__group,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ll(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))) ) ).
% sum.nat_group
tff(fact_3502_prod_Onat__group,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),K: nat,N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_lm(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))) ) ).
% prod.nat_group
tff(fact_3503_Iio__Int__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [K: A,X: A] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),bot_bot(set(A))) ) ).
% Iio_Int_singleton
tff(fact_3504_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).
% ivl_disj_un_singleton(2)
tff(fact_3505_Min_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic643756798350308766er_Min(A,A3) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).
% Min.remove
tff(fact_3506_Min_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).
% Min.insert_remove
tff(fact_3507_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( aa(code_integer,$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),code_nat_of_integer(U))) ) ) ).
% image_atLeastZeroLessThan_integer
tff(fact_3508_power__diff__1__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,X,one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% power_diff_1_eq
tff(fact_3509_one__diff__power__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] : minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,one_one(A),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% one_diff_power_eq
tff(fact_3510_geometric__sum,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N2: nat] :
( ( X != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2),one_one(A)),minus_minus(A,X,one_one(A))) ) ) ) ).
% geometric_sum
tff(fact_3511_prod_OatMost__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gz(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% prod.atMost_shift
tff(fact_3512_sum__gp__strict,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N2: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),N2),divide_divide(A,minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),minus_minus(A,one_one(A),X))) ) ).
% sum_gp_strict
tff(fact_3513_power__diff__sumr2,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat,Y: A] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,X,Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ln(A,fun(nat,fun(A,fun(nat,A))),X),N2),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N2))) ) ).
% power_diff_sumr2
tff(fact_3514_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N2: nat,Y: A] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N2))) = aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,X,Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_lo(A,fun(nat,fun(A,fun(nat,A))),X),N2),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N2)))) ) ).
% diff_power_eq_sum
tff(fact_3515_at__bot__def,axiom,
! [A: $tType] :
( order(A)
=> ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_lp(A,filter(A))),top_top(set(A)))) ) ) ).
% at_bot_def
tff(fact_3516_at__bot__sub,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_lq(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).
% at_bot_sub
tff(fact_3517_finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite5375528669736107172at_top(A,A3) = aa(set(set(A)),filter(set(A)),principal(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),bot_bot(set(set(A))))) ) ) ).
% finite_subsets_at_top_finite
tff(fact_3518_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_3519_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_3520_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A4) = one_one(A) ) ).
% gcd.bottom_left_bottom
tff(fact_3521_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),one_one(A)) = one_one(A) ) ).
% gcd.bottom_right_bottom
tff(fact_3522_is__unit__gcd__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3),one_one(A))
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) = one_one(A) ) ) ) ).
% is_unit_gcd_iff
tff(fact_3523_Gcd__insert,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A,A3: set(A)] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),gcd_Gcd(A,A3)) ) ).
% Gcd_insert
tff(fact_3524_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3)) ) ).
% Gcd_fin.insert
tff(fact_3525_Gcd__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A,B3: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) ) ).
% Gcd_2
tff(fact_3526_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) != bot_bot(filter(set(A))) ).
% finite_subsets_at_top_neq_bot
tff(fact_3527_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N2)) ).
% gcd_mult_distrib_nat
tff(fact_3528_greaterThan__non__empty,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).
% greaterThan_non_empty
tff(fact_3529_gcd__add__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [M: A,K: A,N2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N2) ) ).
% gcd_add_mult
tff(fact_3530_gcd__dvd__prod,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,K: A] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3),aa(A,A,aa(A,fun(A,A),times_times(A),K),B3)) ) ).
% gcd_dvd_prod
tff(fact_3531_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ( at_bot(A) != bot_bot(filter(A)) ) ) ).
% trivial_limit_at_bot_linorder
tff(fact_3532_Gcd__in,axiom,
! [A3: set(nat)] :
( ! [A5: nat,B2: nat] :
( member(nat,A5,A3)
=> ( member(nat,B2,A3)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A5),B2),A3) ) )
=> ( ( A3 != bot_bot(set(nat)) )
=> member(nat,gcd_Gcd(nat,A3),A3) ) ) ).
% Gcd_in
tff(fact_3533_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).
% gcd_mult_unit1
tff(fact_3534_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).
% gcd_mult_unit2
tff(fact_3535_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),divide_divide(A,C2,A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).
% gcd_div_unit2
tff(fact_3536_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),divide_divide(A,B3,A4)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),C2) ) ) ) ).
% gcd_div_unit1
tff(fact_3537_bezout__nat,axiom,
! [A4: nat,B3: nat] :
( ( A4 != zero_zero(nat) )
=> ? [X3: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B3)) ) ).
% bezout_nat
tff(fact_3538_bezout__gcd__nat_H,axiom,
! [B3: nat,A4: nat] :
? [X3: nat,Y2: nat] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3))
& ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),X3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B3) ) )
| ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3))
& ( minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B3) ) ) ) ).
% bezout_gcd_nat'
tff(fact_3539_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(7)
tff(fact_3540_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] :
( member(A,A4,A3)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),aa(set(A),A,semiring_gcd_Gcd_fin(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ) ).
% Gcd_fin.remove
tff(fact_3541_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),aa(set(A),A,semiring_gcd_Gcd_fin(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ).
% Gcd_fin.insert_remove
tff(fact_3542_finite__subsets__at__top__def,axiom,
! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_ls(set(A),fun(set(A),filter(set(A))),A3)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_lt(set(A),fun(set(A),$o),A3)))) ).
% finite_subsets_at_top_def
tff(fact_3543_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
aa(set(A),A,semiring_gcd_Gcd_fin(A),A3) = $ite(aa(set(A),$o,finite_finite2(A),A3),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A3),one_one(A)) ) ).
% Gcd_fin.eq_fold
tff(fact_3544_greaterThan__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = minus_minus(set(nat),aa(nat,set(nat),set_ord_greaterThan(nat),K),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).
% greaterThan_Suc
tff(fact_3545_wf__finite__segments,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
=> ( trans(A,R2)
=> ( ! [X3: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_cv(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),X3)))
=> wf(A,R2) ) ) ) ).
% wf_finite_segments
tff(fact_3546_plus__rat_Orsp,axiom,
aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$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_kv(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_kv(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat.rsp
tff(fact_3547_relation__of__def,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),A3: set(A)] : order_relation_of(A,P,A3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_lu(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),P),A3))) ).
% relation_of_def
tff(fact_3548_filterlim__base__iff,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,I: set(A),F4: fun(A,set(B)),F2: fun(B,C),G5: fun(D,set(C)),J: set(D)] :
( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> ! [J4: A] :
( member(A,J4,I)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J4))
| aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J4)),aa(A,set(B),F4,I3)) ) ) )
=> ( filterlim(B,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_lv(fun(D,set(C)),fun(D,filter(C)),G5)),J)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_lj(fun(A,set(B)),fun(A,filter(B)),F4)),I)))
<=> ! [X2: D] :
( member(D,X2,J)
=> ? [Xa2: A] :
( member(A,Xa2,I)
& ! [Xb3: B] :
( member(B,Xb3,aa(A,set(B),F4,Xa2))
=> member(C,aa(B,C,F2,Xb3),aa(D,set(C),G5,X2)) ) ) ) ) ) ) ).
% filterlim_base_iff
tff(fact_3549_positive_Orsp,axiom,
aa(fun(product_prod(int,int),$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($o)),aTP_Lamp_ky(product_prod(int,int),$o)),aTP_Lamp_ky(product_prod(int,int),$o)) ).
% positive.rsp
tff(fact_3550_ratrel__iff,axiom,
! [X: product_prod(int,int),Y: product_prod(int,int)] :
( aa(product_prod(int,int),$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_3551_filterlim__ident,axiom,
! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_ac(A,A),F4,F4) ).
% filterlim_ident
tff(fact_3552_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F23: filter(A),F2: fun(C,A),F12: filter(C)] :
( filterlim(A,B,G,F32,F23)
=> ( filterlim(C,A,F2,F23,F12)
=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_lw(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F12) ) ) ).
% filterlim_compose
tff(fact_3553_filterlim__mono,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A),F24: filter(B),F13: filter(A)] :
( filterlim(A,B,F2,F23,F12)
=> ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F23),F24)
=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F13),F12)
=> filterlim(A,B,F2,F24,F13) ) ) ) ).
% filterlim_mono
tff(fact_3554_irrefl__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
<=> ! [A7: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7),R2) ) ).
% irrefl_def
tff(fact_3555_irreflI,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [A5: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R3)
=> irrefl(A,R3) ) ).
% irreflI
tff(fact_3556_filterlim__sup,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(B),F12: filter(A),F23: filter(A)] :
( filterlim(A,B,F2,F4,F12)
=> ( filterlim(A,B,F2,F4,F23)
=> filterlim(A,B,F2,F4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F12),F23)) ) ) ).
% filterlim_sup
tff(fact_3557_filterlim__top,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(A)] : filterlim(A,B,F2,top_top(filter(B)),F4) ).
% filterlim_top
tff(fact_3558_filterlim__inf,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F32: filter(B),F12: filter(A)] :
( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F32),F12)
<=> ( filterlim(A,B,F2,F23,F12)
& filterlim(A,B,F2,F32,F12) ) ) ).
% filterlim_inf
tff(fact_3559_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V3: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U3),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V3),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).
% bezout_int
tff(fact_3560_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N2)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N2)) ).
% gcd_mult_distrib_int
tff(fact_3561_filterlim__INF,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G5: fun(C,filter(B)),B5: set(C),F4: filter(A)] :
( filterlim(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G5),B5)),F4)
<=> ! [X2: C] :
( member(C,X2,B5)
=> filterlim(A,B,F2,aa(C,filter(B),G5,X2),F4) ) ) ).
% filterlim_INF
tff(fact_3562_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A3: set(A),F2: fun(B,C),F4: filter(C),G5: fun(A,filter(B))] :
( member(A,X,A3)
=> ( filterlim(B,C,F2,F4,aa(A,filter(B),G5,X))
=> filterlim(B,C,F2,F4,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G5),A3))) ) ) ).
% filterlim_INF'
tff(fact_3563_filterlim__If,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G5: filter(B),F4: filter(A),P: fun(A,$o),G: fun(A,B)] :
( filterlim(A,B,F2,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),P))))
=> ( filterlim(A,B,G,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P)))))
=> filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_lx(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G5,F4) ) ) ).
% filterlim_If
tff(fact_3564_irrefl__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : irrefl(A,minus_minus(set(product_prod(A,A)),R2,id2(A))) ).
% irrefl_diff_Id
tff(fact_3565_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J: set(A),I2: fun(A,C),I: set(C),F4: fun(C,set(D)),F2: fun(D,E),G5: fun(A,set(E))] :
( ! [M3: A,X3: B] :
( member(A,M3,J)
=> member(C,aa(A,C,I2,M3),I) )
=> ( ! [M3: A,X3: D] :
( member(A,M3,J)
=> ( member(D,X3,aa(C,set(D),F4,aa(A,C,I2,M3)))
=> member(E,aa(D,E,F2,X3),aa(A,set(E),G5,M3)) ) )
=> filterlim(D,E,F2,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),aTP_Lamp_ly(fun(A,set(E)),fun(A,filter(E)),G5)),J)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_lz(fun(C,set(D)),fun(C,filter(D)),F4)),I))) ) ) ).
% filterlim_base
tff(fact_3566_ratrel__def,axiom,
! [X4: product_prod(int,int),Xa3: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X4),Xa3)
<=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X4) != zero_zero(int) )
& ( aa(product_prod(int,int),int,product_snd(int,int),Xa3) != zero_zero(int) )
& ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X4)),aa(product_prod(int,int),int,product_snd(int,int),Xa3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa3)),aa(product_prod(int,int),int,product_snd(int,int),X4)) ) ) ) ).
% ratrel_def
tff(fact_3567_times__rat_Orsp,axiom,
aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$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))),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_kw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_kw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat.rsp
tff(fact_3568_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod(int,int),X: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
=> ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_3569_positive_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( aa(product_prod(int,int),$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_3570_times__rat_Oabs__eq,axiom,
! [Xa: 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,Xa),Xa)
=> ( 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),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),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),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% times_rat.abs_eq
tff(fact_3571_pairself__image__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F2)),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),P))) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_ma(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F2),P)) ).
% pairself_image_eq
tff(fact_3572_flat__lub__def,axiom,
! [A: $tType,B3: A,A3: set(A)] :
partial_flat_lub(A,B3,A3) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),B3,the(A,aa(set(A),fun(A,$o),aTP_Lamp_mb(A,fun(set(A),fun(A,$o)),B3),A3))) ).
% flat_lub_def
tff(fact_3573_eq__or__mem__image__simp,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: B,B5: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_mc(fun(B,A),fun(B,fun(set(B),fun(A,$o))),F2),A4),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F2,A4)),aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_md(fun(B,A),fun(set(B),fun(A,$o)),F2),B5))) ).
% eq_or_mem_image_simp
tff(fact_3574_ex__assn__proper,axiom,
! [A: $tType,P: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o))] :
( ! [X3: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),P,X3))
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_me(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ) ).
% ex_assn_proper
tff(fact_3575_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mf(fun(A,fun(B,$o)),fun(A,$o),P)) ).
% Domain_Collect_case_prod
tff(fact_3576_Range__Collect__case__prod,axiom,
! [A: $tType,B: $tType,P: fun(B,fun(A,$o))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mg(fun(B,fun(A,$o)),fun(A,$o),P)) ).
% Range_Collect_case_prod
tff(fact_3577_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_md(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).
% Setcompr_eq_image
tff(fact_3578_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_mh(fun(B,A),fun(fun(B,$o),fun(A,$o)),F2),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),P)) ).
% setcompr_eq_image
tff(fact_3579_fs__contract,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(A,fun(B,C)),S: set(C)] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_mi(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),F2),S))) = aa(fun(A,$o),set(A),collect(A),aa(set(C),fun(A,$o),aTP_Lamp_mj(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),F2),S)) ).
% fs_contract
tff(fact_3580_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mk(fun(B,A),fun(A,$o),F2)) = aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) ).
% full_SetCompr_eq
tff(fact_3581_Id__def,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_ml(product_prod(A,A),$o)) ).
% Id_def
tff(fact_3582_Domain__unfold,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mm(set(product_prod(A,B)),fun(A,$o),R2)) ).
% Domain_unfold
tff(fact_3583_Pow__Compl,axiom,
! [A: $tType,A3: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_mn(set(A),fun(set(A),$o),A3)) ).
% Pow_Compl
tff(fact_3584_Un__interval,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [B1: A,B22: A,B32: A,F2: fun(A,B)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B1),B22)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B22),B32)
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_mo(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B22),F2))),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_mo(A,fun(A,fun(fun(A,B),fun(B,$o))),B22),B32),F2))) = aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_mo(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B32),F2)) ) ) ) ) ).
% Un_interval
tff(fact_3585_relcomp__unfold,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R2,S2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_mp(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S2))) ).
% relcomp_unfold
tff(fact_3586_inf__Sup2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B5)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_mq(set(A),fun(set(A),fun(A,$o)),A3),B5))) ) ) ) ) ) ) ).
% inf_Sup2_distrib
tff(fact_3587_inf__Sup1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic5882676163264333800up_fin(A,A3)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_mr(set(A),fun(A,fun(A,$o)),A3),X))) ) ) ) ) ).
% inf_Sup1_distrib
tff(fact_3588_sup__Inf2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,A3)),lattic7752659483105999362nf_fin(A,B5)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ms(set(A),fun(set(A),fun(A,$o)),A3),B5))) ) ) ) ) ) ) ).
% sup_Inf2_distrib
tff(fact_3589_sup__Inf1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic7752659483105999362nf_fin(A,A3)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_mt(set(A),fun(A,fun(A,$o)),A3),X))) ) ) ) ) ).
% sup_Inf1_distrib
tff(fact_3590_brk__rel__def,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : brk_rel(A,B,R3) = aa(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B))),aa(set(product_prod(product_prod($o,A),product_prod($o,B))),fun(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B)))),sup_sup(set(product_prod(product_prod($o,A),product_prod($o,B)))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_mu(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),R3))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_mv(product_prod(product_prod($o,A),product_prod($o,B)),$o))) ).
% brk_rel_def
tff(fact_3591_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType,A3: set(C),F2: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A3,F2,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_mw(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A3),F2),G)) ).
% image2_def
tff(fact_3592_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [F2: fun(A,C),G: fun(B,C),A3: set(A),B5: set(B)] :
( inj_on(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_mx(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),F2),G)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))
=> ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F2),A3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),B5)) = aa(set(C),C,aa(fun(C,C),fun(set(C),C),groups7311177749621191930dd_sum(C,C),id(C)),aa(fun(C,$o),set(C),collect(C),aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_my(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),F2),G),A3),B5))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_3593_ex__assn__def,axiom,
! [A: $tType,P: fun(A,assn)] : ex_assn(A,P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_mz(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ).
% ex_assn_def
tff(fact_3594_mset__set__Union,axiom,
! [A: $tType,A3: set(A),B5: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A3)),mset_set(A,B5)) ) ) ) ) ).
% mset_set_Union
tff(fact_3595_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_3596_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U)) ).
% finite_greaterThanLessThan_integer
tff(fact_3597_ex__assn__const,axiom,
! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_na(assn,fun(A,assn),C2)) = C2 ).
% ex_assn_const
tff(fact_3598_case__prod__Pair,axiom,
! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).
% case_prod_Pair
tff(fact_3599_apfst__id,axiom,
! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).
% apfst_id
tff(fact_3600_apsnd__id,axiom,
! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).
% apsnd_id
tff(fact_3601_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
=> ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanLessThan_empty
tff(fact_3602_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A4: A,B3: A] :
( ( set_or5935395276787703475ssThan(A,A4,B3) = bot_bot(set(A)) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ).
% greaterThanLessThan_empty_iff
tff(fact_3603_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A4: A,B3: A] :
( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A4,B3) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) ) ) ).
% greaterThanLessThan_empty_iff2
tff(fact_3604_mset__set_Oempty,axiom,
! [A: $tType] : mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ).
% mset_set.empty
tff(fact_3605_swap__comp__swap,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).
% swap_comp_swap
tff(fact_3606_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_ac(A,A),aTP_Lamp_ae(B,B)) = id(product_prod(A,B)) ).
% map_prod.identity
tff(fact_3607_apfst__def,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(A,C)] : product_apfst(A,C,B,F2) = product_map_prod(A,C,B,B,F2,id(B)) ).
% apfst_def
tff(fact_3608_apsnd__def,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2) = product_map_prod(A,A,B,C,id(A),F2) ).
% apsnd_def
tff(fact_3609_ex__one__point__gen,axiom,
! [A: $tType,P: fun(A,assn),V: A] :
( ! [H: product_prod(heap_ext(product_unit),set(nat)),X3: A] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,P,X3)),H)
=> ( X3 = V ) )
=> ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).
% ex_one_point_gen
tff(fact_3610_ex__distrib__star,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_nb(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P)),Q) ).
% ex_distrib_star
tff(fact_3611_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,Abs),Rep) = id(A) ) ) ).
% type_copy_Abs_o_Rep
tff(fact_3612_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(B,A),fun(B,B),comp(A,B,B,Rep),Abs) = id(B) ) ) ).
% type_copy_Rep_o_Abs
tff(fact_3613_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(B,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M2 = id(B) )
=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,aa(fun(B,B),fun(B,A),comp(B,A,B,Abs),M2)),Rep) = id(A) ) ) ) ).
% type_copy_map_id0
tff(fact_3614_ex__distrib__and,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_nc(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P)),Q) ).
% ex_distrib_and
tff(fact_3615_ex__distrib__or,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_nd(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P)),Q) ).
% ex_distrib_or
tff(fact_3616_ex__join__or,axiom,
! [A: $tType,P: fun(A,assn),Q: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ne(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_nf(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).
% ex_join_or
tff(fact_3617_fst__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_ex(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% fst_diag_id
tff(fact_3618_snd__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_ex(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% snd_diag_id
tff(fact_3619_mset__set__empty__iff,axiom,
! [A: $tType,A3: set(A)] :
( ( mset_set(A,A3) = zero_zero(multiset(A)) )
<=> ( ( A3 = bot_bot(set(A)) )
| ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).
% mset_set_empty_iff
tff(fact_3620_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(5)
tff(fact_3621_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(4)
tff(fact_3622_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(1)
tff(fact_3623_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(1)
tff(fact_3624_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F2: fun(B,A),X: B,C2: C,G: fun(B,C),A3: set(B)] :
( ( B3 = aa(B,A,F2,X) )
=> ( ( C2 = aa(B,C,G,X) )
=> ( member(B,X,A3)
=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B3),C2),bNF_Greatest_image2(B,A,C,A3,F2,G)) ) ) ) ).
% image2_eqI
tff(fact_3625_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_3626_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: A,B3: A] : minus_minus(set(A),set_or1337092689740270186AtMost(A,A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A4,B3) ) ).
% atLeastAtMost_diff_ends
tff(fact_3627_positive__def,axiom,
positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_ky(product_prod(int,int),$o)) ).
% positive_def
tff(fact_3628_finite__def,axiom,
! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_ng(fun(set(A),$o),fun(set(A),$o))) ).
% finite_def
tff(fact_3629_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I2: A,C6: set(A),G: fun(A,B),X: A] :
( member(A,I2,C6)
=> ( inj_on(A,B,G,C6)
=> ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_nh(set(A),fun(fun(A,B),fun(A,fun(B,A))),C6),G),X)),G),I2) = aa(A,A,id(A),I2) ) ) ) ).
% If_the_inv_into_f_f
tff(fact_3630_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Z2: A] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,the_inv_into(A,B,top_top(set(A)),F2)),F2),Z2) = aa(A,A,id(A),Z2) ) ) ).
% the_inv_f_o_f_id
tff(fact_3631_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: fun(A,B),C6: set(A),B5: set(A),X: A] :
( inj_on(A,B,G,C6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
=> member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_ni(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C6),X),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).
% If_the_inv_into_in_Func
tff(fact_3632_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set(A),F4: fun(A,filter(B)),P: fun(B,$o)] :
( ( B5 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,B5)
=> ! [B2: A] :
( member(A,B2,B5)
=> ? [X4: A] :
( member(A,X4,B5)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A5)),aa(A,filter(B),F4,B2))) ) ) )
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B5)))
<=> ? [X2: A] :
( member(A,X2,B5)
& aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(A,filter(B),F4,X2)) ) ) ) ) ).
% eventually_INF_base
tff(fact_3633_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( ~ dvd_dvd(A,A4,one_one(A))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,B3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))) ) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_3634_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_3635_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U)) ).
% finite_greaterThanAtMost_integer
tff(fact_3636_eventually__top,axiom,
! [A: $tType,P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),top_top(filter(A)))
<=> ! [X_12: A] : aa(A,$o,P,X_12) ) ).
% eventually_top
tff(fact_3637_eventually__const,axiom,
! [A: $tType,F4: filter(A),P: $o] :
( ( F4 != bot_bot(filter(A)) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_aj($o,fun(A,$o),(P))),F4)
<=> (P) ) ) ).
% eventually_const
tff(fact_3638_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
=> ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanAtMost_empty
tff(fact_3639_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).
% greaterThanAtMost_empty_iff
tff(fact_3640_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_3641_euclidean__size__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).
% euclidean_size_1
tff(fact_3642_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
( ! [X7: set(A)] :
( aa(set(A),$o,finite_finite2(A),X7)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),A3)
=> aa(set(A),$o,P,X7) ) )
=> aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A3)) ) ).
% eventually_finite_subsets_at_top_weakI
tff(fact_3643_eventually__principal,axiom,
! [A: $tType,P: fun(A,$o),S: set(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(A),filter(A),principal(A),S))
<=> ! [X2: A] :
( member(A,X2,S)
=> aa(A,$o,P,X2) ) ) ).
% eventually_principal
tff(fact_3644_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C6: $o,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj($o,fun(fun(A,$o),fun(A,$o)),(C6)),P)),F4)
<=> ( (C6)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ) ).
% eventually_frequently_const_simps(6)
tff(fact_3645_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C6: $o,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_nk($o,fun(fun(A,$o),fun(A,$o)),(C6)),P)),F4)
<=> ( (C6)
| aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ) ).
% eventually_frequently_const_simps(4)
tff(fact_3646_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: fun(A,$o),C6: $o,F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_nl(fun(A,$o),fun($o,fun(A,$o)),P),(C6))),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
| (C6) ) ) ).
% eventually_frequently_const_simps(3)
tff(fact_3647_eventually__mp,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4) ) ) ).
% eventually_mp
tff(fact_3648_eventually__True,axiom,
! [A: $tType,F4: filter(A)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ci(A,$o)),F4) ).
% eventually_True
tff(fact_3649_eventually__conj,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4) ) ) ).
% eventually_conj
tff(fact_3650_eventually__elim2,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o),R3: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4)
=> ( ! [I3: A] :
( aa(A,$o,P,I3)
=> ( aa(A,$o,Q,I3)
=> aa(A,$o,R3,I3) ) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R3),F4) ) ) ) ).
% eventually_elim2
tff(fact_3651_eventually__subst,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_nm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4) ) ) ).
% eventually_subst
tff(fact_3652_eventually__rev__mp,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4) ) ) ).
% eventually_rev_mp
tff(fact_3653_eventually__conj__iff,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4) ) ) ).
% eventually_conj_iff
tff(fact_3654_not__eventually__impI,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4)
=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4) ) ) ).
% not_eventually_impI
tff(fact_3655_eventuallyI,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( ! [X3: A] : aa(A,$o,P,X3)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ).
% eventuallyI
tff(fact_3656_filter__eq__iff,axiom,
! [A: $tType,F4: filter(A),F7: filter(A)] :
( ( F4 = F7 )
<=> ! [P4: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P4),F4)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P4),F7) ) ) ).
% filter_eq_iff
tff(fact_3657_eventually__mono,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F4) ) ) ).
% eventually_mono
tff(fact_3658_not__eventuallyD,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ? [X3: A] : ~ aa(A,$o,P,X3) ) ).
% not_eventuallyD
tff(fact_3659_always__eventually,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( ! [X_1: A] : aa(A,$o,P,X_1)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ).
% always_eventually
tff(fact_3660_le__filter__def,axiom,
! [A: $tType,F4: filter(A),F7: filter(A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
<=> ! [P4: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P4),F7)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P4),F4) ) ) ).
% le_filter_def
tff(fact_3661_filter__leI,axiom,
! [A: $tType,F7: filter(A),F4: filter(A)] :
( ! [P3: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),F7)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),F4) )
=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7) ) ).
% filter_leI
tff(fact_3662_filter__leD,axiom,
! [A: $tType,F4: filter(A),F7: filter(A),P: fun(A,$o)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F7)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ) ).
% filter_leD
tff(fact_3663_eventually__happens_H,axiom,
! [A: $tType,F4: filter(A),P: fun(A,$o)] :
( ( F4 != bot_bot(filter(A)) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).
% eventually_happens'
tff(fact_3664_eventually__happens,axiom,
! [A: $tType,P: fun(A,$o),Net: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),Net)
=> ( ( Net = bot_bot(filter(A)) )
| ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).
% eventually_happens
tff(fact_3665_eventually__bot,axiom,
! [A: $tType,P: fun(A,$o)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),bot_bot(filter(A))) ).
% eventually_bot
tff(fact_3666_eventually__inf,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),F7: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F7))
<=> ? [Q4: fun(A,$o),R7: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q4),F4)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R7),F7)
& ! [X2: A] :
( ( aa(A,$o,Q4,X2)
& aa(A,$o,R7,X2) )
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_inf
tff(fact_3667_eventually__ex,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_mf(fun(A,fun(B,$o)),fun(A,$o),P)),F4)
<=> ? [Y8: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_nn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y8)),F4) ) ).
% eventually_ex
tff(fact_3668_eventually__sup,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),F7: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F7))
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F7) ) ) ).
% eventually_sup
tff(fact_3669_eventually__Sup,axiom,
! [A: $tType,P: fun(A,$o),S: set(filter(A))] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S))
<=> ! [X2: filter(A)] :
( member(filter(A),X2,S)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),X2) ) ) ).
% eventually_Sup
tff(fact_3670_filterlim__iff,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
( filterlim(A,B,F2,F23,F12)
<=> ! [P4: fun(B,$o)] :
( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P4),F23)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_ad(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P4)),F12) ) ) ).
% filterlim_iff
tff(fact_3671_filterlim__cong,axiom,
! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F2: fun(B,A),G: fun(B,A)] :
( ( F12 = F13 )
=> ( ( F23 = F24 )
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_no(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G)),F23)
=> ( filterlim(B,A,F2,F12,F23)
<=> filterlim(B,A,G,F13,F24) ) ) ) ) ).
% filterlim_cong
tff(fact_3672_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),F2: fun(B,A),G5: filter(B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( filterlim(B,A,F2,F4,G5)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_np(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),G5) ) ) ).
% eventually_compose_filterlim
tff(fact_3673_False__imp__not__eventually,axiom,
! [A: $tType,P: fun(A,$o),Net: filter(A)] :
( ! [X3: A] : ~ aa(A,$o,P,X3)
=> ( ( Net != bot_bot(filter(A)) )
=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),Net) ) ) ).
% False_imp_not_eventually
tff(fact_3674_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_aj($o,fun(A,$o),(P))),F4)
<=> ( (P)
| ( F4 = bot_bot(filter(A)) ) ) ) ).
% eventually_const_iff
tff(fact_3675_trivial__limit__def,axiom,
! [A: $tType,F4: filter(A)] :
( ( F4 = bot_bot(filter(A)) )
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_cy(A,$o)),F4) ) ).
% trivial_limit_def
tff(fact_3676_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( linorder(A)
& no_bot(A) )
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_nq(A,fun(A,$o),C2)),at_bot(A)) ) ).
% eventually_at_bot_not_equal
tff(fact_3677_Func__is__emp,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( bNF_Wellorder_Func(A,B,A3,B5) = bot_bot(set(fun(A,B))) )
<=> ( ( A3 != bot_bot(set(A)) )
& ( B5 = bot_bot(set(B)) ) ) ) ).
% Func_is_emp
tff(fact_3678_Func__non__emp,axiom,
! [A: $tType,B: $tType,B5: set(A),A3: set(B)] :
( ( B5 != bot_bot(set(A)) )
=> ( bNF_Wellorder_Func(B,A,A3,B5) != bot_bot(set(fun(B,A))) ) ) ).
% Func_non_emp
tff(fact_3679_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_bot(A))
<=> ? [N7: A] :
! [N4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N4),N7)
=> aa(A,$o,P,N4) ) ) ) ).
% eventually_at_bot_linorder
tff(fact_3680_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( linorder(A)
& no_bot(A) )
=> ! [P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_bot(A))
<=> ? [N7: A] :
! [N4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),N7)
=> aa(A,$o,P,N4) ) ) ) ).
% eventually_at_bot_dense
tff(fact_3681_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A3))
<=> aa(set(A),$o,P,A3) ) ) ).
% eventually_finite_subsets_at_top_finite
tff(fact_3682_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(6)
tff(fact_3683_euclidean__size__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( euclid6346220572633701492n_size(A,A4) = euclid6346220572633701492n_size(A,one_one(A)) ) ) ) ).
% euclidean_size_unit
tff(fact_3684_euclidean__size__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A4: A,B3: A] : euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),euclid6346220572633701492n_size(A,A4)),euclid6346220572633701492n_size(A,B3)) ) ).
% euclidean_size_mult
tff(fact_3685_eventually__le__at__bot,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_nr(A,fun(A,$o)),C2)),at_bot(A)) ) ).
% eventually_le_at_bot
tff(fact_3686_eventually__gt__at__bot,axiom,
! [A: $tType] :
( unboun7993243217541854897norder(A)
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ns(A,fun(A,$o),C2)),at_bot(A)) ) ).
% eventually_gt_at_bot
tff(fact_3687_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(B),G5: filter(A),F7: filter(B),G6: filter(A),F8: fun(A,B)] :
( filterlim(A,B,F2,F4,G5)
=> ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),F7)
=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G6),G5)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_nt(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F8)),G6)
=> filterlim(A,B,F8,F7,G6) ) ) ) ) ).
% filterlim_mono_eventually
tff(fact_3688_filterlim__principal,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),S: set(B),F4: filter(A)] :
( filterlim(A,B,F2,aa(set(B),filter(B),principal(B),S),F4)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aTP_Lamp_cb(fun(A,B),fun(set(B),fun(A,$o)),F2),S)),F4) ) ).
% filterlim_principal
tff(fact_3689_le__principal,axiom,
! [A: $tType,F4: filter(A),A3: set(A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),aa(set(A),filter(A),principal(A),A3))
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),F4) ) ).
% le_principal
tff(fact_3690_eventually__INF1,axiom,
! [B: $tType,A: $tType,I2: A,I: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
( member(A,I2,I)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(A,filter(B),F4,I2))
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I))) ) ) ).
% eventually_INF1
tff(fact_3691_eventually__inf__principal,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),S2: set(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),S2)))
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_nu(fun(A,$o),fun(set(A),fun(A,$o)),P),S2)),F4) ) ).
% eventually_inf_principal
tff(fact_3692_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A3))
<=> ? [X9: set(A)] :
( aa(set(A),$o,finite_finite2(A),X9)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),A3)
& ! [Y8: set(A)] :
( ( aa(set(A),$o,finite_finite2(A),Y8)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),Y8)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Y8),A3) )
=> aa(set(A),$o,P,Y8) ) ) ) ).
% eventually_finite_subsets_at_top
tff(fact_3693_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A4,B3)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A4) ) ) ) ).
% Ioc_disjoint
tff(fact_3694_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(8)
tff(fact_3695_eventually__Inf__base,axiom,
! [A: $tType,B5: set(filter(A)),P: fun(A,$o)] :
( ( B5 != bot_bot(set(filter(A))) )
=> ( ! [F5: filter(A)] :
( member(filter(A),F5,B5)
=> ! [G7: filter(A)] :
( member(filter(A),G7,B5)
=> ? [X4: filter(A)] :
( member(filter(A),X4,B5)
& aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X4),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G7)) ) ) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
<=> ? [X2: filter(A)] :
( member(filter(A),X2,B5)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),X2) ) ) ) ) ).
% eventually_Inf_base
tff(fact_3696_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(3)
tff(fact_3697_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
<=> ( ( euclid6346220572633701492n_size(A,A4) = euclid6346220572633701492n_size(A,one_one(A)) )
& ( A4 != zero_zero(A) ) ) ) ) ).
% unit_iff_euclidean_size
tff(fact_3698_size__mult__mono,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),euclid6346220572633701492n_size(A,A4)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))) ) ) ).
% size_mult_mono
tff(fact_3699_size__mult__mono_H,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),euclid6346220572633701492n_size(A,A4)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4))) ) ) ).
% size_mult_mono'
tff(fact_3700_euclidean__size__times__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = euclid6346220572633701492n_size(A,B3) ) ) ) ).
% euclidean_size_times_unit
tff(fact_3701_eventually__INF__finite,axiom,
! [A: $tType,B: $tType,A3: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A3)))
<=> ? [Q4: fun(A,fun(B,$o))] :
( ! [X2: A] :
( member(A,X2,A3)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),Q4,X2)),aa(A,filter(B),F4,X2)) )
& ! [Y3: B] :
( ! [X2: A] :
( member(A,X2,A3)
=> aa(B,$o,aa(A,fun(B,$o),Q4,X2),Y3) )
=> aa(B,$o,P,Y3) ) ) ) ) ).
% eventually_INF_finite
tff(fact_3702_filterlim__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F2: fun(A,set(B)),A3: set(B),F4: filter(A)] :
( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,A3),F4)
<=> ! [X9: set(B)] :
( ( aa(set(B),$o,finite_finite2(B),X9)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),A3) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_nv(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F2),A3),X9)),F4) ) ) ).
% filterlim_finite_subsets_at_top
tff(fact_3703_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(2)
tff(fact_3704_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(5)
tff(fact_3705_filterlim__at__bot,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),F4: filter(A)] :
( filterlim(A,B,F2,at_bot(B),F4)
<=> ! [Z8: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_nw(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ).
% filterlim_at_bot
tff(fact_3706_filterlim__at__bot__le,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),F4: filter(A),C2: B] :
( filterlim(A,B,F2,at_bot(B),F4)
<=> ! [Z8: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z8),C2)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_nw(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ) ).
% filterlim_at_bot_le
tff(fact_3707_filterlim__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( dense_linorder(B)
& no_bot(B) )
=> ! [F2: fun(A,B),F4: filter(A)] :
( filterlim(A,B,F2,at_bot(B),F4)
<=> ! [Z8: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_nx(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ).
% filterlim_at_bot_dense
tff(fact_3708_prod_Ohead,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,N2))) ) ) ) ).
% prod.head
tff(fact_3709_eventually__INF,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),F4: fun(B,filter(A)),B5: set(B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B5)))
<=> ? [X9: set(B)] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B5)
& aa(set(B),$o,finite_finite2(B),X9)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X9))) ) ) ).
% eventually_INF
tff(fact_3710_filterlim__at__bot__lt,axiom,
! [B: $tType,A: $tType] :
( unboun7993243217541854897norder(B)
=> ! [F2: fun(A,B),F4: filter(A),C2: B] :
( filterlim(A,B,F2,at_bot(B),F4)
<=> ! [Z8: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z8),C2)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ny(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ) ).
% filterlim_at_bot_lt
tff(fact_3711_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: A,B3: A] : set_or3652927894154168847AtMost(A,A4,B3) = minus_minus(set(A),set_or1337092689740270186AtMost(A,A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_3712_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_3713_eventually__Inf,axiom,
! [A: $tType,P: fun(A,$o),B5: set(filter(A))] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
<=> ? [X9: set(filter(A))] :
( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X9),B5)
& aa(set(filter(A)),$o,finite_finite2(filter(A)),X9)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9)) ) ) ).
% eventually_Inf
tff(fact_3714_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_3715_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A13: set(B),B14: set(A),F22: fun(C,D),B23: set(C),A23: set(D)] :
( ( aa(set(B),set(A),image2(B,A,F1),A13) = B14 )
=> ( inj_on(C,D,F22,B23)
=> ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B23)),A23)
=> ( ( ( B23 = bot_bot(set(C)) )
=> ( A23 = bot_bot(set(D)) ) )
=> ( bNF_Wellorder_Func(C,A,B23,B14) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A23,A13)) ) ) ) ) ) ).
% Func_map_surj
tff(fact_3716_interval__cases,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A)] :
( ! [A5: A,B2: A,X3: A] :
( member(A,A5,S)
=> ( member(A,B2,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B2)
=> member(A,X3,S) ) ) ) )
=> ? [A5: A,B2: A] :
( ( S = bot_bot(set(A)) )
| ( S = top_top(set(A)) )
| ( S = aa(A,set(A),set_ord_lessThan(A),B2) )
| ( S = aa(A,set(A),set_ord_atMost(A),B2) )
| ( S = aa(A,set(A),set_ord_greaterThan(A),A5) )
| ( S = set_ord_atLeast(A,A5) )
| ( S = set_or5935395276787703475ssThan(A,A5,B2) )
| ( S = set_or3652927894154168847AtMost(A,A5,B2) )
| ( S = set_or7035219750837199246ssThan(A,A5,B2) )
| ( S = set_or1337092689740270186AtMost(A,A5,B2) ) ) ) ) ).
% interval_cases
tff(fact_3717_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y4: set(B),X5: set(A),F4: filter(B),F2: fun(A,C)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y4)),X5)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_nz(set(B),fun(B,$o),Y4)),F4)
=> ( aa(filter(A),filter(C),aa(fun(A,C),fun(filter(A),filter(C)),map_filter_on(A,C,X5),F2),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,Y4),G),F4)) = aa(filter(B),filter(C),aa(fun(B,C),fun(filter(B),filter(C)),map_filter_on(B,C,Y4),aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G)),F4) ) ) ) ).
% map_filter_on_comp
tff(fact_3718_divmod__cases,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B3: A,A4: A] :
( ( ( B3 != zero_zero(A) )
=> ( ( modulo_modulo(A,A4,B3) = zero_zero(A) )
=> ( A4 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A4,B3)),B3) ) ) )
=> ( ( ( B3 != zero_zero(A) )
=> ! [Q7: A,R: A] :
( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B3))
=> ( ( R != zero_zero(A) )
=> ( ( divide_divide(A,A4,B3) = Q7 )
=> ( ( modulo_modulo(A,A4,B3) = R )
=> ( A4 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q7),B3)),R) ) ) ) ) ) ) )
=> ( B3 = zero_zero(A) ) ) ) ) ).
% divmod_cases
tff(fact_3719_division__segment__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).
% division_segment_1
tff(fact_3720_atLeast__empty__triv,axiom,
! [A: $tType] : set_ord_atLeast(set(A),bot_bot(set(A))) = top_top(set(set(A))) ).
% atLeast_empty_triv
tff(fact_3721_division__segment__numeral,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [K: num] : euclid7384307370059645450egment(A,aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ).
% division_segment_numeral
tff(fact_3722_division__segment__of__nat,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N2: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),N2)) = one_one(A) ) ).
% division_segment_of_nat
tff(fact_3723_division__segment__euclidean__size,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A4)),aa(nat,A,semiring_1_of_nat(A),euclid6346220572633701492n_size(A,A4))) = A4 ) ).
% division_segment_euclidean_size
tff(fact_3724_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A] : bot_bot(set(A)) != set_ord_atLeast(A,L) ) ).
% not_empty_eq_Ici_eq_empty
tff(fact_3725_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] :
( ( set_ord_atLeast(A,X) = top_top(set(A)) )
<=> ( X = bot_bot(A) ) ) ) ).
% atLeast_eq_UNIV_iff
tff(fact_3726_division__segment__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A4)),euclid7384307370059645450egment(A,B3)) ) ) ) ) ).
% division_segment_mult
tff(fact_3727_is__unit__division__segment,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A4: A] : dvd_dvd(A,euclid7384307370059645450egment(A,A4),one_one(A)) ) ).
% is_unit_division_segment
tff(fact_3728_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(8)
tff(fact_3729_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X5: set(A),F4: filter(A),P: fun(B,$o),F2: fun(A,B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),X5)),F4)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),map_filter_on(A,B,X5),F2),F4))
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_oa(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X5),P),F2)),F4) ) ) ).
% eventually_map_filter_on
tff(fact_3730_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(6)
tff(fact_3731_atMost__Int__atLeast,axiom,
! [A: $tType] :
( order(A)
=> ! [N2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),N2)),set_ord_atLeast(A,N2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),N2),bot_bot(set(A))) ) ).
% atMost_Int_atLeast
tff(fact_3732_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = set_ord_atLeast(A,L) ) ).
% ivl_disj_un_singleton(1)
tff(fact_3733_atLeast__Suc,axiom,
! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = minus_minus(set(nat),set_ord_atLeast(nat,K),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ).
% atLeast_Suc
tff(fact_3734_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B3: A,R2: A,Q3: A] :
( ( B3 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B3)),R2),B3) = Q3 ) ) ) ) ) ).
% div_bounded
tff(fact_3735_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B3: A,R2: A,Q3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B3)),R2) = A4 )
=> ( divide_divide(A,A4,B3) = Q3 ) ) ) ) ) ) ).
% div_eqI
tff(fact_3736_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B3: A,R2: A,Q3: A,A4: A] :
( ( B3 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B3)),R2) = A4 )
=> ( modulo_modulo(A,A4,B3) = R2 ) ) ) ) ) ) ).
% mod_eqI
tff(fact_3737_at__top__def,axiom,
! [A: $tType] :
( order(A)
=> ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_ob(A,filter(A))),top_top(set(A)))) ) ) ).
% at_top_def
tff(fact_3738_at__top__sub,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_oc(A,filter(A))),set_ord_atLeast(A,C2))) ) ).
% at_top_sub
tff(fact_3739_filterlim__at__top__gt,axiom,
! [B: $tType,A: $tType] :
( unboun7993243217541854897norder(B)
=> ! [F2: fun(A,B),F4: filter(A),C2: B] :
( filterlim(A,B,F2,at_top(B),F4)
<=> ! [Z8: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z8)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_od(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ) ).
% filterlim_at_top_gt
tff(fact_3740_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat] :
aa(num,A,numeral_numeral(A),num_of_nat(N2)) = $ite(N2 = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).
% numeral_num_of_nat_unfold
tff(fact_3741_eventually__sequentially__Suc,axiom,
! [P: fun(nat,$o)] :
( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_oe(fun(nat,$o),fun(nat,$o),P)),at_top(nat))
<=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).
% eventually_sequentially_Suc
tff(fact_3742_eventually__sequentially__seg,axiom,
! [P: fun(nat,$o),K: nat] :
( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),aTP_Lamp_of(fun(nat,$o),fun(nat,fun(nat,$o)),P),K)),at_top(nat))
<=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).
% eventually_sequentially_seg
tff(fact_3743_filterlim__Suc,axiom,
filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).
% filterlim_Suc
tff(fact_3744_eventually__sequentiallyI,axiom,
! [C2: nat,P: fun(nat,$o)] :
( ! [X3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X3)
=> aa(nat,$o,P,X3) )
=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).
% eventually_sequentiallyI
tff(fact_3745_eventually__sequentially,axiom,
! [P: fun(nat,$o)] :
( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat))
<=> ? [N7: nat] :
! [N4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N4)
=> aa(nat,$o,P,N4) ) ) ).
% eventually_sequentially
tff(fact_3746_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: fun(nat,A),F4: filter(A)] :
( filterlim(nat,A,aTP_Lamp_og(fun(nat,A),fun(nat,A),F2),F4,at_top(nat))
<=> filterlim(nat,A,F2,F4,at_top(nat)) ) ).
% filterlim_sequentially_Suc
tff(fact_3747_le__sequentially,axiom,
! [F4: filter(nat)] :
( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
<=> ! [N7: nat] : aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N7)),F4) ) ).
% le_sequentially
tff(fact_3748_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ( at_top(A) != bot_bot(filter(A)) ) ) ).
% trivial_limit_at_top_linorder
tff(fact_3749_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_oh(A,fun(A,$o),C2)),at_top(A)) ) ).
% eventually_at_top_not_equal
tff(fact_3750_eventually__at__top__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
<=> ? [N7: A] :
! [N4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N7),N4)
=> aa(A,$o,P,N4) ) ) ) ).
% eventually_at_top_linorder
tff(fact_3751_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,P: fun(A,$o)] :
( ! [X3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X3)
=> aa(A,$o,P,X3) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A)) ) ) ).
% eventually_at_top_linorderI
tff(fact_3752_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
<=> ? [N7: A] :
! [N4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),N4)
=> aa(A,$o,P,N4) ) ) ) ).
% eventually_at_top_dense
tff(fact_3753_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_3754_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_3755_eventually__ge__at__top,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less_eq(A),C2)),at_top(A)) ) ).
% eventually_ge_at_top
tff(fact_3756_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [C2: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less(A),C2)),at_top(A)) ) ).
% eventually_gt_at_top
tff(fact_3757_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,Q,X3)
=> ( aa(A,$o,Q,Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) ) ) )
=> ( ! [X3: B] :
( aa(B,$o,P,X3)
=> ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
=> ( ! [X3: B] :
( aa(B,$o,P,X3)
=> aa(A,$o,Q,aa(B,A,G,X3)) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),at_top(A))
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),at_top(B))
=> filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
tff(fact_3758_filterlim__at__top__mono,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
( filterlim(A,B,F2,at_top(B),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_oi(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),F4)
=> filterlim(A,B,G,at_top(B),F4) ) ) ) ).
% filterlim_at_top_mono
tff(fact_3759_filterlim__at__top__ge,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),F4: filter(A),C2: B] :
( filterlim(A,B,F2,at_top(B),F4)
<=> ! [Z8: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z8)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_oj(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ) ).
% filterlim_at_top_ge
tff(fact_3760_filterlim__at__top,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),F4: filter(A)] :
( filterlim(A,B,F2,at_top(B),F4)
<=> ! [Z8: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_oj(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ).
% filterlim_at_top
tff(fact_3761_filterlim__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( unboun7993243217541854897norder(B)
=> ! [F2: fun(A,B),F4: filter(A)] :
( filterlim(A,B,F2,at_top(B),F4)
<=> ! [Z8: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ok(fun(A,B),fun(B,fun(A,$o)),F2),Z8)),F4) ) ) ).
% filterlim_at_top_dense
tff(fact_3762_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J: set(A),I: set(B),F2: fun(D,C),F4: fun(B,filter(D)),G5: fun(A,filter(C))] :
( ! [M3: A] :
( member(A,M3,J)
=> ? [X4: B] :
( member(B,X4,I)
& aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),aa(filter(D),filter(C),aa(fun(D,C),fun(filter(D),filter(C)),filtermap(D,C),F2),aa(B,filter(D),F4,X4))),aa(A,filter(C),G5,M3)) ) )
=> filterlim(D,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),G5),J)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),F4),I))) ) ).
% filterlim_INF_INF
tff(fact_3763_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A3: set(A)] :
( ? [A7: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),bot_bot(set(A))))
<=> uniq(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)) ) ).
% subset_singleton_iff_Uniq
tff(fact_3764_Gcd__fin__def,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).
% Gcd_fin_def
tff(fact_3765_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F2: fun(A,B)] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ? [X_1: A] : lattic501386751177426532rg_min(A,B,F2,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S),X_1) ) ) ) ).
% ex_is_arg_min_if_finite
tff(fact_3766_trivial__limit__sequentially,axiom,
at_top(nat) != bot_bot(filter(nat)) ).
% trivial_limit_sequentially
tff(fact_3767_filtermap__id_H,axiom,
! [A: $tType,X4: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_ac(A,A)),X4) = X4 ).
% filtermap_id'
tff(fact_3768_filtermap__id,axiom,
! [A: $tType] : aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),id(A)) = id(filter(A)) ).
% filtermap_id
tff(fact_3769_filtermap__bot,axiom,
! [B: $tType,A: $tType,F2: fun(B,A)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),bot_bot(filter(B))) = bot_bot(filter(A)) ).
% filtermap_bot
tff(fact_3770_filtermap__principal,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(B),filter(B),principal(B),A3)) = aa(set(A),filter(A),principal(A),aa(set(B),set(A),image2(B,A,F2),A3)) ).
% filtermap_principal
tff(fact_3771_eventually__False__sequentially,axiom,
~ aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_ol(nat,$o)),at_top(nat)) ).
% eventually_False_sequentially
tff(fact_3772_filtermap__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),F4: filter(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F4)) = aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),aa(fun(C,B),fun(C,A),aTP_Lamp_cl(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),F4) ).
% filtermap_filtermap
tff(fact_3773_filtermap__ident,axiom,
! [A: $tType,F4: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_ac(A,A)),F4) = F4 ).
% filtermap_ident
tff(fact_3774_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F2: fun(nat,A)] : aa(filter(nat),filter(A),aa(fun(nat,A),fun(filter(nat),filter(A)),filtermap(nat,A),F2),at_top(nat)) != bot_bot(filter(A)) ).
% filtermap_sequentually_ne_bot
tff(fact_3775_filtermap__mono,axiom,
! [B: $tType,A: $tType,F4: filter(A),F7: filter(A),F2: fun(A,B)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F4)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F7)) ) ).
% filtermap_mono
tff(fact_3776_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),F4: filter(B)] :
( ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F4) = bot_bot(filter(A)) )
<=> ( F4 = bot_bot(filter(B)) ) ) ).
% filtermap_bot_iff
tff(fact_3777_filtermap__sup,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),F12: filter(B),F23: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F23)) ).
% filtermap_sup
tff(fact_3778_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),F4: filter(B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F4))
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_np(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),F4) ) ).
% eventually_filtermap
tff(fact_3779_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F12: filter(B),G: fun(C,A),F23: filter(C)] :
( filterlim(A,B,F2,F12,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),G),F23))
<=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_lw(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),F12,F23) ) ).
% filterlim_filtermap
tff(fact_3780_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(A),G5: filter(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( ( aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F4) = aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G5) )
<=> ( F4 = G5 ) ) ) ).
% filtermap_eq_strong
tff(fact_3781_filterlim__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
( filterlim(A,B,F2,F23,F12)
<=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F12)),F23) ) ).
% filterlim_def
tff(fact_3782_map__filter__on__UNIV,axiom,
! [B: $tType,A: $tType] : map_filter_on(A,B,top_top(set(A))) = filtermap(A,B) ).
% map_filter_on_UNIV
tff(fact_3783_filtermap__inf,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),F12: filter(B),F23: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F23))),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F12)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F23))) ).
% filtermap_inf
tff(fact_3784_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F4: filter(B),G5: filter(A),F2: fun(B,A)] :
( filterlim(A,B,G,F4,G5)
=> ( filterlim(B,A,F2,G5,F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,A),fun(A,$o),aTP_Lamp_om(fun(A,B),fun(fun(B,A),fun(A,$o)),G),F2)),G5)
=> ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F4) = G5 ) ) ) ) ).
% filtermap_fun_inverse
tff(fact_3785_filtermap__SUP,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F4: fun(C,filter(B)),B5: set(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_on(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B5)) ).
% filtermap_SUP
tff(fact_3786_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(A),G5: filter(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F4)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G5))
<=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),G5) ) ) ).
% filtermap_mono_strong
tff(fact_3787_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
( inj_on(A,B,F2,A3)
=> ( aa(filter(set(A)),filter(set(B)),aa(fun(set(A),set(B)),fun(filter(set(A)),filter(set(B))),filtermap(set(A),set(B)),image2(A,B,F2)),finite5375528669736107172at_top(A,A3)) = finite5375528669736107172at_top(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).
% filtermap_image_finite_subsets_at_top
tff(fact_3788_filtermap__INF,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F4: fun(C,filter(B)),B5: set(C)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_on(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B5))) ).
% filtermap_INF
tff(fact_3789_pairwise__disjnt__iff,axiom,
! [A: $tType,A14: set(set(A))] :
( pairwise(set(A),disjnt(A),A14)
<=> ! [X2: A] : uniq(set(A),aa(A,fun(set(A),$o),aTP_Lamp_oo(set(set(A)),fun(A,fun(set(A),$o)),A14),X2)) ) ).
% pairwise_disjnt_iff
tff(fact_3790_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: A,A3: set(A)] :
( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
=> ( member(A,A4,A3)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ) ).
% bounded_quasi_semilattice_set.remove
tff(fact_3791_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: A,A3: set(A)] :
( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
tff(fact_3792_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F4: filter(A),X: B] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),aa(set(B),filter(B),principal(B),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = aa(filter(A),filter(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(filter(A),filter(product_prod(A,B))),filtermap(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))),X)),F4) ).
% prod_filter_principal_singleton2
tff(fact_3793_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_kv(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat_def
tff(fact_3794_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F4: filter(A),G5: filter(B),H7: filter(C)] : aa(filter(C),filter(product_prod(product_prod(A,B),C)),aa(filter(product_prod(A,B)),fun(filter(C),filter(product_prod(product_prod(A,B),C))),prod_filter(product_prod(A,B),C),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5)),H7) = aa(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C)),aa(fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),fun(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C))),filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_oq(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))))),aa(filter(product_prod(B,C)),filter(product_prod(A,product_prod(B,C))),aa(filter(A),fun(filter(product_prod(B,C)),filter(product_prod(A,product_prod(B,C)))),prod_filter(A,product_prod(B,C)),F4),aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),G5),H7))) ).
% prod_filter_assoc
tff(fact_3795_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F4: filter(A),F7: filter(A),G5: filter(B),G6: filter(B)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
=> ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G5),G6)
=> aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F7),G6)) ) ) ).
% prod_filter_mono
tff(fact_3796_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A3: filter(A),B5: filter(B)] :
( ( aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),A3),B5) = bot_bot(filter(product_prod(A,B))) )
<=> ( ( A3 = bot_bot(filter(A)) )
| ( B5 = bot_bot(filter(B)) ) ) ) ).
% prod_filter_eq_bot
tff(fact_3797_prod__filter__commute,axiom,
! [B: $tType,A: $tType,F4: filter(A),G5: filter(B)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5) = aa(filter(product_prod(B,A)),filter(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(filter(product_prod(B,A)),filter(product_prod(A,B))),filtermap(product_prod(B,A),product_prod(A,B)),product_swap(B,A)),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),G5),F4)) ).
% prod_filter_commute
tff(fact_3798_eventually__prod__same,axiom,
! [A: $tType,P: fun(product_prod(A,A),$o),F4: filter(A)] :
( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),P),aa(filter(A),filter(product_prod(A,A)),aa(filter(A),fun(filter(A),filter(product_prod(A,A))),prod_filter(A,A),F4),F4))
<=> ? [Q4: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q4),F4)
& ! [X2: A,Y3: A] :
( aa(A,$o,Q4,X2)
=> ( aa(A,$o,Q4,Y3)
=> aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3)) ) ) ) ) ).
% eventually_prod_same
tff(fact_3799_eventually__prod__filter,axiom,
! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F4: filter(A),G5: filter(B)] :
( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),P),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5))
<=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pf),F4)
& aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pg),G5)
& ! [X2: A,Y3: B] :
( aa(A,$o,Pf,X2)
=> ( aa(B,$o,Pg,Y3)
=> aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3)) ) ) ) ) ).
% eventually_prod_filter
tff(fact_3800_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter(A),B5: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),A3),B5))),A3) ).
% filtermap_fst_prod_filter
tff(fact_3801_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter(B),B5: filter(A)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(B,A)),filter(A),aa(fun(product_prod(B,A),A),fun(filter(product_prod(B,A)),filter(A)),filtermap(product_prod(B,A),A),product_snd(B,A)),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),A3),B5))),B5) ).
% filtermap_snd_prod_filter
tff(fact_3802_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter(A),B5: filter(B),C6: filter(A),D4: filter(B)] :
( ( A3 != bot_bot(filter(A)) )
=> ( ( B5 != bot_bot(filter(B)) )
=> ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),A3),B5)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),C6),D4))
<=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),A3),C6)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B5),D4) ) ) ) ) ).
% prod_filter_mono_iff
tff(fact_3803_prod__filtermap1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),F4: filter(C),G5: filter(B)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),F2),F4)),G5) = aa(filter(product_prod(C,B)),filter(product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(filter(product_prod(C,B)),filter(product_prod(A,B))),filtermap(product_prod(C,B),product_prod(A,B)),product_apfst(C,A,B,F2)),aa(filter(B),filter(product_prod(C,B)),aa(filter(C),fun(filter(B),filter(product_prod(C,B))),prod_filter(C,B),F4),G5)) ).
% prod_filtermap1
tff(fact_3804_prod__filtermap2,axiom,
! [B: $tType,A: $tType,C: $tType,F4: filter(A),G: fun(C,B),G5: filter(C)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),G5)) = aa(filter(product_prod(A,C)),filter(product_prod(A,B)),aa(fun(product_prod(A,C),product_prod(A,B)),fun(filter(product_prod(A,C)),filter(product_prod(A,B))),filtermap(product_prod(A,C),product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G)),aa(filter(C),filter(product_prod(A,C)),aa(filter(A),fun(filter(C),filter(product_prod(A,C))),prod_filter(A,C),F4),G5)) ).
% prod_filtermap2
tff(fact_3805_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G5: filter(B),F4: filter(A),G: fun(A,C),H7: filter(C)] :
( filterlim(A,B,F2,G5,F4)
=> ( filterlim(A,C,G,H7,F4)
=> filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_or(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),G5),H7),F4) ) ) ).
% filterlim_Pair
tff(fact_3806_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(C),filter(product_prod(A,B)),aa(fun(C,product_prod(A,B)),fun(filter(C),filter(product_prod(A,B))),filtermap(C,product_prod(A,B)),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gn(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),F4)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),F2),F4)),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F4))) ).
% filtermap_Pair
tff(fact_3807_principal__prod__principal,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(set(A),filter(A),principal(A),A3)),aa(set(B),filter(B),principal(B),B5)) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),principal(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) ).
% principal_prod_principal
tff(fact_3808_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),bot_bot(set(A))) = Top ) ) ).
% bounded_quasi_semilattice_set.empty
tff(fact_3809_bounded__quasi__semilattice__set_Oinsert,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: A,A3: set(A)] :
( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A3)) ) ) ).
% bounded_quasi_semilattice_set.insert
tff(fact_3810_prod__filter__def,axiom,
! [A: $tType,B: $tType,F4: filter(A),G5: filter(B)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,$o),fun(B,$o))),set(filter(product_prod(A,B))),image2(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B)),aa(fun(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),fun(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B))),product_case_prod(fun(A,$o),fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_os(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,$o),fun(B,$o)),$o),set(product_prod(fun(A,$o),fun(B,$o))),collect(product_prod(fun(A,$o),fun(B,$o))),aa(fun(fun(A,$o),fun(fun(B,$o),$o)),fun(product_prod(fun(A,$o),fun(B,$o)),$o),product_case_prod(fun(A,$o),fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_ot(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),F4),G5))))) ).
% prod_filter_def
tff(fact_3811_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F4: filter(product_prod(A,B)),A3: filter(A),B5: filter(B)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),F4)),A3)
=> ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),filtermap(product_prod(A,B),B),product_snd(A,B)),F4)),B5)
=> aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),F4),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),A3),B5)) ) ) ).
% le_prod_filterI
tff(fact_3812_eventually__prod__sequentially,axiom,
! [P: fun(product_prod(nat,nat),$o)] :
( aa(filter(product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),$o),fun(filter(product_prod(nat,nat)),$o),eventually(product_prod(nat,nat)),P),aa(filter(nat),filter(product_prod(nat,nat)),aa(filter(nat),fun(filter(nat),filter(product_prod(nat,nat))),prod_filter(nat,nat),at_top(nat)),at_top(nat)))
<=> ? [N7: nat] :
! [M4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M4)
=> ! [N4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N4)
=> aa(product_prod(nat,nat),$o,P,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_3813_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),Q: fun(B,$o),G5: filter(B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q),G5)
=> aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_ou(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),P),Q)),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),F4),G5)) ) ) ).
% eventually_prodI
tff(fact_3814_eventually__prod1,axiom,
! [A: $tType,B: $tType,B5: filter(A),P: fun(B,$o),A3: filter(B)] :
( ( B5 != bot_bot(filter(A)) )
=> ( aa(filter(product_prod(B,A)),$o,aa(fun(product_prod(B,A),$o),fun(filter(product_prod(B,A)),$o),eventually(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_ov(fun(B,$o),fun(B,fun(A,$o)),P))),aa(filter(A),filter(product_prod(B,A)),aa(filter(B),fun(filter(A),filter(product_prod(B,A))),prod_filter(B,A),A3),B5))
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),A3) ) ) ).
% eventually_prod1
tff(fact_3815_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter(A),P: fun(B,$o),B5: filter(B)] :
( ( A3 != bot_bot(filter(A)) )
=> ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ow(fun(B,$o),fun(A,fun(B,$o)),P))),aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),A3),B5))
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),B5) ) ) ).
% eventually_prod2
tff(fact_3816_prod__filter__INF,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,I: set(A),J: set(B),A3: fun(A,filter(C)),B5: fun(B,filter(D))] :
( ( I != bot_bot(set(A)) )
=> ( ( J != bot_bot(set(B)) )
=> ( aa(filter(D),filter(product_prod(C,D)),aa(filter(C),fun(filter(D),filter(product_prod(C,D))),prod_filter(C,D),aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A3),I))),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B5),J))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_oy(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J),A3),B5)),I)) ) ) ) ).
% prod_filter_INF
tff(fact_3817_prod__filter__INF1,axiom,
! [C: $tType,B: $tType,A: $tType,I: set(A),A3: fun(A,filter(B)),B5: filter(C)] :
( ( I != bot_bot(set(A)) )
=> ( aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A3),I))),B5) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_oz(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A3),B5)),I)) ) ) ).
% prod_filter_INF1
tff(fact_3818_prod__filter__INF2,axiom,
! [C: $tType,B: $tType,A: $tType,J: set(A),A3: filter(B),B5: fun(A,filter(C))] :
( ( J != bot_bot(set(A)) )
=> ( aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),A3),aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B5),J))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_pa(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A3),B5)),J)) ) ) ).
% prod_filter_INF2
tff(fact_3819_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_kw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat_def
tff(fact_3820_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F4: filter(B)] : aa(filter(B),filter(product_prod(A,B)),aa(filter(A),fun(filter(B),filter(product_prod(A,B))),prod_filter(A,B),aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),F4) = aa(filter(B),filter(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(filter(B),filter(product_prod(A,B))),filtermap(B,product_prod(A,B)),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),F4) ).
% prod_filter_principal_singleton
tff(fact_3821_relImage__def,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,B)),F2: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R3,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_pb(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R3),F2)) ).
% relImage_def
tff(fact_3822_add__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),fract(A4,B3)),fract(C2,D3)) = fract(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A4),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3)) ) ) ) ).
% add_rat
tff(fact_3823_mono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( condit1219197933456340205attice(B)
& condit1219197933456340205attice(A) )
=> ! [F2: fun(A,B),A3: fun(C,A),I: set(C)] :
( order_mono(A,B,F2)
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A3),I))
=> ( ( I != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I))) ) ) ) ) ).
% mono_cINF
tff(fact_3824_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F2: fun(A,B),A3: set(A)] :
( order_mono(A,B,F2)
=> ( condit1013018076250108175_below(A,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).
% mono_cInf
tff(fact_3825_divide__rat,axiom,
! [A4: int,B3: int,C2: int,D3: int] : divide_divide(rat,fract(A4,B3),fract(C2,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A4),D3),aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) ).
% divide_rat
tff(fact_3826_sgn__rat,axiom,
! [A4: int,B3: int] : aa(rat,rat,sgn_sgn(rat),fract(A4,B3)) = aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A4)),aa(int,int,sgn_sgn(int),B3))) ).
% sgn_rat
tff(fact_3827_mult__rat,axiom,
! [A4: int,B3: int,C2: int,D3: int] : aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),fract(A4,B3)),fract(C2,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A4),C2),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3)) ).
% mult_rat
tff(fact_3828_le__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A4,B3)),fract(C2,D3))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A4),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3))) ) ) ) ).
% le_rat
tff(fact_3829_less__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A4,B3)),fract(C2,D3))
<=> aa(int,$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),A4),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3))) ) ) ) ).
% less_rat
tff(fact_3830_diff__rat,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( minus_minus(rat,fract(A4,B3),fract(C2,D3)) = fract(minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),A4),D3),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D3)) ) ) ) ).
% diff_rat
tff(fact_3831_mono__add,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A4: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A4)) ) ).
% mono_add
tff(fact_3832_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).
% mono_strict_invE
tff(fact_3833_monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) ) ) ) ).
% monoD
tff(fact_3834_monoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) ) ) ) ).
% monoE
tff(fact_3835_monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F2: fun(A,B)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y2)) )
=> order_mono(A,B,F2) ) ) ).
% monoI
tff(fact_3836_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F2: fun(A,B)] :
( order_mono(A,B,F2)
<=> ! [X2: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y3)) ) ) ) ).
% mono_def
tff(fact_3837_mono__Int,axiom,
! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B5: set(A)] :
( order_mono(set(A),set(B),F2)
=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B5))) ) ).
% mono_Int
tff(fact_3838_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mono_invE
tff(fact_3839_mono__Un,axiom,
! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B5: set(A)] :
( order_mono(set(A),set(B),F2)
=> aa(set(B),$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)),sup_sup(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B5))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) ) ).
% mono_Un
tff(fact_3840_eq__rat_I1_J,axiom,
! [B3: int,D3: int,A4: int,C2: int] :
( ( B3 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( ( fract(A4,B3) = fract(C2,D3) )
<=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A4),D3) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3) ) ) ) ) ).
% eq_rat(1)
tff(fact_3841_mult__rat__cancel,axiom,
! [C2: int,A4: int,B3: int] :
( ( C2 != zero_zero(int) )
=> ( fract(aa(int,int,aa(int,fun(int,int),times_times(int),C2),A4),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B3)) = fract(A4,B3) ) ) ).
% mult_rat_cancel
tff(fact_3842_Rings_Omono__mult,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> order_mono(A,A,aa(A,fun(A,A),times_times(A),A4)) ) ) ).
% Rings.mono_mult
tff(fact_3843_mono__times__nat,axiom,
! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2)) ) ).
% mono_times_nat
tff(fact_3844_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F2: fun(A,B),S: set(A)] :
( order_mono(A,B,F2)
=> ( ? [X4: A] :
( member(A,X4,S)
& ! [Xa4: A] :
( member(A,Xa4,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa4) ) )
=> ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_pd(fun(A,B),fun(set(A),fun(B,$o)),F2),S)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_pe(set(A),fun(A,$o),S))) ) ) ) ) ).
% Least_mono
tff(fact_3845_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),A3: set(A)] :
( order_mono(A,B,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,B,F2,lattic643756798349783984er_Max(A,A3)) = lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).
% mono_Max_commute
tff(fact_3846_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),A3: set(A)] :
( order_mono(A,B,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,B,F2,lattic643756798350308766er_Min(A,A3)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).
% mono_Min_commute
tff(fact_3847_positive__rat,axiom,
! [A4: int,B3: int] :
( aa(rat,$o,positive,fract(A4,B3))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A4),B3)) ) ).
% positive_rat
tff(fact_3848_finite_Omono,axiom,
! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_ng(fun(set(A),$o),fun(set(A),$o))) ).
% finite.mono
tff(fact_3849_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F2: fun(A,B),A3: set(A)] :
( order_mono(A,B,F2)
=> ( condit941137186595557371_above(A,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).
% mono_cSup
tff(fact_3850_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F2: fun(A,B),A3: fun(C,A),I: set(C)] :
( order_mono(A,B,F2)
=> ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A3),I))
=> ( ( I != bot_bot(set(C)) )
=> aa(B,$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,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I)))) ) ) ) ) ).
% mono_cSUP
tff(fact_3851_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),F4: filter(A),G5: filter(B)] :
( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R3),F4),G5)
=> ~ ! [Z9: filter(product_prod(A,B))] :
( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3)),Z9)
=> ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_fst(A,B)),Z9) = F4 )
=> ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_snd(A,B)),Z9) != G5 ) ) ) ) ).
% rel_filter.cases
tff(fact_3852_rel__filter_Osimps,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),F4: filter(A),G5: filter(B)] :
( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R3),F4),G5)
<=> ? [Z8: filter(product_prod(A,B))] :
( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3)),Z8)
& ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_fst(A,B)),Z8) = F4 )
& ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_snd(A,B)),Z8) = G5 ) ) ) ).
% rel_filter.simps
tff(fact_3853_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),Z5: filter(product_prod(A,B)),F4: filter(A),G5: filter(B)] :
( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3)),Z5)
=> ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_fst(A,B)),Z5) = F4 )
=> ( ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))),product_snd(A,B)),Z5) = G5 )
=> aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R3),F4),G5) ) ) ) ).
% rel_filter.intros
tff(fact_3854_mono__compose,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Q: fun(A,fun(B,C)),F2: fun(D,B)] :
( order_mono(A,fun(B,C),Q)
=> order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_pf(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F2)) ) ) ).
% mono_compose
tff(fact_3855_rel__filter__mono,axiom,
! [B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A3),B5)
=> aa(fun(filter(A),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(B),$o)),fun(fun(filter(A),fun(filter(B),$o)),$o),ord_less_eq(fun(filter(A),fun(filter(B),$o))),rel_filter(A,B,A3)),rel_filter(A,B,B5)) ) ).
% rel_filter_mono
tff(fact_3856_rel__filter__eq,axiom,
! [A: $tType] : rel_filter(A,A,fequal(A)) = fequal(filter(A)) ).
% rel_filter_eq
tff(fact_3857_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A3),bot_bot(filter(A))),bot_bot(filter(B))) ).
% bot_filter_parametric
tff(fact_3858_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A3),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A3),rel_filter(A,B,A3))),sup_sup(filter(A))),sup_sup(filter(B))) ).
% sup_filter_parametric
tff(fact_3859_eventually__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,A3,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A3),fequal($o))),eventually(A)),eventually(B)) ).
% eventually_parametric
tff(fact_3860_filtermap__parametric,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: fun(A,fun(C,$o)),B5: fun(B,fun(D,$o))] : aa(fun(fun(C,D),fun(filter(C),filter(D))),$o,aa(fun(fun(A,B),fun(filter(A),filter(B))),fun(fun(fun(C,D),fun(filter(C),filter(D))),$o),bNF_rel_fun(fun(A,B),fun(C,D),fun(filter(A),filter(B)),fun(filter(C),filter(D)),bNF_rel_fun(A,C,B,D,A3,B5),bNF_rel_fun(filter(A),filter(C),filter(B),filter(D),rel_filter(A,C,A3),rel_filter(B,D,B5))),filtermap(A,B)),filtermap(C,D)) ).
% filtermap_parametric
tff(fact_3861_member__product,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A3: set(A),B5: set(B)] :
( member(product_prod(A,B),X,product_product(A,B,A3,B5))
<=> member(product_prod(A,B),X,product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) ) ).
% member_product
tff(fact_3862_Product__Type_Oproduct__def,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] : product_product(A,B,A3,B5) = product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)) ).
% Product_Type.product_def
tff(fact_3863_cInf__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,S)
=> ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_pg(set(A),fun(A,$o),S))) ) ) ) ) ).
% cInf_cSup
tff(fact_3864_cSup__cInf,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,S)
=> ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ph(set(A),fun(A,$o),S))) ) ) ) ) ).
% cSup_cInf
tff(fact_3865_ball__empty,axiom,
! [A: $tType,P: fun(A,$o),X4: A] :
( member(A,X4,bot_bot(set(A)))
=> aa(A,$o,P,X4) ) ).
% ball_empty
tff(fact_3866_Ball__Collect,axiom,
! [A: $tType,A3: set(A),P: fun(A,$o)] :
( ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,P,X2) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)) ) ).
% Ball_Collect
tff(fact_3867_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),G: fun(A,$o)] :
( ! [X2: A] :
( member(A,X2,aa(set(B),set(A),image2(B,A,F2),A3))
=> aa(A,$o,G,X2) )
<=> ! [X2: B] :
( member(B,X2,A3)
=> aa(B,$o,aa(fun(B,A),fun(B,$o),comp(A,$o,B,G),F2),X2) ) ) ).
% Ball_image_comp
tff(fact_3868_eventually__ball__finite__distrib,axiom,
! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_pi(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P)),Net)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2)),Net) ) ) ) ).
% eventually_ball_finite_distrib
tff(fact_3869_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X3)),Net) )
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_pi(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P)),Net) ) ) ).
% eventually_ball_finite
tff(fact_3870_Field__not__elem,axiom,
! [A: $tType,V: A,R3: set(product_prod(A,A))] :
( ~ member(A,V,aa(set(product_prod(A,A)),set(A),field2(A),R3))
=> ! [X4: product_prod(A,A)] :
( member(product_prod(A,A),X4,R3)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_pj(A,fun(A,fun(A,$o)),V)),X4) ) ) ).
% Field_not_elem
tff(fact_3871_irrefl__distinct,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
<=> ! [X2: product_prod(A,A)] :
( member(product_prod(A,A),X2,R2)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_pk(A,fun(A,$o))),X2) ) ) ).
% irrefl_distinct
tff(fact_3872_Chains__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_pl(set(product_prod(A,A)),fun(set(A),$o),R2)) ).
% Chains_def
tff(fact_3873_trans__join,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
<=> ! [X2: product_prod(A,A)] :
( member(product_prod(A,A),X2,R2)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_pn(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X2) ) ) ).
% trans_join
tff(fact_3874_refl__on__def_H,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,A3,R2)
<=> ( ! [X2: product_prod(A,A)] :
( member(product_prod(A,A),X2,R2)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_po(set(A),fun(A,fun(A,$o)),A3)),X2) )
& ! [X2: A] :
( member(A,X2,A3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2) ) ) ) ).
% refl_on_def'
tff(fact_3875_Inf__filter__def,axiom,
! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),$o),set(filter(A)),collect(filter(A)),aTP_Lamp_pp(set(filter(A)),fun(filter(A),$o),S))) ).
% Inf_filter_def
tff(fact_3876_UnderS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_UnderS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_pq(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A3)) ).
% UnderS_def
tff(fact_3877_Under__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_Under(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_pr(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A3)) ).
% Under_def
tff(fact_3878_Above__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_Above(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ps(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A3)) ).
% Above_def
tff(fact_3879_min__ext__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_pt(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).
% min_ext_def
tff(fact_3880_bex__empty,axiom,
! [A: $tType,P: fun(A,$o)] :
~ ? [X4: A] :
( member(A,X4,bot_bot(set(A)))
& aa(A,$o,P,X4) ) ).
% bex_empty
tff(fact_3881_bex__UNIV,axiom,
! [A: $tType,P: fun(A,$o)] :
( ? [X2: A] :
( member(A,X2,top_top(set(A)))
& aa(A,$o,P,X2) )
<=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).
% bex_UNIV
tff(fact_3882_Image__Collect__case__prod,axiom,
! [A: $tType,B: $tType,P: fun(B,fun(A,$o)),A3: set(B)] : image(B,A,aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P)),A3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_pu(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),P),A3)) ).
% Image_Collect_case_prod
tff(fact_3883_image__def,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(B),set(A),image2(B,A,F2),A3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_pv(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) ).
% image_def
tff(fact_3884_Image__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(B)] : image(B,A,R2,S2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_pw(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S2)) ).
% Image_def
tff(fact_3885_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] : vimage(A,B,F2,aa(set(A),set(B),image2(A,B,F2),A3)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_px(fun(A,B),fun(set(A),fun(A,$o)),F2),A3)) ).
% vimage_image_eq
tff(fact_3886_max__extp_Omax__extI,axiom,
! [A: $tType,X5: set(A),Y4: set(A),R3: fun(A,fun(A,$o))] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( aa(set(A),$o,finite_finite2(A),Y4)
=> ( ( Y4 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,Y4)
& aa(A,$o,aa(A,fun(A,$o),R3,X3),Xa3) ) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),X5),Y4) ) ) ) ) ).
% max_extp.max_extI
tff(fact_3887_max__extp_Osimps,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),A1),A22)
<=> ( aa(set(A),$o,finite_finite2(A),A1)
& aa(set(A),$o,finite_finite2(A),A22)
& ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
& ! [X2: A] :
( member(A,X2,A1)
=> ? [Xa2: A] :
( member(A,Xa2,A22)
& aa(A,$o,aa(A,fun(A,$o),R3,X2),Xa2) ) ) ) ) ).
% max_extp.simps
tff(fact_3888_max__extp_Ocases,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),A1),A22)
=> ~ ( aa(set(A),$o,finite_finite2(A),A1)
=> ( aa(set(A),$o,finite_finite2(A),A22)
=> ( ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
=> ~ ! [X4: A] :
( member(A,X4,A1)
=> ? [Xa4: A] :
( member(A,Xa4,A22)
& aa(A,$o,aa(A,fun(A,$o),R3,X4),Xa4) ) ) ) ) ) ) ).
% max_extp.cases
tff(fact_3889_max__ext__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : max_ext(A,R3) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_py(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R3))) ).
% max_ext_eq
tff(fact_3890_Sup__filter__def,axiom,
! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_pz(set(filter(A)),fun(fun(A,$o),$o),S)) ).
% Sup_filter_def
tff(fact_3891_frequently__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,A3,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A3),fequal($o))),frequently(A)),frequently(B)) ).
% frequently_parametric
tff(fact_3892_AboveS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_AboveS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_qa(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A3)) ).
% AboveS_def
tff(fact_3893_frequently__const,axiom,
! [A: $tType,F4: filter(A),P: $o] :
( ( F4 != bot_bot(filter(A)) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_aj($o,fun(A,$o),(P))),F4)
<=> (P) ) ) ).
% frequently_const
tff(fact_3894_ball__UNIV,axiom,
! [A: $tType,P: fun(A,$o)] :
( ! [X2: A] :
( member(A,X2,top_top(set(A)))
=> aa(A,$o,P,X2) )
<=> ! [X_12: A] : aa(A,$o,P,X_12) ) ).
% ball_UNIV
tff(fact_3895_frequently__mono,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_mono
tff(fact_3896_frequentlyE,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> ~ ! [X3: A] : ~ aa(A,$o,P,X3) ) ).
% frequentlyE
tff(fact_3897_not__frequently__False,axiom,
! [A: $tType,F4: filter(A)] : ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_cy(A,$o)),F4) ).
% not_frequently_False
tff(fact_3898_frequently__disj__iff,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
| aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_disj_iff
tff(fact_3899_frequently__elim1,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> ( ! [I3: A] :
( aa(A,$o,P,I3)
=> aa(A,$o,Q,I3) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_elim1
tff(fact_3900_frequently__disj,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4) ) ) ).
% frequently_disj
tff(fact_3901_frequently__all,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_qb(fun(A,fun(B,$o)),fun(A,$o),P)),F4)
<=> ! [Y8: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_nn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y8)),F4) ) ).
% frequently_all
tff(fact_3902_frequently__ex,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> ? [X_1: A] : aa(A,$o,P,X_1) ) ).
% frequently_ex
tff(fact_3903_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: fun(A,$o),C6: $o,F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa($o,fun(A,$o),aTP_Lamp_qc(fun(A,$o),fun($o,fun(A,$o)),P),(C6))),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
& (C6) ) ) ).
% eventually_frequently_const_simps(1)
tff(fact_3904_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C6: $o,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_qd($o,fun(fun(A,$o),fun(A,$o)),(C6)),P)),F4)
<=> ( (C6)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4) ) ) ).
% eventually_frequently_const_simps(2)
tff(fact_3905_top__filter__def,axiom,
! [A: $tType] : top_top(filter(A)) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),fAll(A)) ).
% top_filter_def
tff(fact_3906_eventually__all__finite,axiom,
! [A: $tType,B: $tType] :
( finite_finite(A)
=> ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
( ! [Y2: A] : aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_qe(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2)),Net)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_qf(fun(B,fun(A,$o)),fun(B,$o),P)),Net) ) ) ).
% eventually_all_finite
tff(fact_3907_frequently__imp__iff,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_imp_iff
tff(fact_3908_frequently__rev__mp,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_rev_mp
tff(fact_3909_not__frequently,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P)),F4) ) ).
% not_frequently
tff(fact_3910_not__eventually,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P)),F4) ) ).
% not_eventually
tff(fact_3911_frequently__def,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
<=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P)),F4) ) ).
% frequently_def
tff(fact_3912_frequently__mp,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ).
% frequently_mp
tff(fact_3913_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: fun(A,$o),C6: $o,F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_qg(fun(A,$o),fun($o,fun(A,$o)),P),(C6))),F4)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4)
=> (C6) ) ) ).
% eventually_frequently_const_simps(5)
tff(fact_3914_frequently__const__iff,axiom,
! [A: $tType,P: $o,F4: filter(A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_aj($o,fun(A,$o),(P))),F4)
<=> ( (P)
& ( F4 != bot_bot(filter(A)) ) ) ) ).
% frequently_const_iff
tff(fact_3915_inf__filter__def,axiom,
! [A: $tType,F4: filter(A),F7: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F7) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_qh(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F4),F7)) ).
% inf_filter_def
tff(fact_3916_rel__fun__def,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: fun(A,fun(C,$o)),B5: fun(B,fun(D,$o)),X4: fun(A,B),Xa3: fun(C,D)] :
( aa(fun(C,D),$o,aa(fun(A,B),fun(fun(C,D),$o),bNF_rel_fun(A,C,B,D,A3,B5),X4),Xa3)
<=> ! [Xb3: A,Y3: C] :
( aa(C,$o,aa(A,fun(C,$o),A3,Xb3),Y3)
=> aa(D,$o,aa(B,fun(D,$o),B5,aa(A,B,X4,Xb3)),aa(C,D,Xa3,Y3)) ) ) ).
% rel_fun_def
tff(fact_3917_bot__filter__def,axiom,
! [A: $tType] : bot_bot(filter(A)) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_qi(fun(A,$o),$o)) ).
% bot_filter_def
tff(fact_3918_eventually__frequently,axiom,
! [A: $tType,F4: filter(A),P: fun(A,$o)] :
( ( F4 != bot_bot(filter(A)) )
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F4) ) ) ).
% eventually_frequently
tff(fact_3919_frequently__bex__finite__distrib,axiom,
! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),F4: filter(B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_qj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P)),F4)
<=> ? [X2: A] :
( member(A,X2,A3)
& aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2)),F4) ) ) ) ).
% frequently_bex_finite_distrib
tff(fact_3920_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),F4: filter(B)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_qj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P)),F4)
=> ? [X3: A] :
( member(A,X3,A3)
& aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X3)),F4) ) ) ) ).
% frequently_bex_finite
tff(fact_3921_eventually__frequentlyE,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_qk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F4)
=> ( ( F4 != bot_bot(filter(A)) )
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F4) ) ) ) ).
% eventually_frequentlyE
tff(fact_3922_AboveS__disjoint,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),order_AboveS(A,R2,A3)) = bot_bot(set(A)) ).
% AboveS_disjoint
tff(fact_3923_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ql(fun(A,$o),fun(A,$o),P)),at_top(A)) ) ) ).
% eventually_all_ge_at_top
tff(fact_3924_Least__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_qm(fun(A,$o),fun(A,$o),P)) ) ).
% Least_def
tff(fact_3925_sup__filter__def,axiom,
! [A: $tType,F4: filter(A),F7: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F7) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_qn(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F4),F7)) ).
% sup_filter_def
tff(fact_3926_principal__def,axiom,
! [A: $tType,S: set(A)] : aa(set(A),filter(A),principal(A),S) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),ball(A,S)) ).
% principal_def
tff(fact_3927_filtermap__def,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),F4: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F4) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_qp(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),F2),F4)) ).
% filtermap_def
tff(fact_3928_map__filter__on__def,axiom,
! [A: $tType,B: $tType,X5: set(B),F2: fun(B,A),F4: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,X5),F2),F4) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_qr(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),X5),F2),F4)) ).
% map_filter_on_def
tff(fact_3929_Greatest__def,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_qs(fun(A,$o),fun(A,$o),P)) ) ).
% Greatest_def
tff(fact_3930_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A),B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( order_AboveS(A,R2,B5) != bot_bot(set(A)) )
=> ( member(A,B3,B5)
=> ( ( bNF_Wellorder_wo_suc(A,R2,B5) != B3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_Wellorder_wo_suc(A,R2,B5)),R2) ) ) ) ) ) ).
% wo_rel.suc_greater
tff(fact_3931_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( order_AboveS(A,R2,B5) != bot_bot(set(A)) )
=> member(A,bNF_Wellorder_wo_suc(A,R2,B5),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).
% wo_rel.suc_inField
tff(fact_3932_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( order_AboveS(A,R2,B5) != bot_bot(set(A)) )
=> member(A,bNF_Wellorder_wo_suc(A,R2,B5),order_AboveS(A,R2,B5)) ) ) ) ).
% wo_rel.suc_AboveS
tff(fact_3933_GreatestI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> ( ! [Y5: A] :
( aa(A,$o,P,Y5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),X3) )
=> aa(A,$o,Q,X3) ) )
=> aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).
% GreatestI2_order
tff(fact_3934_Greatest__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
=> ( order_Greatest(A,P) = X ) ) ) ) ).
% Greatest_equality
tff(fact_3935_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A,B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A4,order_AboveS(A,R2,B5))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R2,B5)),A4),R2) ) ) ).
% wo_rel.suc_least_AboveS
tff(fact_3936_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B5: set(A),A4: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,A4,order_AboveS(A,R2,B5))
=> ( ! [A16: A] :
( member(A,A16,order_AboveS(A,R2,B5))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A16),R2) )
=> ( A4 = bNF_Wellorder_wo_suc(A,R2,B5) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
tff(fact_3937_ord_OLeast__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_qt(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).
% ord.Least_def
tff(fact_3938_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( order_ofilter(A,R2,A3)
=> ( ( order_AboveS(A,R2,A3) != bot_bot(set(A)) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),bNF_Wellorder_wo_suc(A,R2,A3)),R2)
=> ( ( B3 != bNF_Wellorder_wo_suc(A,R2,A3) )
=> member(A,B3,A3) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
tff(fact_3939_filtercomap__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F4: filter(B)] : filtercomap(A,B,F2,F4) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_qu(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),F2),F4)) ).
% filtercomap_def
tff(fact_3940_init__seg__of__def,axiom,
! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),$o),aTP_Lamp_qv(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ).
% init_seg_of_def
tff(fact_3941_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F4: filter(B)] : filterlim(A,B,F2,F4,filtercomap(A,B,F2,F4)) ).
% filterlim_filtercomap
tff(fact_3942_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : filtercomap(A,B,F2,bot_bot(filter(B))) = bot_bot(filter(A)) ).
% filtercomap_bot
tff(fact_3943_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),F4: filter(A),F2: fun(B,A)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4)
=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_np(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),filtercomap(B,A,F2,F4)) ) ).
% eventually_filtercomapI
tff(fact_3944_filtercomap__top,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : filtercomap(A,B,F2,top_top(filter(B))) = top_top(filter(A)) ).
% filtercomap_top
tff(fact_3945_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : filtercomap(A,B,F2,aa(set(B),filter(B),principal(B),A3)) = aa(set(A),filter(A),principal(A),vimage(A,B,F2,A3)) ).
% filtercomap_principal
tff(fact_3946_eventually__filtercomap,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,B),F4: filter(B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,F4))
<=> ? [Q4: fun(B,$o)] :
( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q4),F4)
& ! [X2: A] :
( aa(B,$o,Q4,aa(A,B,F2,X2))
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_filtercomap
tff(fact_3947_ord_OLeast_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).
% ord.Least.cong
tff(fact_3948_filtercomap__filtercomap,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),G: fun(B,C),F4: filter(C)] : filtercomap(A,B,F2,filtercomap(B,C,G,F4)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qw(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),F4) ).
% filtercomap_filtercomap
tff(fact_3949_filtercomap__ident,axiom,
! [A: $tType,F4: filter(A)] : filtercomap(A,A,aTP_Lamp_ac(A,A),F4) = F4 ).
% filtercomap_ident
tff(fact_3950_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F4: filter(A),F7: filter(A),F2: fun(B,A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F7)
=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),filtercomap(B,A,F2,F4)),filtercomap(B,A,F2,F7)) ) ).
% filtercomap_mono
tff(fact_3951_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F12: filter(B),F23: filter(B)] : filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F12),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtercomap(A,B,F2,F12)),filtercomap(A,B,F2,F23)) ).
% filtercomap_inf
tff(fact_3952_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(B,C),G5: filter(C),F4: filter(A)] :
( filterlim(A,B,F2,filtercomap(B,C,G,G5),F4)
<=> filterlim(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2),G5,F4) ) ).
% filterlim_filtercomap_iff
tff(fact_3953_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
( ! [P3: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),F4)
=> ? [X4: B] : aa(A,$o,P3,aa(B,A,F2,X4)) )
=> ( filtercomap(B,A,F2,F4) != bot_bot(filter(B)) ) ) ).
% filtercomap_neq_bot
tff(fact_3954_filterlim__iff__le__filtercomap,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F4: filter(B),G5: filter(A)] :
( filterlim(A,B,F2,F4,G5)
<=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),filtercomap(A,B,F2,F4)) ) ).
% filterlim_iff_le_filtercomap
tff(fact_3955_filtercomap__filtermap,axiom,
! [B: $tType,A: $tType,F4: filter(A),F2: fun(A,B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),filtercomap(A,B,F2,aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F4))) ).
% filtercomap_filtermap
tff(fact_3956_filtermap__filtercomap,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),F4: filter(A)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),filtercomap(B,A,F2,F4))),F4) ).
% filtermap_filtercomap
tff(fact_3957_filtermap__le__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),F4: filter(B),G5: filter(A)] :
( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F4)),G5)
<=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),filtercomap(B,A,F2,G5)) ) ).
% filtermap_le_iff_le_filtercomap
tff(fact_3958_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),F12: filter(B),F23: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtercomap(A,B,F2,F12)),filtercomap(A,B,F2,F23))),filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F12),F23))) ).
% filtercomap_sup
tff(fact_3959_filtercomap__INF,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F4: fun(C,filter(B)),B5: set(C)] : filtercomap(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_qx(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B5)) ).
% filtercomap_INF
tff(fact_3960_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
( ( F4 != bot_bot(filter(A)) )
=> ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
=> ( filtercomap(B,A,F2,F4) != bot_bot(filter(B)) ) ) ) ).
% filtercomap_neq_bot_surj
tff(fact_3961_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [P: fun(A,$o),F2: fun(A,B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_top(B)))
<=> ? [N7: B] :
! [X2: A] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N7),aa(A,B,F2,X2))
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_filtercomap_at_top_linorder
tff(fact_3962_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( linorder(B)
& no_top(B) )
=> ! [P: fun(A,$o),F2: fun(A,B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_top(B)))
<=> ? [N7: B] :
! [X2: A] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N7),aa(A,B,F2,X2))
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_filtercomap_at_top_dense
tff(fact_3963_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [P: fun(A,$o),F2: fun(A,B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_bot(B)))
<=> ? [N7: B] :
! [X2: A] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),N7)
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
tff(fact_3964_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( linorder(B)
& no_bot(B) )
=> ! [P: fun(A,$o),F2: fun(A,B)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_bot(B)))
<=> ? [N7: B] :
! [X2: A] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),N7)
=> aa(A,$o,P,X2) ) ) ) ).
% eventually_filtercomap_at_bot_dense
tff(fact_3965_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(A,C),F4: fun(B,filter(C)),B5: set(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_qy(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F2),F4)),B5))),filtercomap(A,C,F2,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F4),B5)))) ).
% filtercomap_SUP
tff(fact_3966_flip__pred,axiom,
! [A: $tType,B: $tType,A3: set(product_prod(A,B)),R3: fun(B,fun(A,$o))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),conversep(B,A,R3))))
=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),set(product_prod(B,A)),image2(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_is(A,fun(B,product_prod(B,A))))),A3)),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),R3))) ) ).
% flip_pred
tff(fact_3967_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R2))
<=> single_valued(A,B,R2) ) ).
% single_valuedp_single_valued_eq
tff(fact_3968_set__encode__empty,axiom,
nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).
% set_encode_empty
tff(fact_3969_mset__set_Oinsert__remove,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = add_mset(A,X,mset_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).
% mset_set.insert_remove
tff(fact_3970_conversep__eq,axiom,
! [A: $tType] : conversep(A,A,fequal(A)) = fequal(A) ).
% conversep_eq
tff(fact_3971_conversep__iff,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: B,B3: A] :
( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A4),B3)
<=> aa(B,$o,aa(A,fun(B,$o),R2,B3),A4) ) ).
% conversep_iff
tff(fact_3972_conversep__inject,axiom,
! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] :
( ( conversep(B,A,R2) = conversep(B,A,S2) )
<=> ( R2 = S2 ) ) ).
% conversep_inject
tff(fact_3973_conversep__conversep,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] : conversep(B,A,conversep(A,B,R2)) = R2 ).
% conversep_conversep
tff(fact_3974_conversep__noteq,axiom,
! [A: $tType,X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),conversep(A,A,aTP_Lamp_pk(A,fun(A,$o))),X4),Xa3)
<=> ( X4 != Xa3 ) ) ).
% conversep_noteq
tff(fact_3975_conversep__mono,axiom,
! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2))
<=> aa(fun(B,fun(A,$o)),$o,aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),$o),ord_less_eq(fun(B,fun(A,$o))),R2),S2) ) ).
% conversep_mono
tff(fact_3976_mset__set_Oinsert,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = add_mset(A,X,mset_set(A,A3)) ) ) ) ).
% mset_set.insert
tff(fact_3977_conversep__le__swap,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(A,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R2),conversep(B,A,S2))
<=> aa(fun(B,fun(A,$o)),$o,aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),$o),ord_less_eq(fun(B,fun(A,$o))),conversep(A,B,R2)),S2) ) ).
% conversep_le_swap
tff(fact_3978_leq__conversepI,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( ( R3 = fequal(A) )
=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R3),conversep(A,A,R3)) ) ).
% leq_conversepI
tff(fact_3979_mset__union__2__elem,axiom,
! [A: $tType,A4: A,B3: A,C2: A,M2: multiset(A)] :
( ( add_mset(A,A4,add_mset(A,B3,zero_zero(multiset(A)))) = add_mset(A,C2,M2) )
=> ( ( ( add_mset(A,A4,zero_zero(multiset(A))) = M2 )
& ( B3 = C2 ) )
| ( ( A4 = C2 )
& ( add_mset(A,B3,zero_zero(multiset(A))) = M2 ) ) ) ) ).
% mset_union_2_elem
tff(fact_3980_conversep_Osimps,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: B,A22: A] :
( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A1),A22)
<=> ? [A7: A,B7: B] :
( ( A1 = B7 )
& ( A22 = A7 )
& aa(B,$o,aa(A,fun(B,$o),R2,A7),B7) ) ) ).
% conversep.simps
tff(fact_3981_conversepD,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),B3: B,A4: A] :
( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),B3),A4)
=> aa(B,$o,aa(A,fun(B,$o),R2,A4),B3) ) ).
% conversepD
tff(fact_3982_conversepE,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: B,A22: A] :
( aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),A1),A22)
=> aa(B,$o,aa(A,fun(B,$o),R2,A22),A1) ) ).
% conversepE
tff(fact_3983_conversepI,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,A4),B3)
=> aa(A,$o,aa(B,fun(A,$o),conversep(A,B,R2),B3),A4) ) ).
% conversepI
tff(fact_3984_single__valuedpD,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),X: A,Y: B,Z2: B] :
( single_valuedp(A,B,R2)
=> ( aa(B,$o,aa(A,fun(B,$o),R2,X),Y)
=> ( aa(B,$o,aa(A,fun(B,$o),R2,X),Z2)
=> ( Y = Z2 ) ) ) ) ).
% single_valuedpD
tff(fact_3985_single__valuedpI,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
( ! [X3: A,Y2: B,Z4: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,X3),Y2)
=> ( aa(B,$o,aa(A,fun(B,$o),R2,X3),Z4)
=> ( Y2 = Z4 ) ) )
=> single_valuedp(A,B,R2) ) ).
% single_valuedpI
tff(fact_3986_single__valuedp__def,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] :
( single_valuedp(A,B,R2)
<=> ! [X2: A,Y3: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,X2),Y3)
=> ! [Z3: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,X2),Z3)
=> ( Y3 = Z3 ) ) ) ) ).
% single_valuedp_def
tff(fact_3987_converse__meet,axiom,
! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] : conversep(B,A,aa(fun(B,fun(A,$o)),fun(B,fun(A,$o)),aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),fun(B,fun(A,$o))),inf_inf(fun(B,fun(A,$o))),R2),S2)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2)) ).
% converse_meet
tff(fact_3988_converse__join,axiom,
! [A: $tType,B: $tType,R2: fun(B,fun(A,$o)),S2: fun(B,fun(A,$o))] : conversep(B,A,aa(fun(B,fun(A,$o)),fun(B,fun(A,$o)),aa(fun(B,fun(A,$o)),fun(fun(B,fun(A,$o)),fun(B,fun(A,$o))),sup_sup(fun(B,fun(A,$o))),R2),S2)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),conversep(B,A,R2)),conversep(B,A,S2)) ).
% converse_join
tff(fact_3989_rel__filter__conversep,axiom,
! [A: $tType,B: $tType,A3: fun(B,fun(A,$o))] : rel_filter(A,B,conversep(B,A,A3)) = conversep(filter(B),filter(A),rel_filter(B,A,A3)) ).
% rel_filter_conversep
tff(fact_3990_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),conversep(B,A,aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),converse(B,A,R2)) ) ).
% conversep_converse_eq
tff(fact_3991_single__valuedp__less__eq,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),S2: fun(A,fun(B,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R2),S2)
=> ( single_valuedp(A,B,S2)
=> single_valuedp(A,B,R2) ) ) ).
% single_valuedp_less_eq
tff(fact_3992_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : single_valuedp(A,B,bot_bot(fun(A,fun(B,$o)))) ).
% single_valuedp_bot
tff(fact_3993_converse__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : converse(A,B,X4) = aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),X4)))) ).
% converse_def
tff(fact_3994_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B5: multiset(A),C6: multiset(A),S2: A,A3: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),A3) )
=> ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),minus_minus(multiset(A),B5,add_mset(A,S2,zero_zero(multiset(A))))) )
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B5,add_mset(A,S2,zero_zero(multiset(A))))),C6) ) )
=> ~ ( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),minus_minus(multiset(A),C6,add_mset(A,S2,zero_zero(multiset(A))))) )
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),minus_minus(multiset(A),C6,add_mset(A,S2,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases2
tff(fact_3995_mset__unplusm__dist__cases,axiom,
! [A: $tType,S2: A,A3: multiset(A),B5: multiset(A),C6: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),A3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C6) )
=> ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),minus_minus(multiset(A),B5,add_mset(A,S2,zero_zero(multiset(A))))) )
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B5,add_mset(A,S2,zero_zero(multiset(A))))),C6) ) )
=> ~ ( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),minus_minus(multiset(A),C6,add_mset(A,S2,zero_zero(multiset(A))))) )
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),minus_minus(multiset(A),C6,add_mset(A,S2,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases
tff(fact_3996_mset__single__cases2_H,axiom,
! [A: $tType,S2: A,C2: multiset(A),R5: A,C10: multiset(A)] :
( ( add_mset(A,S2,C2) = add_mset(A,R5,C10) )
=> ( ( ( S2 = R5 )
=> ( C2 != C10 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C10 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,S2,zero_zero(multiset(A)))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,R5,zero_zero(multiset(A)))) )
=> ( ( minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A)))) = Cc )
=> ( minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases2'
tff(fact_3997_mset__single__cases2,axiom,
! [A: $tType,S2: A,C2: multiset(A),R5: A,C10: multiset(A)] :
( ( add_mset(A,S2,C2) = add_mset(A,R5,C10) )
=> ( ( ( S2 = R5 )
=> ( C2 != C10 ) )
=> ~ ( ( C10 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A))))),add_mset(A,S2,zero_zero(multiset(A)))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A))))),add_mset(A,R5,zero_zero(multiset(A)))) )
=> ( minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A)))) != minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases2
tff(fact_3998_mset__single__cases_H,axiom,
! [A: $tType,S2: A,C2: multiset(A),R5: A,C10: multiset(A)] :
( ( add_mset(A,S2,C2) = add_mset(A,R5,C10) )
=> ( ( ( S2 = R5 )
=> ( C2 != C10 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C10 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),Cc) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R5,zero_zero(multiset(A)))),Cc) )
=> ( ( minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A)))) = Cc )
=> ( minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases'
tff(fact_3999_mset__single__cases,axiom,
! [A: $tType,S2: A,C2: multiset(A),R5: A,C10: multiset(A)] :
( ( add_mset(A,S2,C2) = add_mset(A,R5,C10) )
=> ( ( ( S2 = R5 )
=> ( C2 != C10 ) )
=> ~ ( ( C10 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S2,zero_zero(multiset(A)))),minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A))))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R5,zero_zero(multiset(A)))),minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A))))) )
=> ( minus_minus(multiset(A),C2,add_mset(A,R5,zero_zero(multiset(A)))) != minus_minus(multiset(A),C10,add_mset(A,S2,zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases
tff(fact_4000_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
( single_valuedp(A,B,R2)
<=> ! [X2: A] : uniq(B,aa(A,fun(B,$o),R2,X2)) ) ).
% single_valuedp_iff_Uniq
tff(fact_4001_mset__set_Oremove,axiom,
! [A: $tType,A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( mset_set(A,A3) = add_mset(A,X,mset_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% mset_set.remove
tff(fact_4002_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A4: A,M2: multiset(A)] :
( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),minus_minus(multiset(A),M2,add_mset(A,A4,zero_zero(multiset(A))))))
=> ( aa(multiset(A),set(A),set_mset(A),minus_minus(multiset(A),M2,add_mset(A,A4,zero_zero(multiset(A))))) = minus_minus(set(A),aa(multiset(A),set(A),set_mset(A),M2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ) ) ).
% at_most_one_mset_mset_diff
tff(fact_4003_normalize__crossproduct,axiom,
! [Q3: int,S2: int,P2: int,R2: int] :
( ( Q3 != 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),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S2)) )
=> ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q3) ) ) ) ) ).
% normalize_crossproduct
tff(fact_4004_su__rel__fun_Of__def,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A] :
( su_rel_fun(A,B,F4,F2)
=> ( aa(A,B,F2,A3) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A3)) ) ) ).
% su_rel_fun.f_def
tff(fact_4005_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B)] :
( ! [A10: A,B6: B,B8: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B6),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B8),F4)
=> ( B6 = B8 ) ) )
=> ( ! [A10: A,P3: $o] :
( ! [B15: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B15),F4)
=> (P3) )
=> (P3) )
=> ( ! [A10: A] : aa(A,B,F2,A10) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A10))
=> su_rel_fun(A,B,F4,F2) ) ) ) ).
% su_rel_fun.intro
tff(fact_4006_set__mset__empty,axiom,
! [A: $tType] : aa(multiset(A),set(A),set_mset(A),zero_zero(multiset(A))) = bot_bot(set(A)) ).
% set_mset_empty
tff(fact_4007_set__mset__eq__empty__iff,axiom,
! [A: $tType,M2: multiset(A)] :
( ( aa(multiset(A),set(A),set_mset(A),M2) = bot_bot(set(A)) )
<=> ( M2 = zero_zero(multiset(A)) ) ) ).
% set_mset_eq_empty_iff
tff(fact_4008_set__mset__add__mset__insert,axiom,
! [A: $tType,A4: A,A3: multiset(A)] : aa(multiset(A),set(A),set_mset(A),add_mset(A,A4,A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(multiset(A),set(A),set_mset(A),A3)) ).
% set_mset_add_mset_insert
tff(fact_4009_mset__diff__cancel1elem,axiom,
! [A: $tType,A4: A,B5: multiset(A)] :
( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),B5))
=> ( minus_minus(multiset(A),add_mset(A,A4,zero_zero(multiset(A))),B5) = add_mset(A,A4,zero_zero(multiset(A))) ) ) ).
% mset_diff_cancel1elem
tff(fact_4010_mset__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset(A),B5: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)))
=> ( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),A3))
=> member(A,A4,aa(multiset(A),set(A),set_mset(A),B5)) ) ) ).
% mset_un_cases
tff(fact_4011_mset__left__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset(A),B5: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)))
=> ( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),A3))
=> member(A,A4,aa(multiset(A),set(A),set_mset(A),B5)) ) ) ).
% mset_left_cancel_union
tff(fact_4012_mset__right__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset(A),B5: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)))
=> ( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),B5))
=> member(A,A4,aa(multiset(A),set(A),set_mset(A),A3)) ) ) ).
% mset_right_cancel_union
tff(fact_4013_ex__Melem__conv,axiom,
! [A: $tType,A3: multiset(A)] :
( ? [X2: A] : member(A,X2,aa(multiset(A),set(A),set_mset(A),A3))
<=> ( A3 != zero_zero(multiset(A)) ) ) ).
% ex_Melem_conv
tff(fact_4014_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A] :
( su_rel_fun(A,B,F4,F2)
=> ~ ! [B6: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B6),F4) ) ).
% su_rel_fun.surjective
tff(fact_4015_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A,B5: B,B12: B] :
( su_rel_fun(A,B,F4,F2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B12),F4)
=> ( B5 = B12 ) ) ) ) ).
% su_rel_fun.unique
tff(fact_4016_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A,B5: B] :
( su_rel_fun(A,B,F4,F2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5),F4)
=> ( B5 = aa(A,B,F2,A3) ) ) ) ).
% su_rel_fun.repr2
tff(fact_4017_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A] :
( su_rel_fun(A,B,F4,F2)
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),aa(A,B,F2,A3)),F4) ) ).
% su_rel_fun.repr1
tff(fact_4018_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B),A3: A,B5: B] :
( su_rel_fun(A,B,F4,F2)
=> ( ( aa(A,B,F2,A3) = B5 )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5),F4) ) ) ).
% su_rel_fun.repr
tff(fact_4019_in__Inf__multiset__iff,axiom,
! [A: $tType,A3: set(multiset(A)),X: A] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( member(A,X,aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)))
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,A3)
=> member(A,X,aa(multiset(A),set(A),set_mset(A),X2)) ) ) ) ).
% in_Inf_multiset_iff
tff(fact_4020_mset__left__cancel__elem,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,B3,zero_zero(multiset(A)))),A3)))
=> ( ( A4 != B3 )
=> member(A,A4,aa(multiset(A),set(A),set_mset(A),A3)) ) ) ).
% mset_left_cancel_elem
tff(fact_4021_mset__right__cancel__elem,axiom,
! [A: $tType,A4: A,A3: multiset(A),B3: A] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),add_mset(A,B3,zero_zero(multiset(A))))))
=> ( ( A4 != B3 )
=> member(A,A4,aa(multiset(A),set(A),set_mset(A),A3)) ) ) ).
% mset_right_cancel_elem
tff(fact_4022_infinite__set__mset__mset__set,axiom,
! [A: $tType,A3: set(A)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(multiset(A),set(A),set_mset(A),mset_set(A,A3)) = bot_bot(set(A)) ) ) ).
% infinite_set_mset_mset_set
tff(fact_4023_set__mset__Inf,axiom,
! [A: $tType,A3: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A3)) ) ) ).
% set_mset_Inf
tff(fact_4024_set__mset__single,axiom,
! [A: $tType,B3: A] : aa(multiset(A),set(A),set_mset(A),add_mset(A,B3,zero_zero(multiset(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) ).
% set_mset_single
tff(fact_4025_mset__contains__eq,axiom,
! [A: $tType,M: A,M2: multiset(A)] :
( member(A,M,aa(multiset(A),set(A),set_mset(A),M2))
<=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,M,zero_zero(multiset(A)))),minus_minus(multiset(A),M2,add_mset(A,M,zero_zero(multiset(A))))) = M2 ) ) ).
% mset_contains_eq
tff(fact_4026_mset__union__diff__comm,axiom,
! [A: $tType,T5: A,S: multiset(A),T3: multiset(A)] :
( member(A,T5,aa(multiset(A),set(A),set_mset(A),S))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T3),minus_minus(multiset(A),S,add_mset(A,T5,zero_zero(multiset(A))))) = minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T3),S),add_mset(A,T5,zero_zero(multiset(A)))) ) ) ).
% mset_union_diff_comm
tff(fact_4027_diff__union__single__conv2,axiom,
! [A: $tType,A4: A,J: multiset(A),I: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),J))
=> ( minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),J),I),add_mset(A,A4,zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),J,add_mset(A,A4,zero_zero(multiset(A))))),I) ) ) ).
% diff_union_single_conv2
tff(fact_4028_mset__un__single__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset(A),B5: multiset(A),C6: multiset(A)] :
( ( add_mset(A,A4,A3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C6) )
=> ( ( member(A,A4,aa(multiset(A),set(A),set_mset(A),B5))
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B5,add_mset(A,A4,zero_zero(multiset(A))))),C6) ) )
=> ~ ( member(A,A4,aa(multiset(A),set(A),set_mset(A),C6))
=> ( A3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),minus_minus(multiset(A),C6,add_mset(A,A4,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_un_single_un_cases
tff(fact_4029_su__rel__fun__def,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F2: fun(A,B)] :
( su_rel_fun(A,B,F4,F2)
<=> ( ! [A11: A,B10: B,B16: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B10),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B16),F4)
=> ( B10 = B16 ) ) )
& ! [A11: A,P4: $o] :
( ! [B10: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A11),B10),F4)
=> (P4) )
=> (P4) )
& ! [A11: A] : aa(A,B,F2,A11) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A11)) ) ) ).
% su_rel_fun_def
tff(fact_4030_mult1__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : mult1(A,R2) = aa(fun(product_prod(multiset(A),multiset(A)),$o),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),$o)),fun(product_prod(multiset(A),multiset(A)),$o),product_case_prod(multiset(A),multiset(A),$o),aTP_Lamp_qz(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),R2))) ).
% mult1_def
tff(fact_4031_rat__minus__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(minus_minus(rat,P2,Q3)) = 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_rb(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).
% rat_minus_code
tff(fact_4032_rat__plus__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q3)) = 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_rd(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).
% rat_plus_code
tff(fact_4033_rat__times__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q3)) = 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_rf(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).
% rat_times_code
tff(fact_4034_rat__less__code,axiom,
! [P2: rat,Q3: rat] :
( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q3)
<=> 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_rh(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).
% rat_less_code
tff(fact_4035_rat__less__eq__code,axiom,
! [P2: rat,Q3: rat] :
( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q3)
<=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_rj(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).
% rat_less_eq_code
tff(fact_4036_mult1E,axiom,
! [A: $tType,N: multiset(A),M2: multiset(A),R2: set(product_prod(A,A))] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N),M2),mult1(A,R2))
=> ~ ! [A5: A,M0: multiset(A)] :
( ( M2 = add_mset(A,A5,M0) )
=> ! [K6: multiset(A)] :
( ( N = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K6) )
=> ~ ! [B13: A] :
( member(A,B13,aa(multiset(A),set(A),set_mset(A),K6))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),A5),R2) ) ) ) ) ).
% mult1E
tff(fact_4037_mult1I,axiom,
! [A: $tType,M2: multiset(A),A4: A,M02: multiset(A),N: multiset(A),K2: multiset(A),R2: set(product_prod(A,A))] :
( ( M2 = add_mset(A,A4,M02) )
=> ( ( N = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K2) )
=> ( ! [B2: A] :
( member(A,B2,aa(multiset(A),set(A),set_mset(A),K2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A4),R2) )
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N),M2),mult1(A,R2)) ) ) ) ).
% mult1I
tff(fact_4038_less__add,axiom,
! [A: $tType,N: multiset(A),A4: A,M02: multiset(A),R2: set(product_prod(A,A))] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N),add_mset(A,A4,M02)),mult1(A,R2))
=> ( ? [M5: multiset(A)] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M5),M02),mult1(A,R2))
& ( N = add_mset(A,A4,M5) ) )
| ? [K6: multiset(A)] :
( ! [B13: A] :
( member(A,B13,aa(multiset(A),set(A),set_mset(A),K6))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),A4),R2) )
& ( N = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K6) ) ) ) ) ).
% less_add
tff(fact_4039_rat__divide__code,axiom,
! [P2: rat,Q3: rat] : quotient_of(divide_divide(rat,P2,Q3)) = 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_rl(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).
% rat_divide_code
tff(fact_4040_rat__inverse__code,axiom,
! [P2: rat] : quotient_of(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_rm(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).
% rat_inverse_code
tff(fact_4041_mult__implies__one__step,axiom,
! [A: $tType,R2: set(product_prod(A,A)),M2: multiset(A),N: multiset(A)] :
( trans(A,R2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M2),N),mult(A,R2))
=> ? [I5: multiset(A),J5: multiset(A)] :
( ( N = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),J5) )
& ? [K6: multiset(A)] :
( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),K6) )
& ( J5 != zero_zero(multiset(A)) )
& ! [X4: A] :
( member(A,X4,aa(multiset(A),set(A),set_mset(A),K6))
=> ? [Xa4: A] :
( member(A,Xa4,aa(multiset(A),set(A),set_mset(A),J5))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4),R2) ) ) ) ) ) ) ).
% mult_implies_one_step
tff(fact_4042_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N2: nat,X: A] :
aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N2,X)) = $ite(N2 = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% set_mset_replicate_mset_subset
tff(fact_4043_size__diff__se,axiom,
! [A: $tType,T5: A,S: multiset(A)] :
( member(A,T5,aa(multiset(A),set(A),set_mset(A),S))
=> ( aa(multiset(A),nat,size_size(multiset(A)),S) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),minus_minus(multiset(A),S,add_mset(A,T5,zero_zero(multiset(A)))))),one_one(nat)) ) ) ).
% size_diff_se
tff(fact_4044_inverse__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] : inverse_inverse(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A4)),inverse_inverse(A,B3)) ) ).
% inverse_mult_distrib
tff(fact_4045_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( inverse_inverse(A,one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_4046_inverse__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A] :
( ( inverse_inverse(A,X) = one_one(A) )
<=> ( X = one_one(A) ) ) ) ).
% inverse_eq_1_iff
tff(fact_4047_left__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A4)),A4) = one_one(A) ) ) ) ).
% left_inverse
tff(fact_4048_right__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),inverse_inverse(A,A4)) = one_one(A) ) ) ) ).
% right_inverse
tff(fact_4049_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : inverse_inverse(A,aa(num,A,numeral_numeral(A),W)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W)) ) ).
% inverse_eq_divide_numeral
tff(fact_4050_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : inverse_inverse(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = divide_divide(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).
% inverse_eq_divide_neg_numeral
tff(fact_4051_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Y: A,X: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),inverse_inverse(A,Y)) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_4052_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( inverse_inverse(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,B3)),inverse_inverse(A,A4)) ) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_4053_inverse__unique,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3) = one_one(A) )
=> ( inverse_inverse(A,A4) = B3 ) ) ) ).
% inverse_unique
tff(fact_4054_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] : divide_divide(A,A4,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),inverse_inverse(A,B3)) ) ).
% field_class.field_divide_inverse
tff(fact_4055_divide__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A] : divide_divide(A,A4,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),inverse_inverse(A,B3)) ) ).
% divide_inverse
tff(fact_4056_divide__inverse__commute,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] : divide_divide(A,A4,B3) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,B3)),A4) ) ).
% divide_inverse_commute
tff(fact_4057_inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] : inverse_inverse(A,A4) = divide_divide(A,one_one(A),A4) ) ).
% inverse_eq_divide
tff(fact_4058_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: nat,N2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),inverse_inverse(A,X)),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),inverse_inverse(A,X)),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).
% power_mult_power_inverse_commute
tff(fact_4059_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),inverse_inverse(A,X)) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).
% power_mult_inverse_distrib
tff(fact_4060_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Xa: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,aa(nat,A,semiring_1_of_nat(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),inverse_inverse(A,aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).
% mult_inverse_of_nat_commute
tff(fact_4061_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Xa: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,aa(int,A,ring_1_of_int(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),inverse_inverse(A,aa(int,A,ring_1_of_int(A),Xa))) ) ).
% mult_inverse_of_int_commute
tff(fact_4062_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),inverse_inverse(A,X)),one_one(A))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).
% inverse_le_1_iff
tff(fact_4063_one__less__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),inverse_inverse(A,A4)) ) ) ) ).
% one_less_inverse
tff(fact_4064_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),inverse_inverse(A,X))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).
% one_less_inverse_iff
tff(fact_4065_inverse__add,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),inverse_inverse(A,A4)),inverse_inverse(A,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3)),inverse_inverse(A,A4))),inverse_inverse(A,B3)) ) ) ) ) ).
% inverse_add
tff(fact_4066_division__ring__inverse__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),inverse_inverse(A,A4)),inverse_inverse(A,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3))),inverse_inverse(A,B3)) ) ) ) ) ).
% division_ring_inverse_add
tff(fact_4067_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A4)),A4) = one_one(A) ) ) ) ).
% field_class.field_inverse
tff(fact_4068_division__ring__inverse__diff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( minus_minus(A,inverse_inverse(A,A4),inverse_inverse(A,B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A4)),minus_minus(A,B3,A4))),inverse_inverse(A,B3)) ) ) ) ) ).
% division_ring_inverse_diff
tff(fact_4069_divide__rat__def,axiom,
! [Q3: rat,R2: rat] : divide_divide(rat,Q3,R2) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q3),inverse_inverse(rat,R2)) ).
% divide_rat_def
tff(fact_4070_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( inverse_inverse(A,A4) = divide_divide(A,one_one(A),A4) ) ) ) ).
% nonzero_inverse_eq_divide
tff(fact_4071_inverse__less__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),inverse_inverse(A,A4)),inverse_inverse(A,B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A4) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3) ) ) ) ) ).
% inverse_less_iff
tff(fact_4072_inverse__le__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),inverse_inverse(A,A4)),inverse_inverse(A,B3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A4) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3) ) ) ) ) ).
% inverse_le_iff
tff(fact_4073_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),inverse_inverse(A,X))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ) ).
% one_le_inverse_iff
tff(fact_4074_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),inverse_inverse(A,X)),one_one(A))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).
% inverse_less_1_iff
tff(fact_4075_one__le__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),inverse_inverse(A,A4)) ) ) ) ).
% one_le_inverse
tff(fact_4076_power__diff__conv__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: nat,N2: nat] :
( ( X != zero_zero(A) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),minus_minus(nat,N2,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),inverse_inverse(A,X)),M)) ) ) ) ) ).
% power_diff_conv_inverse
tff(fact_4077_mset__size__le1__cases,axiom,
! [A: $tType,M2: multiset(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M2)),aa(nat,nat,suc,zero_zero(nat)))
=> ( ( M2 != zero_zero(multiset(A)) )
=> ~ ! [M3: A] : M2 != add_mset(A,M3,zero_zero(multiset(A))) ) ) ).
% mset_size_le1_cases
tff(fact_4078_mset__size1elem,axiom,
! [A: $tType,P: multiset(A),Q3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),one_one(nat))
=> ( member(A,Q3,aa(multiset(A),set(A),set_mset(A),P))
=> ( P = add_mset(A,Q3,zero_zero(multiset(A))) ) ) ) ).
% mset_size1elem
tff(fact_4079_one__step__implies__mult,axiom,
! [A: $tType,J: multiset(A),K2: multiset(A),R2: set(product_prod(A,A)),I: multiset(A)] :
( ( J != zero_zero(multiset(A)) )
=> ( ! [X3: A] :
( member(A,X3,aa(multiset(A),set(A),set_mset(A),K2))
=> ? [Xa3: A] :
( member(A,Xa3,aa(multiset(A),set(A),set_mset(A),J))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R2) ) )
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I),K2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I),J)),mult(A,R2)) ) ) ).
% one_step_implies_mult
tff(fact_4080_multp__code__iff__mult,axiom,
! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N: multiset(A),M2: multiset(A)] :
( irrefl(A,R3)
=> ( trans(A,R3)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R3) )
=> ( multp_code(A,P,N,M2)
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N),M2),mult(A,R3)) ) ) ) ) ).
% multp_code_iff_mult
tff(fact_4081_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R3: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N: multiset(A),M2: multiset(A)] :
( irrefl(A,R3)
=> ( trans(A,R3)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R3) )
=> ( multeqp_code(A,P,N,M2)
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N),M2),aa(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A))),aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A)))),sup_sup(set(product_prod(multiset(A),multiset(A)))),mult(A,R3)),id2(multiset(A)))) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
tff(fact_4082_mset__size2elem,axiom,
! [A: $tType,P: multiset(A),Q3: A,Q5: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q3,zero_zero(multiset(A)))),add_mset(A,Q5,zero_zero(multiset(A))))),P)
=> ( P = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q3,zero_zero(multiset(A)))),add_mset(A,Q5,zero_zero(multiset(A)))) ) ) ) ).
% mset_size2elem
tff(fact_4083_prod__mset_Oremove,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,A3: multiset(A)] :
( member(A,X,aa(multiset(A),set(A),set_mset(A),A3))
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),minus_minus(multiset(A),A3,add_mset(A,X,zero_zero(multiset(A)))))) ) ) ) ).
% prod_mset.remove
tff(fact_4084_prod__mset__empty,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),zero_zero(multiset(A))) = one_one(A) ) ) ).
% prod_mset_empty
tff(fact_4085_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,N: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),add_mset(A,X,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N)) ) ).
% prod_mset.add_mset
tff(fact_4086_prod__mset__Un,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: multiset(A),B5: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A3)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),B5)) ) ).
% prod_mset_Un
tff(fact_4087_prod__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: multiset(A),N: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M2)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N)) ) ).
% prod_mset.union
tff(fact_4088_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,A3: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ? [X3: multiset(A)] :
( member(multiset(A),X3,A3)
& ! [Xa3: multiset(A)] :
( member(multiset(A),Xa3,A3)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa3),X3)
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
tff(fact_4089_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,A3: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ? [X3: multiset(A)] :
( member(multiset(A),X3,A3)
& ! [Xa3: multiset(A)] :
( member(multiset(A),Xa3,A3)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),Xa3)
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
tff(fact_4090_mset__le__incr__right2,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),C2: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B3)) ) ).
% mset_le_incr_right2
tff(fact_4091_mset__le__incr__right1,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),C2: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),C2)) ) ).
% mset_le_incr_right1
tff(fact_4092_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset(A),A4: multiset(A),B3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A4)),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3) ) ).
% mset_le_decr_left2
tff(fact_4093_mset__le__decr__left1,axiom,
! [A: $tType,A4: multiset(A),C2: multiset(A),B3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C2)),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3) ) ).
% mset_le_decr_left1
tff(fact_4094_mset__union__subset,axiom,
! [A: $tType,A3: multiset(A),B5: multiset(A),C6: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)),C6)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),C6)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),C6) ) ) ).
% mset_union_subset
tff(fact_4095_mset__le__distrib,axiom,
! [A: $tType,X5: multiset(A),A3: multiset(A),B5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5))
=> ~ ! [Xa5: multiset(A),Xb4: multiset(A)] :
( ( X5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa5),Xb4) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa5),A3)
=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xb4),B5) ) ) ) ).
% mset_le_distrib
tff(fact_4096_mset__le__addE,axiom,
! [A: $tType,Xs: multiset(A),Ys: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xs),Ys)
=> ~ ! [Zs: multiset(A)] : Ys != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs) ) ).
% mset_le_addE
tff(fact_4097_mset__le__subtract,axiom,
! [A: $tType,A3: multiset(A),B5: multiset(A),C6: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),minus_minus(multiset(A),A3,C6)),minus_minus(multiset(A),B5,C6)) ) ).
% mset_le_subtract
tff(fact_4098_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A4: multiset(A),B3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,A4)),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3) ) ).
% mset_le_add_mset_decr_left1
tff(fact_4099_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A4: multiset(A),B3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,A4)),B3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C2,zero_zero(multiset(A)))),B3) ) ).
% mset_le_add_mset_decr_left2
tff(fact_4100_mset__le__single__cases,axiom,
! [A: $tType,M2: multiset(A),A4: A] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M2),add_mset(A,A4,zero_zero(multiset(A))))
=> ( ( M2 != zero_zero(multiset(A)) )
=> ( M2 = add_mset(A,A4,zero_zero(multiset(A))) ) ) ) ).
% mset_le_single_cases
tff(fact_4101_mset__le__add__mset,axiom,
! [A: $tType,X: A,B5: multiset(A),C6: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,B5)),C6)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),C6)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),C6) ) ) ).
% mset_le_add_mset
tff(fact_4102_mset__le__subtract__left,axiom,
! [A: $tType,A3: multiset(A),B5: multiset(A),X5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)),X5)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),minus_minus(multiset(A),X5,A3))
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),X5) ) ) ).
% mset_le_subtract_left
tff(fact_4103_mset__le__subtract__right,axiom,
! [A: $tType,A3: multiset(A),B5: multiset(A),X5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)),X5)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),minus_minus(multiset(A),X5,B5))
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),X5) ) ) ).
% mset_le_subtract_right
tff(fact_4104_prod__mset_Oneutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: multiset(A)] :
( ! [X3: A] :
( member(A,X3,aa(multiset(A),set(A),set_mset(A),A3))
=> ( X3 = one_one(A) ) )
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A3) = one_one(A) ) ) ) ).
% prod_mset.neutral
tff(fact_4105_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),X3) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ).
% subset_mset.cInf_greatest
tff(fact_4106_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),X3) )
=> ( ! [Y2: multiset(A)] :
( ! [X4: multiset(A)] :
( member(multiset(A),X4,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),X4) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),A4) )
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = A4 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
tff(fact_4107_subset__mset_OcSup__least,axiom,
! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),Z2) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)),Z2) ) ) ).
% subset_mset.cSup_least
tff(fact_4108_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),A4) )
=> ( ! [Y2: multiset(A)] :
( ! [X4: multiset(A)] :
( member(multiset(A),X4,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),Y2) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),Y2) )
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = A4 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
tff(fact_4109_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B5: multiset(A),X5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,B5)),X5)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),minus_minus(multiset(A),X5,B5))
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),X5) ) ) ).
% mset_le_subtract_add_mset_right
tff(fact_4110_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B5: multiset(A),X5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,B5)),X5)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),minus_minus(multiset(A),X5,add_mset(A,X,zero_zero(multiset(A)))))
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),X5) ) ) ).
% mset_le_subtract_add_mset_left
tff(fact_4111_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: multiset(A)] :
( dvd_dvd(A,aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A3),one_one(A))
<=> ! [X2: A] :
( member(A,X2,aa(multiset(A),set(A),set_mset(A),A3))
=> dvd_dvd(A,X2,one_one(A)) ) ) ) ).
% is_unit_prod_mset_iff
tff(fact_4112_subset__mset_OcINF__greatest,axiom,
! [B: $tType,A: $tType,A3: set(A),M: multiset(B),F2: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(A,multiset(B),F2,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))) ) ) ).
% subset_mset.cINF_greatest
tff(fact_4113_subset__mset_OcSUP__least,axiom,
! [A: $tType,B: $tType,A3: set(A),F2: fun(A,multiset(B)),M2: multiset(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X3)),M2) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),M2) ) ) ).
% subset_mset.cSUP_least
tff(fact_4114_mset__le__mono__add__single,axiom,
! [A: $tType,A4: A,Ys: multiset(A),B3: A,Ws: multiset(A)] :
( member(A,A4,aa(multiset(A),set(A),set_mset(A),Ys))
=> ( member(A,B3,aa(multiset(A),set(A),set_mset(A),Ws))
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys),Ws)) ) ) ).
% mset_le_mono_add_single
tff(fact_4115_mset__union__subset__s,axiom,
! [A: $tType,A4: A,B5: multiset(A),C6: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),B5)),C6)
=> ( member(A,A4,aa(multiset(A),set(A),set_mset(A),C6))
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),C6) ) ) ).
% mset_union_subset_s
tff(fact_4116_mset__2dist2__cases,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset(A),B5: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5))
=> ( ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),A3)
=> ( ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),add_mset(A,B3,zero_zero(multiset(A))))),B5)
=> ( ( member(A,A4,aa(multiset(A),set(A),set_mset(A),A3))
=> ~ member(A,B3,aa(multiset(A),set(A),set_mset(A),B5)) )
=> ~ ( member(A,A4,aa(multiset(A),set(A),set_mset(A),B5))
=> ~ member(A,B3,aa(multiset(A),set(A),set_mset(A),A3)) ) ) ) ) ) ).
% mset_2dist2_cases
tff(fact_4117_subset__mset_OatLeastAtMost__singleton,axiom,
! [A: $tType,A4: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,A4) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),bot_bot(set(multiset(A)))) ).
% subset_mset.atLeastAtMost_singleton
tff(fact_4118_subset__mset_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),C2: multiset(A)] :
( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),C2),bot_bot(set(multiset(A)))) )
<=> ( ( A4 = B3 )
& ( B3 = C2 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
tff(fact_4119_subset__mset_OcINF__superset__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),G: fun(A,multiset(B)),B5: set(A),F2: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B5))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [X3: A] :
( member(A,X3,B5)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),G,X3)),aa(A,multiset(B),F2,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B5))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))) ) ) ) ) ).
% subset_mset.cINF_superset_mono
tff(fact_4120_subset__mset_OcSUP__subset__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),G: fun(A,multiset(B)),B5: set(A),F2: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B5))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X3)),aa(A,multiset(B),G,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B5))) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
tff(fact_4121_subset__mset_Obdd__above__empty,axiom,
! [A: $tType] : condit8047198070973881523_above(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).
% subset_mset.bdd_above_empty
tff(fact_4122_subset__mset_Obdd__below__empty,axiom,
! [A: $tType] : condit8119078960628432327_below(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).
% subset_mset.bdd_below_empty
tff(fact_4123_subset__mset_OatLeastatMost__empty__iff,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] :
( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3) ) ).
% subset_mset.atLeastatMost_empty_iff
tff(fact_4124_subset__mset_OatLeastatMost__empty__iff2,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),B3) ) ).
% subset_mset.atLeastatMost_empty_iff2
tff(fact_4125_subset__mset_OcInf__le__cSup,axiom,
! [A: $tType,A3: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)) ) ) ) ).
% subset_mset.cInf_le_cSup
tff(fact_4126_subset__mset_OcSup__mono,axiom,
! [A: $tType,B5: set(multiset(A)),A3: set(multiset(A))] :
( ( B5 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( ! [B2: multiset(A)] :
( member(multiset(A),B2,B5)
=> ? [X4: multiset(A)] :
( member(multiset(A),X4,A3)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),X4) ) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)) ) ) ) ).
% subset_mset.cSup_mono
tff(fact_4127_subset__mset_OcSup__le__iff,axiom,
! [A: $tType,S: set(multiset(A)),A4: multiset(A)] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S)),A4)
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,S)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),A4) ) ) ) ) ).
% subset_mset.cSup_le_iff
tff(fact_4128_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B5)
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A3),B5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5)) ) ) ) ).
% subset_mset.cSup_subset_mono
tff(fact_4129_subset__mset_Ole__cInf__iff,axiom,
! [A: $tType,S: set(multiset(A)),A4: multiset(A)] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S))
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,S)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),X2) ) ) ) ) ).
% subset_mset.le_cInf_iff
tff(fact_4130_subset__mset_OcInf__mono,axiom,
! [A: $tType,B5: set(multiset(A)),A3: set(multiset(A))] :
( ( B5 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A3)
=> ( ! [B2: multiset(A)] :
( member(multiset(A),B2,B5)
=> ? [X4: multiset(A)] :
( member(multiset(A),X4,A3)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),B2) ) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5)) ) ) ) ).
% subset_mset.cInf_mono
tff(fact_4131_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B5)
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A3),B5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)) ) ) ) ).
% subset_mset.cInf_superset_mono
tff(fact_4132_subset__mset_OatLeastAtMost__singleton_H,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] :
( ( A4 = B3 )
=> ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3) = aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),bot_bot(set(multiset(A)))) ) ) ).
% subset_mset.atLeastAtMost_singleton'
tff(fact_4133_subset__mset_OcSUP__mono,axiom,
! [A: $tType,B: $tType,C: $tType,A3: set(A),G: fun(C,multiset(B)),B5: set(C),F2: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),B5))
=> ( ! [N3: A] :
( member(A,N3,A3)
=> ? [X4: C] :
( member(C,X4,B5)
& aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,N3)),aa(C,multiset(B),G,X4)) ) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),B5))) ) ) ) ).
% subset_mset.cSUP_mono
tff(fact_4134_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A3: set(A),F2: fun(A,multiset(B)),U: multiset(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),U)
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F2,X2)),U) ) ) ) ) ).
% subset_mset.cSUP_le_iff
tff(fact_4135_subset__mset_OcINF__mono,axiom,
! [C: $tType,B: $tType,A: $tType,B5: set(A),F2: fun(C,multiset(B)),A3: set(C),G: fun(A,multiset(B))] :
( ( B5 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F2),A3))
=> ( ! [M3: A] :
( member(A,M3,B5)
=> ? [X4: C] :
( member(C,X4,A3)
& aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(C,multiset(B),F2,X4)),aa(A,multiset(B),G,M3)) ) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),B5))) ) ) ) ).
% subset_mset.cINF_mono
tff(fact_4136_subset__mset_Ole__cINF__iff,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),U: multiset(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3)))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(A,multiset(B),F2,X2)) ) ) ) ) ).
% subset_mset.le_cINF_iff
tff(fact_4137_subset__mset_OcSup__cInf,axiom,
! [A: $tType,S: set(multiset(A))] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_rn(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).
% subset_mset.cSup_cInf
tff(fact_4138_subset__mset_OcInf__cSup,axiom,
! [A: $tType,S: set(multiset(A))] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_ro(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).
% subset_mset.cInf_cSup
tff(fact_4139_subset__mset_Omono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F2: fun(multiset(A),B),A3: fun(C,multiset(A)),I: set(C)] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A3),I))
=> ( ( I != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A3),I)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F2),A3)),I))) ) ) ) ) ).
% subset_mset.mono_cINF
tff(fact_4140_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F2: fun(multiset(A),B),A3: fun(C,multiset(A)),I: set(C)] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A3),I))
=> ( ( I != bot_bot(set(C)) )
=> 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,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F2),A3)),I))),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A3),I)))) ) ) ) ) ).
% subset_mset.mono_cSUP
tff(fact_4141_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [F2: fun(multiset(A),B),A3: set(multiset(A))] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F2),A3))) ) ) ) ) ).
% subset_mset.mono_cInf
tff(fact_4142_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [F2: fun(multiset(A),B),A3: set(multiset(A))] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F2)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F2),A3))),aa(multiset(A),B,F2,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3))) ) ) ) ) ).
% subset_mset.mono_cSup
tff(fact_4143_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [Less_eq: fun(A,fun(A,$o))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).
% order.mono.cong
tff(fact_4144_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,A3: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),aTP_Lamp_rr(multiset(A),set(A))),A3))) ) ) ).
% bdd_above_multiset_imp_finite_support
tff(fact_4145_Sup__multiset__in__multiset,axiom,
! [A: $tType,A3: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_rt(set(multiset(A)),fun(A,$o),A3))) ) ) ).
% Sup_multiset_in_multiset
tff(fact_4146_subset__mset_OcINF__union,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B5))
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B5))) ) ) ) ) ) ).
% subset_mset.cINF_union
tff(fact_4147_subset__mset_OcINF__insert,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),A4: A] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),F2,A4)),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))) ) ) ) ).
% subset_mset.cINF_insert
tff(fact_4148_mset__empty__count,axiom,
! [A: $tType,M2: multiset(A)] :
( ! [P6: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),P6) = zero_zero(nat)
<=> ( M2 = zero_zero(multiset(A)) ) ) ).
% mset_empty_count
tff(fact_4149_count__ne__remove,axiom,
! [A: $tType,X: A,T5: A,S: multiset(A)] :
( ( X != T5 )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),S),X) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),minus_minus(multiset(A),S,add_mset(A,T5,zero_zero(multiset(A))))),X) ) ) ).
% count_ne_remove
tff(fact_4150_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set(multiset(A)),X4: A] :
aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X4) = $ite(aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_ru(A,fun(fun(A,nat),nat),X4)),aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X)))) ).
% Inf_multiset.rep_eq
tff(fact_4151_count__Inf__multiset__nonempty,axiom,
! [A: $tType,A3: set(multiset(A)),X: A] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_rs(A,fun(multiset(A),nat),X)),A3)) ) ) ).
% count_Inf_multiset_nonempty
tff(fact_4152_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set(A),A4: A] :
( aa(set(A),$o,finite_finite2(A),S)
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A4) = $ite(member(A,A4,S),one_one(nat),zero_zero(nat)) ) ) ).
% count_mset_set_finite_iff
tff(fact_4153_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A3)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B5)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A3),B5) != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A3),B5))) ) ) ) ).
% subset_mset.less_eq_cInf_inter
tff(fact_4154_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A3)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B5)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A3),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5)) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
tff(fact_4155_subset__mset_OcInf__insert,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A4),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ) ).
% subset_mset.cInf_insert
tff(fact_4156_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A4,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A4),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5))) ) ) ).
% subset_mset.cInf_insert_If
tff(fact_4157_subset__mset_OcINF__inf__distrib,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A3))
=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A3))) = aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_rv(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F2),G)),A3)) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
tff(fact_4158_count__Sup__multiset__nonempty,axiom,
! [A: $tType,A3: set(multiset(A)),X: A] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_rs(A,fun(multiset(A),nat),X)),A3)) ) ) ) ).
% count_Sup_multiset_nonempty
tff(fact_4159_Sup__multiset__def,axiom,
! [A: $tType,A3: set(multiset(A))] :
aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3) = $ite(
( ( A3 != bot_bot(set(multiset(A))) )
& condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3) ),
aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_rw(set(multiset(A)),fun(A,nat),A3)),
zero_zero(multiset(A)) ) ).
% Sup_multiset_def
tff(fact_4160_size__multiset__eq,axiom,
! [A: $tType,F2: fun(A,nat),M2: multiset(A)] : size_multiset(A,F2,M2) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(multiset(A),fun(A,nat),aTP_Lamp_rx(fun(A,nat),fun(multiset(A),fun(A,nat)),F2),M2)),aa(multiset(A),set(A),set_mset(A),M2)) ).
% size_multiset_eq
tff(fact_4161_count__image__mset,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A3: multiset(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F2),A3)),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),A3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),A3))) ).
% count_image_mset
tff(fact_4162_subset__mset_OcSUP__union,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),B5: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( ( B5 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B5))
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),B5))) ) ) ) ) ) ).
% subset_mset.cSUP_union
tff(fact_4163_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_ak(B,A)),A3)) = one_one(A) ) ).
% prod_mset.neutral_const
tff(fact_4164_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),X: B,A3: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),add_mset(B,X,A3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A3))) ) ).
% prod_mset.insert
tff(fact_4165_mset__map__id,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),G: fun(A,B),X5: multiset(A)] :
( ! [X3: A] : aa(B,A,F2,aa(A,B,G,X3)) = X3
=> ( aa(multiset(B),multiset(A),image_mset(B,A,F2),aa(multiset(A),multiset(B),image_mset(A,B,G),X5)) = X5 ) ) ).
% mset_map_id
tff(fact_4166_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),P: multiset(B),M1: multiset(A),M22: multiset(A)] :
( ( aa(multiset(B),multiset(A),image_mset(B,A,F2),P) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M1),M22) )
=> ~ ! [P12: multiset(B),P22: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
=> ( ( aa(multiset(B),multiset(A),image_mset(B,A,F2),P12) = M1 )
=> ( aa(multiset(B),multiset(A),image_mset(B,A,F2),P22) != M22 ) ) ) ) ).
% mset_map_split_orig
tff(fact_4167_prod__mset_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [A3: multiset(A),B5: multiset(A),G: fun(A,B)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),B5) = zero_zero(multiset(A)) )
=> ( aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),B5))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),A3))),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),aa(multiset(A),multiset(B),image_mset(A,B,G),B5))) ) ) ) ).
% prod_mset.union_disjoint
tff(fact_4168_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),P: multiset(B),M1: multiset(A),M22: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F2),P)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M1),M22))
=> ~ ! [P12: multiset(B),P22: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F2),P12)),M1)
=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F2),P22)),M22) ) ) ) ).
% mset_map_split_orig_le
tff(fact_4169_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M2: multiset(product_prod(A,B)),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C))] :
( ! [X3: A,Y2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M2))
=> ( aa(B,C,aa(A,fun(B,C),F2,X3),Y2) = aa(B,C,aa(A,fun(B,C),G,X3),Y2) ) )
=> ( aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)),M2) = aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),M2) ) ) ).
% image_mset_cong_pair
tff(fact_4170_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Ha: fun(B,A),A3: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cn(fun(B,A),fun(fun(B,A),fun(B,A)),G),Ha)),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A3))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,Ha),A3))) ) ).
% prod_mset.distrib
tff(fact_4171_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A3: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ry(B,fun(A,fun(B,A)),Y),C2)),A3)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A3),Y)) ) ).
% prod_mset_delta
tff(fact_4172_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A3: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_rz(B,fun(A,fun(B,A)),Y),C2)),A3)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A3),Y)) ) ).
% prod_mset_delta'
tff(fact_4173_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( ( A3 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B5)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A3),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5)) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
tff(fact_4174_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A3)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B5)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A3),B5) != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A3),B5))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A3)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5))) ) ) ) ).
% subset_mset.cSup_inter_less_eq
tff(fact_4175_subset__mset_OcSup__insert,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A4),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)) ) ) ) ).
% subset_mset.cSup_insert
tff(fact_4176_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X5: set(multiset(A)),A4: multiset(A)] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A4,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A4),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5))) ) ) ).
% subset_mset.cSup_insert_If
tff(fact_4177_subset__mset_OSUP__sup__distrib,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A3))
=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),A3))) = aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_sa(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F2),G)),A3)) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
tff(fact_4178_type__definition__multiset,axiom,
! [A: $tType] : type_definition(multiset(A),fun(A,nat),count(A),abs_multiset(A),aa(fun(fun(A,nat),$o),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_sc(fun(A,nat),$o))) ).
% type_definition_multiset
tff(fact_4179_subset__mset_OcSUP__insert,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),A4: A] :
( ( A3 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),F2,A4)),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F2),A3))) ) ) ) ).
% subset_mset.cSUP_insert
tff(fact_4180_Inf__multiset__def,axiom,
! [A: $tType] : complete_Inf_Inf(multiset(A)) = aa(fun(set(fun(A,nat)),fun(A,nat)),fun(set(multiset(A)),multiset(A)),map_fun(set(multiset(A)),set(fun(A,nat)),fun(A,nat),multiset(A),image2(multiset(A),fun(A,nat),count(A)),abs_multiset(A)),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat))) ).
% Inf_multiset_def
tff(fact_4181_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_se(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),A3),X))) ) ) ) ).
% subset_mset.inf_Sup1_distrib
tff(fact_4182_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B5)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_sf(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),A3),B5))) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
tff(fact_4183_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_sg(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),A3),X))) ) ) ) ).
% subset_mset.sup_Inf1_distrib
tff(fact_4184_subset__mset_OInf__fin_Osingleton,axiom,
! [A: $tType,X: multiset(A)] : lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.Inf_fin.singleton
tff(fact_4185_subset__mset_OSup__fin_Osingleton,axiom,
! [A: $tType,X: multiset(A)] : lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.Sup_fin.singleton
tff(fact_4186_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) ) ) ) ).
% subset_mset.Inf_fin.insert
tff(fact_4187_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)) ) ) ) ).
% subset_mset.Sup_fin.insert
tff(fact_4188_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,A3: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
tff(fact_4189_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,Ha: fun(multiset(A),multiset(A)),N: set(multiset(A))] :
( ! [X3: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),Ha,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),Ha,X3)),aa(multiset(A),multiset(A),Ha,Y2))
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),N)
=> ( ( N != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),Ha,lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),N)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),Ha),N)) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
tff(fact_4190_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,Ha: fun(multiset(A),multiset(A)),N: set(multiset(A))] :
( ! [X3: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),Ha,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),Ha,X3)),aa(multiset(A),multiset(A),Ha,Y2))
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),N)
=> ( ( N != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),Ha,lattic4630905495605216202up_fin(multiset(A),union_mset(A),N)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),Ha),N)) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
tff(fact_4191_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3))
=> ! [A12: multiset(A)] :
( member(multiset(A),A12,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A12) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
tff(fact_4192_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( ! [A5: multiset(A)] :
( member(multiset(A),A5,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A5) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) ) ) ) ).
% subset_mset.Inf_fin.boundedI
tff(fact_4193_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3))
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),X2) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
tff(fact_4194_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),X)
=> ! [A12: multiset(A)] :
( member(multiset(A),A12,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A12),X) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
tff(fact_4195_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( ! [A5: multiset(A)] :
( member(multiset(A),A5,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),X) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),X) ) ) ) ).
% subset_mset.Sup_fin.boundedI
tff(fact_4196_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),X)
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,A3)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),X) ) ) ) ) ).
% subset_mset.Sup_fin.bounded_iff
tff(fact_4197_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
=> ( ( X5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),X5) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
tff(fact_4198_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),X5)
=> ( ( X5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),X5) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
tff(fact_4199_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = $ite(minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ).
% subset_mset.Inf_fin.insert_remove
tff(fact_4200_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( member(multiset(A),X,A3)
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3) = $ite(minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ) ).
% subset_mset.Inf_fin.remove
tff(fact_4201_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ~ member(multiset(A),X,A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
tff(fact_4202_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,A3: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A),Y2: multiset(A)] : member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y2),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),Y2),bot_bot(set(multiset(A))))))
=> member(multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3),A3) ) ) ) ).
% subset_mset.Inf_fin.closed
tff(fact_4203_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = $ite(minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ).
% subset_mset.Sup_fin.insert_remove
tff(fact_4204_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( member(multiset(A),X,A3)
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3) = $ite(minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),minus_minus(set(multiset(A)),A3,aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))))))) ) ) ) ).
% subset_mset.Sup_fin.remove
tff(fact_4205_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,A3: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ~ member(multiset(A),X,A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)) ) ) ) ) ).
% subset_mset.Sup_fin.insert_not_elem
tff(fact_4206_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,A3: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A),Y2: multiset(A)] : member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y2),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),Y2),bot_bot(set(multiset(A))))))
=> member(multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3),A3) ) ) ) ).
% subset_mset.Sup_fin.closed
tff(fact_4207_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),B5),A3)
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B5)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3) ) ) ) ) ).
% subset_mset.Inf_fin.subset
tff(fact_4208_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),B5),A3)
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B5)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3) ) ) ) ) ).
% subset_mset.Sup_fin.subset
tff(fact_4209_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A3),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B5)) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
tff(fact_4210_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A3),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B5)) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
tff(fact_4211_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A3),B5)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B5)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
tff(fact_4212_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),A3),B5)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),A3)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),B5)) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
tff(fact_4213_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A3: set(multiset(A)),B5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),A3)
=> ( ( A3 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite2(multiset(A)),B5)
=> ( ( B5 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),A3)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),B5)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_sh(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),A3),B5))) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
tff(fact_4214_multiset_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),A4: multiset(A),B3: multiset(B)] :
( aa(multiset(B),$o,aa(multiset(A),fun(multiset(B),$o),rel_mset(A,B,R3),A4),B3)
<=> ? [Z3: multiset(product_prod(A,B))] :
( member(multiset(product_prod(A,B)),Z3,aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_si(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R3)))
& ( aa(multiset(product_prod(A,B)),multiset(A),image_mset(product_prod(A,B),A,product_fst(A,B)),Z3) = A4 )
& ( aa(multiset(product_prod(A,B)),multiset(B),image_mset(product_prod(A,B),B,product_snd(A,B)),Z3) = B3 ) ) ) ).
% multiset.in_rel
tff(fact_4215_sum__mset__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,A3: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_bd(A,fun(B,A),Y)),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(multiset(B),nat,size_size(multiset(B)),A3))),Y) ) ).
% sum_mset_constant
tff(fact_4216_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,A3: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_sj(B,fun(A,fun(B,A)),Y),C2)),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A3),Y))) ) ).
% sum_mset_delta
tff(fact_4217_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,A3: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_sk(B,fun(A,fun(B,A)),Y),C2)),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A3),Y))) ) ).
% sum_mset_delta'
tff(fact_4218_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat,A4: A] : comm_m7189776963980413722m_mset(A,replicate_mset(A,N2,A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),A4) ) ).
% sum_mset_replicate_mset
tff(fact_4219_sum__mset__product,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add(B)
& times(B)
& semiring_0(A) )
=> ! [F2: fun(B,A),A3: multiset(B),G: fun(C,A),B5: multiset(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F2),A3))),comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,G),B5))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_sm(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),F2),G),B5)),A3)) ) ).
% sum_mset_product
tff(fact_4220_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),M2: multiset(B),C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F2),M2))),C2) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F2),C2)),M2)) ) ).
% sum_mset_distrib_right
tff(fact_4221_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F2: fun(B,A),M2: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F2),M2))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cr(A,fun(fun(B,A),fun(B,A)),C2),F2)),M2)) ) ).
% sum_mset_distrib_left
tff(fact_4222_wcount__def,axiom,
! [A: $tType,F2: fun(A,nat),M2: multiset(A),X4: A] : wcount(A,F2,M2,X4) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),X4)),aa(nat,nat,suc,aa(A,nat,F2,X4))) ).
% wcount_def
tff(fact_4223_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X4: A] :
( aa(A,$o,rangep(B,A,aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X4)
<=> member(A,X4,aa(set(product_prod(B,A)),set(A),range2(B,A),R2)) ) ).
% Rangep_Range_eq
tff(fact_4224_Range__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(B),range2(A,B),X4) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),X4))) ).
% Range_def
tff(fact_4225_aboveS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] : order_aboveS(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_sn(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A4)) ).
% aboveS_def
tff(fact_4226_RangepE,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),B3: B] :
( aa(B,$o,rangep(A,B,R2),B3)
=> ~ ! [A5: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A5),B3) ) ).
% RangepE
tff(fact_4227_RangePI,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,A4),B3)
=> aa(B,$o,rangep(A,B,R2),B3) ) ).
% RangePI
tff(fact_4228_Rangep_Osimps,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: B] :
( aa(B,$o,rangep(A,B,R2),A4)
<=> ? [A7: A,B7: B] :
( ( A4 = B7 )
& aa(B,$o,aa(A,fun(B,$o),R2,A7),B7) ) ) ).
% Rangep.simps
tff(fact_4229_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: B] :
( aa(B,$o,rangep(A,B,R2),A4)
=> ~ ! [A5: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A5),A4) ) ).
% Rangep.cases
tff(fact_4230_relInvImage__def,axiom,
! [B: $tType,A: $tType,A3: set(A),R3: set(product_prod(B,B)),F2: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A3,R3,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_so(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),A3),R3),F2)) ).
% relInvImage_def
tff(fact_4231_scomp__unfold,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,X4: fun(A,product_prod(B,C)),Xa3: fun(B,fun(C,D)),Xb2: A] : aa(A,D,product_scomp(A,B,C,D,X4,Xa3),Xb2) = aa(C,D,aa(B,fun(C,D),Xa3,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X4,Xb2))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X4,Xb2))) ).
% scomp_unfold
tff(fact_4232_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( antisymp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> antisym(A,R2) ) ).
% antisymp_antisym_eq
tff(fact_4233_sub__BitM__One__eq,axiom,
! [N2: num] : neg_numeral_sub(int,bitM(N2),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,N2,one2)) ).
% sub_BitM_One_eq
tff(fact_4234_scomp__apply,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(B,product_prod(C,D)),G: fun(C,fun(D,A)),X: B] : aa(B,A,product_scomp(B,C,D,A,F2,G),X) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G),aa(B,product_prod(C,D),F2,X)) ).
% scomp_apply
tff(fact_4235_scomp__scomp,axiom,
! [A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,C: $tType,F2: fun(A,product_prod(E,F3)),G: fun(E,fun(F3,product_prod(C,D))),Ha: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F3,product_prod(C,D),F2,G),Ha) = product_scomp(A,E,F3,B,F2,aa(fun(C,fun(D,B)),fun(E,fun(F3,B)),aTP_Lamp_sp(fun(E,fun(F3,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F3,B))),G),Ha)) ).
% scomp_scomp
tff(fact_4236_antisympD,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),A4: A,B3: A] :
( antisymp(A,R2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,B3),A4)
=> ( A4 = B3 ) ) ) ) ).
% antisympD
tff(fact_4237_antisympI,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y2),X3)
=> ( X3 = Y2 ) ) )
=> antisymp(A,R2) ) ).
% antisympI
tff(fact_4238_antisymp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( antisymp(A,R2)
<=> ! [X2: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,X2),Y3)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y3),X2)
=> ( X2 = Y3 ) ) ) ) ).
% antisymp_def
tff(fact_4239_antisymp__equality,axiom,
! [A: $tType] : antisymp(A,fequal(A)) ).
% antisymp_equality
tff(fact_4240_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X,product_Pair(B,C)) = X ).
% scomp_Pair
tff(fact_4241_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F2: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F2) = aa(C,fun(A,B),F2,X) ).
% Pair_scomp
tff(fact_4242_scomp__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F2: fun(A,product_prod(C,D)),G: fun(C,fun(D,B)),X4: A] : aa(A,B,product_scomp(A,C,D,B,F2,G),X4) = aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),G),aa(A,product_prod(C,D),F2,X4)) ).
% scomp_def
tff(fact_4243_antisymp__less__eq,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R2),S2)
=> ( antisymp(A,S2)
=> antisymp(A,R2) ) ) ).
% antisymp_less_eq
tff(fact_4244_antisym__bot,axiom,
! [A: $tType] : antisymp(A,bot_bot(fun(A,fun(A,$o)))) ).
% antisym_bot
tff(fact_4245_numeral__BitM,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N2: num] : aa(num,A,numeral_numeral(A),bitM(N2)) = minus_minus(A,aa(num,A,numeral_numeral(A),bit0(N2)),one_one(A)) ) ).
% numeral_BitM
tff(fact_4246_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,product_prod(B,C)),G: fun(A,B),Ha: fun(A,C)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F2) = G )
=> ( ( Ha = aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F2) )
<=> ( bNF_convol(A,B,C,G,Ha) = F2 ) ) ) ).
% convol_expand_snd'
tff(fact_4247_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,product_prod(B,C)),G: fun(A,B)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F2) = G )
=> ( bNF_convol(A,B,C,G,aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F2)) = F2 ) ) ).
% convol_expand_snd
tff(fact_4248_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A3: set(B),F2: fun(B,A)] : bind(B,A,A3,aTP_Lamp_ez(fun(B,A),fun(B,set(A)),F2)) = aa(set(B),set(A),image2(B,A,F2),A3) ).
% bind_singleton_conv_image
tff(fact_4249_set__decode__zero,axiom,
nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).
% set_decode_zero
tff(fact_4250_empty__bind,axiom,
! [B: $tType,A: $tType,F2: fun(B,set(A))] : bind(B,A,bot_bot(set(B)),F2) = bot_bot(set(A)) ).
% empty_bind
tff(fact_4251_Set_Obind__bind,axiom,
! [C: $tType,A: $tType,B: $tType,A3: set(C),B5: fun(C,set(B)),C6: fun(B,set(A))] : bind(B,A,bind(C,B,A3,B5),C6) = bind(C,A,A3,aa(fun(B,set(A)),fun(C,set(A)),aTP_Lamp_sq(fun(C,set(B)),fun(fun(B,set(A)),fun(C,set(A))),B5),C6)) ).
% Set.bind_bind
tff(fact_4252_convol__def,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(A,B),G: fun(A,C),X4: A] : aa(A,product_prod(B,C),bNF_convol(A,B,C,F2,G),X4) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,F2,X4)),aa(A,C,G,X4)) ).
% convol_def
tff(fact_4253_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(C,B),G: fun(C,A),X: C] : aa(product_prod(B,A),A,product_snd(B,A),aa(C,product_prod(B,A),bNF_convol(C,B,A,F2,G),X)) = aa(C,A,G,X) ).
% snd_convol'
tff(fact_4254_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( bind(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)) = B5 ) ) ).
% nonempty_bind_const
tff(fact_4255_bind__const,axiom,
! [B: $tType,A: $tType,A3: set(B),B5: set(A)] :
bind(B,A,A3,aTP_Lamp_au(set(A),fun(B,set(A)),B5)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),B5) ).
% bind_const
tff(fact_4256_Set_Obind__def,axiom,
! [A: $tType,B: $tType,A3: set(B),F2: fun(B,set(A))] : bind(B,A,A3,F2) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,set(A)),fun(A,$o),aTP_Lamp_sr(set(B),fun(fun(B,set(A)),fun(A,$o)),A3),F2)) ).
% Set.bind_def
tff(fact_4257_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(A,C)] : aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),bNF_convol(A,B,C,F2,G)) = F2 ).
% fst_convol
tff(fact_4258_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,C),G: fun(A,B)] : aa(fun(A,product_prod(C,B)),fun(A,B),comp(product_prod(C,B),B,A,product_snd(C,B)),bNF_convol(A,C,B,F2,G)) = G ).
% snd_convol
tff(fact_4259_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F2: fun(C,A),G: fun(D,B),R3: fun(A,fun(B,$o))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),bNF_convol(product_prod(C,D),A,B,aa(fun(product_prod(C,D),C),fun(product_prod(C,D),A),comp(C,A,product_prod(C,D),F2),product_fst(C,D)),aa(fun(product_prod(C,D),D),fun(product_prod(C,D),B),comp(D,B,product_prod(C,D),G),product_snd(C,D)))),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),bNF_vimage2p(C,A,D,B,$o,F2,G,R3))))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3))) ).
% convol_image_vimage2p
tff(fact_4260_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.semilattice_neutr_axioms
tff(fact_4261_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M2) = fold_mset(A,A,times_times(A),one_one(A),M2) ) ).
% prod_mset.eq_fold
tff(fact_4262_Func__empty,axiom,
! [B: $tType,A: $tType,B5: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B5) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert2(fun(A,B)),aTP_Lamp_ss(A,B)),bot_bot(set(fun(A,B)))) ).
% Func_empty
tff(fact_4263_mod__h__bot__normalize,axiom,
! [A: $tType,Ha: heap_ext(product_unit),P: assn] :
( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),Ha)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),bot_bot(set(nat))))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_normalize
tff(fact_4264_predicate2D__vimage2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,$o)),F2: fun(A,C),G: fun(B,D),S: fun(C,fun(D,$o)),X: A,Y: B] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R3),bNF_vimage2p(A,C,B,D,$o,F2,G,S))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,X),Y)
=> aa(D,$o,aa(C,fun(D,$o),S,aa(A,C,F2,X)),aa(B,D,G,Y)) ) ) ).
% predicate2D_vimage2p
tff(fact_4265_vimage2pI,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(A,fun(B,$o)),F2: fun(C,A),X: C,G: fun(D,B),Y: D] :
( aa(B,$o,aa(A,fun(B,$o),R3,aa(C,A,F2,X)),aa(D,B,G,Y))
=> aa(D,$o,aa(C,fun(D,$o),bNF_vimage2p(C,A,D,B,$o,F2,G,R3),X),Y) ) ).
% vimage2pI
tff(fact_4266_vimage2p__def,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F2: fun(A,D),G: fun(B,E),R3: fun(D,fun(E,C)),X4: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),bNF_vimage2p(A,D,B,E,C,F2,G,R3),X4),Xa3) = aa(E,C,aa(D,fun(E,C),R3,aa(A,D,F2,X4)),aa(B,E,G,Xa3)) ).
% vimage2p_def
tff(fact_4267_rel__fun__iff__leq__vimage2p,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,R3: fun(A,fun(B,$o)),S: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D)] :
( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,R3,S),F2),G)
<=> aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R3),bNF_vimage2p(A,C,B,D,$o,F2,G,S)) ) ).
% rel_fun_iff_leq_vimage2p
tff(fact_4268_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_4269_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: fun(A,B),Abs1: fun(B,A),Rep22: fun(C,D),Abs22: fun(D,C),R3: fun(B,fun(D,$o))] :
( type_definition(A,B,Rep1,Abs1,top_top(set(B)))
=> ( type_definition(C,D,Rep22,Abs22,top_top(set(D)))
=> aa(fun(D,C),$o,aa(fun(B,A),fun(fun(D,C),$o),bNF_rel_fun(B,D,A,C,R3,bNF_vimage2p(A,B,C,D,$o,Rep1,Rep22,R3)),Abs1),Abs22) ) ) ).
% Abs_transfer
tff(fact_4270_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),G: fun(C,D),P: fun(D,$o),Ha: fun(D,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( bNF_vimage2p(C,D,B,A,$o,G,Abs,bNF_Grp(D,A,aa(fun(D,$o),set(D),collect(D),P),Ha)) = bNF_Grp(C,B,aa(fun(C,$o),set(C),collect(C),aa(fun(D,$o),fun(C,$o),aTP_Lamp_st(fun(C,D),fun(fun(D,$o),fun(C,$o)),G),P)),aa(fun(C,D),fun(C,B),comp(D,B,C,aa(fun(D,A),fun(D,B),comp(A,B,D,Rep),Ha)),G)) ) ) ).
% type_copy_vimage2p_Grp_Abs
tff(fact_4271_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F2: fun(C,D),P: fun(D,$o),Ha: fun(D,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( bNF_vimage2p(C,D,A,B,$o,F2,Rep,bNF_Grp(D,B,aa(fun(D,$o),set(D),collect(D),P),Ha)) = bNF_Grp(C,A,aa(fun(C,$o),set(C),collect(C),aa(fun(D,$o),fun(C,$o),aTP_Lamp_st(fun(C,D),fun(fun(D,$o),fun(C,$o)),F2),P)),aa(fun(C,D),fun(C,A),comp(D,A,C,aa(fun(D,B),fun(D,A),comp(B,A,D,Abs),Ha)),F2)) ) ) ).
% type_copy_vimage2p_Grp_Rep
tff(fact_4272_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z2: product_prod(A,B),A3: set(A),F2: fun(A,B)] :
( member(product_prod(A,B),Z2,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),bNF_Grp(A,B,A3,F2))))
=> ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),B),comp(A,B,product_prod(A,B),F2),product_fst(A,B)),Z2) = aa(product_prod(A,B),B,product_snd(A,B),Z2) ) ) ).
% Collect_case_prod_Grp_eqD
tff(fact_4273_power__int__add__1_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int] :
( ( ( X != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).
% power_int_add_1'
tff(fact_4274_power__int__1__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N2: int] : power_int(A,one_one(A),N2) = one_one(A) ) ).
% power_int_1_left
tff(fact_4275_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( field(A)
=> ! [W: num,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),M)),power_int(A,Y,M)) ) ).
% power_int_mult_distrib_numeral1
tff(fact_4276_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,W: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W)),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,aa(num,A,numeral_numeral(A),W),M)) ) ).
% power_int_mult_distrib_numeral2
tff(fact_4277_power__int__0__right,axiom,
! [A: $tType] :
( ( inverse(A)
& power(A) )
=> ! [X: A] : power_int(A,X,zero_zero(int)) = one_one(A) ) ).
% power_int_0_right
tff(fact_4278_power__int__mult__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N2: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N2)) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),M),N2))) ) ).
% power_int_mult_numeral
tff(fact_4279_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),B3)) = B3 ) ).
% power_int_minus_one_mult_self'
tff(fact_4280_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)) = one_one(A) ) ).
% power_int_minus_one_mult_self
tff(fact_4281_power__int__add__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),power_int(A,X,aa(num,int,numeral_numeral(int),N2))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2))) ) ).
% power_int_add_numeral
tff(fact_4282_power__int__add__numeral2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N2: num,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N2))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N2)))),B3) ) ).
% power_int_add_numeral2
tff(fact_4283_GrpI,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),X: B,Y: A,A3: set(B)] :
( ( aa(B,A,F2,X) = Y )
=> ( member(B,X,A3)
=> aa(A,$o,aa(B,fun(A,$o),bNF_Grp(B,A,A3,F2),X),Y) ) ) ).
% GrpI
tff(fact_4284_GrpE,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X: A,Y: B] :
( aa(B,$o,aa(A,fun(B,$o),bNF_Grp(A,B,A3,F2),X),Y)
=> ~ ( ( aa(A,B,F2,X) = Y )
=> ~ member(A,X,A3) ) ) ).
% GrpE
tff(fact_4285_Grp__def,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),bNF_Grp(A,B,A3,F2),X4),Xa3)
<=> ( ( Xa3 = aa(A,B,F2,X4) )
& member(A,X4,A3) ) ) ).
% Grp_def
tff(fact_4286_power__int__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y,M)) ) ).
% power_int_mult_distrib
tff(fact_4287_power__int__commutes,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N2: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N2)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N2)) ) ).
% power_int_commutes
tff(fact_4288_power__int__mult,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int,N2: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M),N2)) = power_int(A,power_int(A,X,M),N2) ) ).
% power_int_mult
tff(fact_4289_rel__filter__Grp,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : rel_filter(A,B,bNF_Grp(A,B,top_top(set(A)),F2)) = bNF_Grp(filter(A),filter(B),top_top(set(filter(A))),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2)) ).
% rel_filter_Grp
tff(fact_4290_power__int__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N2: int] : power_int(A,divide_divide(A,one_one(A),X),N2) = divide_divide(A,one_one(A),power_int(A,X,N2)) ) ).
% power_int_one_over
tff(fact_4291_Grp__UNIV__idI,axiom,
! [A: $tType,X: A,Y: A] :
( ( X = Y )
=> aa(A,$o,aa(A,fun(A,$o),bNF_Grp(A,A,top_top(set(A)),id(A)),X),Y) ) ).
% Grp_UNIV_idI
tff(fact_4292_eq__alt,axiom,
! [A: $tType] : fequal(A) = bNF_Grp(A,A,top_top(set(A)),id(A)) ).
% eq_alt
tff(fact_4293_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z2: product_prod(A,B),A3: set(A),F2: fun(A,B)] :
( member(product_prod(A,B),Z2,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),bNF_Grp(A,B,A3,F2))))
=> member(A,aa(product_prod(A,B),A,product_fst(A,B),Z2),A3) ) ).
% Collect_case_prod_Grp_in
tff(fact_4294_Grp__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(A),F2: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),bNF_Grp(A,B,A3,F2)),bNF_Grp(A,B,B5,F2)) ) ).
% Grp_mono
tff(fact_4295_power__int__0__left__If,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int] :
power_int(A,zero_zero(A),M) = $ite(M = zero_zero(int),one_one(A),zero_zero(A)) ) ).
% power_int_0_left_If
tff(fact_4296_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N2: int,N: int,A4: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A4,N2)),power_int(A,A4,N)) ) ) ) ).
% power_int_increasing
tff(fact_4297_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N2: int,N: int,A4: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A4,N2)),power_int(A,A4,N)) ) ) ) ).
% power_int_strict_increasing
tff(fact_4298_power__int__minus__one__minus,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N2) ) ).
% power_int_minus_one_minus
tff(fact_4299_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: int,B3: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),minus_minus(int,A4,B3)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),minus_minus(int,B3,A4)) ) ).
% power_int_minus_one_diff_commute
tff(fact_4300_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N2: int,N: int,A4: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A4,N)),power_int(A,A4,N2)) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_4301_one__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,X,N2)) ) ) ) ).
% one_le_power_int
tff(fact_4302_one__less__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A4: A,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A4)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A4,N2)) ) ) ) ).
% one_less_power_int
tff(fact_4303_power__int__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int,N2: int] :
( ( ( X != zero_zero(A) )
| ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N2) != zero_zero(int) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N2)) ) ) ) ).
% power_int_add
tff(fact_4304_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X: A,A3: set(A),G: fun(A,B)] :
( member(A,X,A3)
=> member(product_prod(A,B),aa(A,product_prod(A,B),bNF_convol(A,A,B,id(A),G),X),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),bNF_Grp(A,B,A3,G)))) ) ).
% convol_mem_GrpI
tff(fact_4305_power__int__minus__left__distrib,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( division_ring(C)
& one(A)
& uminus(A) )
=> ! [X: B,A4: C,N2: int] :
( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),X)
=> ( power_int(C,aa(C,C,uminus_uminus(C),A4),N2) = aa(C,C,aa(C,fun(C,C),times_times(C),power_int(C,aa(C,C,uminus_uminus(C),one_one(C)),N2)),power_int(C,A4,N2)) ) ) ) ).
% power_int_minus_left_distrib
tff(fact_4306_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N2: int,N: int,A4: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N2),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A4)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),one_one(A))
=> ( ( ( A4 != zero_zero(A) )
| ( N != zero_zero(int) )
| ( N2 = zero_zero(int) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A4,N)),power_int(A,A4,N2)) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_4307_power__int__le__one,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,N2)),one_one(A)) ) ) ) ) ).
% power_int_le_one
tff(fact_4308_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M: int,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N2))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),N2) ) ) ) ) ).
% power_int_le_imp_le_exp
tff(fact_4309_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M: int,N2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,X,M)),power_int(A,X,N2))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),N2) ) ) ) ) ).
% power_int_le_imp_less_exp
tff(fact_4310_power__int__minus__mult,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N2: int] :
( ( ( X != zero_zero(A) )
| ( N2 != zero_zero(int) ) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,minus_minus(int,N2,one_one(int)))),X) = power_int(A,X,N2) ) ) ) ).
% power_int_minus_mult
tff(fact_4311_power__int__add__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int] :
( ( ( X != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).
% power_int_add_1
tff(fact_4312_multiset_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : rel_mset(A,B,R3) = relcompp(multiset(A),multiset(product_prod(A,B)),multiset(B),conversep(multiset(product_prod(A,B)),multiset(A),bNF_Grp(multiset(product_prod(A,B)),multiset(A),aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_si(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R3)),image_mset(product_prod(A,B),A,product_fst(A,B)))),bNF_Grp(multiset(product_prod(A,B)),multiset(B),aa(fun(multiset(product_prod(A,B)),$o),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_si(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),R3)),image_mset(product_prod(A,B),B,product_snd(A,B)))) ).
% multiset.rel_compp_Grp
tff(fact_4313_fun_Orel__compp__Grp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(B,fun(C,$o))] : bNF_rel_fun(A,A,B,C,fequal(A),R3) = relcompp(fun(A,B),fun(A,product_prod(B,C)),fun(A,C),conversep(fun(A,product_prod(B,C)),fun(A,B),bNF_Grp(fun(A,product_prod(B,C)),fun(A,B),aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_kt(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R3)),comp(product_prod(B,C),B,A,product_fst(B,C)))),bNF_Grp(fun(A,product_prod(B,C)),fun(A,C),aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_kt(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R3)),comp(product_prod(B,C),C,A,product_snd(B,C)))) ).
% fun.rel_compp_Grp
tff(fact_4314_numeral__unfold__funpow,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).
% numeral_unfold_funpow
tff(fact_4315_add_Ogroup__axioms,axiom,
! [A: $tType] :
( group_add(A)
=> group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).
% add.group_axioms
tff(fact_4316_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(C,$o)),S: fun(C,fun(B,$o)),T3: fun(C,fun(B,$o))] : relcompp(A,C,B,R3,aa(fun(C,fun(B,$o)),fun(C,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(fun(C,fun(B,$o)),fun(C,fun(B,$o))),sup_sup(fun(C,fun(B,$o))),S),T3)) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),relcompp(A,C,B,R3,S)),relcompp(A,C,B,R3,T3)) ).
% relcompp_distrib
tff(fact_4317_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: fun(A,fun(C,$o)),T3: fun(A,fun(C,$o)),R3: fun(C,fun(B,$o))] : relcompp(A,C,B,aa(fun(A,fun(C,$o)),fun(A,fun(C,$o)),aa(fun(A,fun(C,$o)),fun(fun(A,fun(C,$o)),fun(A,fun(C,$o))),sup_sup(fun(A,fun(C,$o))),S),T3),R3) = aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),relcompp(A,C,B,S,R3)),relcompp(A,C,B,T3,R3)) ).
% relcompp_distrib2
tff(fact_4318_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R3: fun(A,fun(C,$o))] : relcompp(A,C,B,R3,bot_bot(fun(C,fun(B,$o)))) = bot_bot(fun(A,fun(B,$o))) ).
% relcompp_bot2
tff(fact_4319_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R3: fun(C,fun(B,$o))] : relcompp(A,C,B,bot_bot(fun(A,fun(C,$o))),R3) = bot_bot(fun(A,fun(B,$o))) ).
% relcompp_bot1
tff(fact_4320_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: fun(B,fun(C,$o)),S2: fun(C,fun(A,$o))] : conversep(B,A,relcompp(B,C,A,R2,S2)) = relcompp(A,C,B,conversep(C,A,S2),conversep(B,C,R2)) ).
% converse_relcompp
tff(fact_4321_funpow__mult,axiom,
! [A: $tType,N2: nat,M: nat,F2: fun(A,A)] : compow(fun(A,A),N2,compow(fun(A,A),M,F2)) = compow(fun(A,A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2),F2) ).
% funpow_mult
tff(fact_4322_group_Oleft__cancel,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A,B3: A,C2: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,A4),B3) = aa(A,A,aa(A,fun(A,A),F2,A4),C2) )
<=> ( B3 = C2 ) ) ) ).
% group.left_cancel
tff(fact_4323_group_Oleft__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A4)),A4) = Z2 ) ) ).
% group.left_inverse
tff(fact_4324_group_Oright__cancel,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B3: A,A4: A,C2: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,B3),A4) = aa(A,A,aa(A,fun(A,A),F2,C2),A4) )
<=> ( B3 = C2 ) ) ) ).
% group.right_cancel
tff(fact_4325_group_Oright__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,Inverse,A4)) = Z2 ) ) ).
% group.right_inverse
tff(fact_4326_group_Oinverse__unique,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A,B3: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,A4),B3) = Z2 )
=> ( aa(A,A,Inverse,A4) = B3 ) ) ) ).
% group.inverse_unique
tff(fact_4327_group_Oinverse__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,Inverse,A4)) = A4 ) ) ).
% group.inverse_inverse
tff(fact_4328_group_Oinverse__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,Z2) = Z2 ) ) ).
% group.inverse_neutral
tff(fact_4329_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,Z2),A4) = A4 ) ) ).
% group.group_left_neutral
tff(fact_4330_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A4: A,B3: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F2,A4),B3)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,B3)),aa(A,A,Inverse,A4)) ) ) ).
% group.inverse_distrib_swap
tff(fact_4331_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(C,$o)),A1: A,A22: C] :
( aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,R2,S2),A1),A22)
=> ~ ! [B2: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,A1),B2)
=> ~ aa(C,$o,aa(B,fun(C,$o),S2,B2),A22) ) ) ).
% relcompp.cases
tff(fact_4332_relcompp_Osimps,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(C,$o)),A1: A,A22: C] :
( aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,R2,S2),A1),A22)
<=> ? [A7: A,B7: B,C5: C] :
( ( A1 = A7 )
& ( A22 = C5 )
& aa(B,$o,aa(A,fun(B,$o),R2,A7),B7)
& aa(C,$o,aa(B,fun(C,$o),S2,B7),C5) ) ) ).
% relcompp.simps
tff(fact_4333_OO__eq,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : relcompp(A,B,B,R3,fequal(B)) = R3 ).
% OO_eq
tff(fact_4334_eq__OO,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : relcompp(A,A,B,fequal(A),R3) = R3 ).
% eq_OO
tff(fact_4335_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(B,$o)),S2: fun(B,fun(C,$o)),A4: A,C2: C] :
( aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,R2,S2),A4),C2)
=> ~ ! [B2: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,A4),B2)
=> ~ aa(C,$o,aa(B,fun(C,$o),S2,B2),C2) ) ) ).
% relcomppE
tff(fact_4336_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(B,$o)),A4: A,B3: B,S2: fun(B,fun(C,$o)),C2: C] :
( aa(B,$o,aa(A,fun(B,$o),R2,A4),B3)
=> ( aa(C,$o,aa(B,fun(C,$o),S2,B3),C2)
=> aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,R2,S2),A4),C2) ) ) ).
% relcomppI
tff(fact_4337_relcompp__apply,axiom,
! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,$o)),S: fun(B,fun(C,$o)),A4: A,C2: C] :
( aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,R3,S),A4),C2)
<=> ? [B7: B] :
( aa(B,$o,aa(A,fun(B,$o),R3,A4),B7)
& aa(C,$o,aa(B,fun(C,$o),S,B7),C2) ) ) ).
% relcompp_apply
tff(fact_4338_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R2: fun(A,fun(D,$o)),S2: fun(D,fun(C,$o)),T5: fun(C,fun(B,$o))] : relcompp(A,C,B,relcompp(A,D,C,R2,S2),T5) = relcompp(A,D,B,R2,relcompp(D,C,B,S2,T5)) ).
% relcompp_assoc
tff(fact_4339_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F2: fun(B,A),R2: fun(C,fun(A,$o)),S2: fun(A,fun(D,$o)),A4: C,C2: D] :
( ! [Y2: A] :
? [X4: B] : Y2 = aa(B,A,F2,X4)
=> ( aa(D,$o,aa(C,fun(D,$o),relcompp(C,A,D,R2,S2),A4),C2)
=> ~ ! [B2: B] :
( aa(A,$o,aa(C,fun(A,$o),R2,A4),aa(B,A,F2,B2))
=> ~ aa(D,$o,aa(A,fun(D,$o),S2,aa(B,A,F2,B2)),C2) ) ) ) ).
% nchotomy_relcomppE
tff(fact_4340_rel__filter__distr,axiom,
! [A: $tType,B: $tType,C: $tType,A3: fun(A,fun(C,$o)),B5: fun(C,fun(B,$o))] : relcompp(filter(A),filter(C),filter(B),rel_filter(A,C,A3),rel_filter(C,B,B5)) = rel_filter(A,B,relcompp(A,C,B,A3,B5)) ).
% rel_filter_distr
tff(fact_4341_leq__OOI,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( ( R3 = fequal(A) )
=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R3),relcompp(A,A,A,R3,R3)) ) ).
% leq_OOI
tff(fact_4342_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: fun(A,fun(B,$o)),R2: fun(A,fun(B,$o)),S3: fun(B,fun(C,$o)),S2: fun(B,fun(C,$o))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R5),R2)
=> ( aa(fun(B,fun(C,$o)),$o,aa(fun(B,fun(C,$o)),fun(fun(B,fun(C,$o)),$o),ord_less_eq(fun(B,fun(C,$o))),S3),S2)
=> aa(fun(A,fun(C,$o)),$o,aa(fun(A,fun(C,$o)),fun(fun(A,fun(C,$o)),$o),ord_less_eq(fun(A,fun(C,$o))),relcompp(A,B,C,R5,S3)),relcompp(A,B,C,R2,S2)) ) ) ).
% relcompp_mono
tff(fact_4343_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: fun(A,fun(B,$o)),Q: fun(B,fun(C,$o)),A4: A,C2: C] :
( aa(C,$o,aa(A,fun(C,$o),relcompp(A,B,C,P,Q),A4),C2)
=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),bNF_pick_middlep(A,B,C,P,Q,A4,C2))
& aa(C,$o,aa(B,fun(C,$o),Q,bNF_pick_middlep(A,B,C,P,Q,A4,C2)),C2) ) ) ).
% pick_middlep
tff(fact_4344_funpow__times__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [F2: fun(A,nat),X: A] : compow(fun(A,A),aa(A,nat,F2,X),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F2,X))) ) ).
% funpow_times_power
tff(fact_4345_relcompp__relcomp__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B)),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),relcompp(A,C,B,aTP_Lamp_su(set(product_prod(A,C)),fun(A,fun(C,$o)),R2),aTP_Lamp_sv(set(product_prod(C,B)),fun(C,fun(B,$o)),S2)),X4),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3),relcomp(A,C,B,R2,S2)) ) ).
% relcompp_relcomp_eq
tff(fact_4346_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac: product_prod(A,B),P: fun(A,fun(C,$o)),Q: fun(C,fun(B,$o))] :
( member(product_prod(A,B),Ac,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),relcompp(A,C,B,P,Q))))
=> member(product_prod(A,C),aa(product_prod(A,B),product_prod(A,C),bNF_fstOp(A,C,B,P,Q),Ac),aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),P))) ) ).
% fstOp_in
tff(fact_4347_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac: product_prod(A,B),P: fun(A,fun(C,$o)),Q: fun(C,fun(B,$o))] :
( member(product_prod(A,B),Ac,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),relcompp(A,C,B,P,Q))))
=> member(product_prod(C,B),aa(product_prod(A,B),product_prod(C,B),bNF_sndOp(A,C,B,P,Q),Ac),aa(fun(product_prod(C,B),$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),Q))) ) ).
% sndOp_in
tff(fact_4348_relcompp__SUP__distrib2,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R2: fun(D,fun(A,fun(C,$o))),I: set(D),S2: fun(C,fun(B,$o))] : relcompp(A,C,B,aa(set(fun(A,fun(C,$o))),fun(A,fun(C,$o)),complete_Sup_Sup(fun(A,fun(C,$o))),aa(set(D),set(fun(A,fun(C,$o))),image2(D,fun(A,fun(C,$o)),R2),I)),S2) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(D),set(fun(A,fun(B,$o))),image2(D,fun(A,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o))),aTP_Lamp_sw(fun(D,fun(A,fun(C,$o))),fun(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o)))),R2),S2)),I)) ).
% relcompp_SUP_distrib2
tff(fact_4349_relcompp__SUP__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,S2: fun(A,fun(C,$o)),R2: fun(D,fun(C,fun(B,$o))),I: set(D)] : relcompp(A,C,B,S2,aa(set(fun(C,fun(B,$o))),fun(C,fun(B,$o)),complete_Sup_Sup(fun(C,fun(B,$o))),aa(set(D),set(fun(C,fun(B,$o))),image2(D,fun(C,fun(B,$o)),R2),I))) = aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(D),set(fun(A,fun(B,$o))),image2(D,fun(A,fun(B,$o)),aa(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o))),aTP_Lamp_sx(fun(A,fun(C,$o)),fun(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o)))),S2),R2)),I)) ).
% relcompp_SUP_distrib
tff(fact_4350_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [F2: fun(A,A),P2: A,K: nat] :
( order_mono(A,A,F2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P2)),P2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),K,F2),bot_bot(A))),P2) ) ) ) ).
% Kleene_iter_lpfp
tff(fact_4351_relpowp__bot,axiom,
! [A: $tType,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( compow(fun(A,fun(A,$o)),N2,bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).
% relpowp_bot
tff(fact_4352_relcomp__def,axiom,
! [A: $tType,C: $tType,B: $tType,X4: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X4,Xa3) = aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),X4),aTP_Lamp_sy(set(product_prod(B,C)),fun(B,fun(C,$o)),Xa3)))) ).
% relcomp_def
tff(fact_4353_OO__Grp__alt,axiom,
! [A: $tType,C: $tType,B: $tType,A3: set(C),F2: fun(C,A),G: fun(C,B),X4: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),relcompp(A,C,B,conversep(C,A,bNF_Grp(C,A,A3,F2)),bNF_Grp(C,B,A3,G)),X4),Xa3)
<=> ? [Z3: C] :
( member(C,Z3,A3)
& ( aa(C,A,F2,Z3) = X4 )
& ( aa(C,B,G,Z3) = Xa3 ) ) ) ).
% OO_Grp_alt
tff(fact_4354_of__nat__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat] : aa(nat,A,semiring_1_of_nat(A),N2) = aa(A,A,compow(fun(A,A),N2,aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).
% of_nat_def
tff(fact_4355_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [K: num,A4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A4) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),A4) ) ).
% numeral_add_unfold_funpow
tff(fact_4356_Grp__UNIV__id,axiom,
! [A: $tType,F2: fun(A,A)] :
( ( F2 = id(A) )
=> ( relcompp(A,A,A,conversep(A,A,bNF_Grp(A,A,top_top(set(A)),F2)),bNF_Grp(A,A,top_top(set(A)),F2)) = bNF_Grp(A,A,top_top(set(A)),F2) ) ) ).
% Grp_UNIV_id
tff(fact_4357_mono__funpow,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Q: fun(A,A)] :
( order_mono(A,A,Q)
=> order_mono(nat,A,aTP_Lamp_sz(fun(A,A),fun(nat,A),Q)) ) ) ).
% mono_funpow
tff(fact_4358_funpow__decreasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [M: nat,N2: nat,F2: fun(A,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
=> ( order_mono(A,A,F2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),M,F2),bot_bot(A))),aa(A,A,compow(fun(A,A),N2,F2),bot_bot(A))) ) ) ) ).
% funpow_decreasing
tff(fact_4359_vimage2p__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(A,C),G: fun(B,D),P: fun(C,fun(D,$o))] : bNF_vimage2p(A,C,B,D,$o,F2,G,P) = relcompp(A,C,B,bNF_Grp(A,C,top_top(set(A)),F2),relcompp(C,D,B,P,conversep(B,D,bNF_Grp(B,D,top_top(set(B)),G)))) ).
% vimage2p_Grp
tff(fact_4360_vimage2p__relcompp__converse,axiom,
! [E: $tType,C: $tType,D: $tType,A: $tType,F3: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),F2: fun(C,E),G: fun(D,F3),R3: fun(B,fun(E,$o)),S: fun(B,fun(F3,$o))] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( bNF_vimage2p(C,E,D,F3,$o,F2,G,relcompp(E,B,F3,conversep(B,E,R3),S)) = relcompp(C,A,D,conversep(A,C,bNF_vimage2p(A,B,C,E,$o,Rep,F2,R3)),bNF_vimage2p(A,B,D,F3,$o,Rep,G,S)) ) ) ).
% vimage2p_relcompp_converse
tff(fact_4361_lfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(A,A),K: nat] :
( order_mono(A,A,F2)
=> ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F2),bot_bot(A)) = aa(A,A,compow(fun(A,A),K,F2),bot_bot(A)) )
=> ( complete_lattice_lfp(A,F2) = aa(A,A,compow(fun(A,A),K,F2),bot_bot(A)) ) ) ) ) ).
% lfp_Kleene_iter
tff(fact_4362_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(fun(A,fun(B,$o)),fun(product_prod(A,B),$o)),P: fun(A,fun(C,$o)),Q: fun(C,fun(B,$o))] : bNF_csquare(product_prod(A,B),product_prod(A,C),C,product_prod(C,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),F2,relcompp(A,C,B,P,Q))),product_snd(A,C),product_fst(C,B),bNF_fstOp(A,C,B,P,Q),bNF_sndOp(A,C,B,P,Q)) ).
% csquare_fstOp_sndOp
tff(fact_4363_and_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.monoid_axioms
tff(fact_4364_less__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_unique(A,B,A3)
=> aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,A3),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A3),fequal($o))),ord_less(filter(A))),ord_less(filter(B))) ) ).
% less_filter_parametric
tff(fact_4365_Powp__Pow__eq,axiom,
! [A: $tType,A3: set(A),X4: set(A)] :
( aa(set(A),$o,powp(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),X4)
<=> member(set(A),X4,pow2(A,A3)) ) ).
% Powp_Pow_eq
tff(fact_4366_typedef__bi__unique,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),A3: set(B),T3: fun(B,fun(A,$o))] :
( type_definition(A,B,Rep,Abs,A3)
=> ( ! [X3: B,Xa4: A] :
( aa(A,$o,aa(B,fun(A,$o),T3,X3),Xa4)
<=> ( X3 = aa(A,B,Rep,Xa4) ) )
=> bi_unique(B,A,T3) ) ) ).
% typedef_bi_unique
tff(fact_4367_bi__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A3: fun(A,fun(B,$o))] :
( bi_unique(A,B,A3)
=> bi_unique(filter(A),filter(B),rel_filter(A,B,A3)) ) ).
% bi_unique_rel_filter
tff(fact_4368_monoid_Oright__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: A] :
( monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A4),Z2) = A4 ) ) ).
% monoid.right_neutral
tff(fact_4369_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: A] :
( monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,Z2),A4) = A4 ) ) ).
% monoid.left_neutral
tff(fact_4370_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,A3: set(A),F1: fun(B,C),F22: fun(D,C),P13: fun(A,B),P23: fun(A,D)] :
( bNF_csquare(A,B,C,D,A3,F1,F22,P13,P23)
<=> ! [X2: A] :
( member(A,X2,A3)
=> ( aa(B,C,F1,aa(A,B,P13,X2)) = aa(D,C,F22,aa(A,D,P23,X2)) ) ) ) ).
% csquare_def
tff(fact_4371_Grp__fst__snd,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : relcompp(A,product_prod(A,B),B,conversep(product_prod(A,B),A,bNF_Grp(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3)),product_fst(A,B))),bNF_Grp(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R3)),product_snd(A,B))) = R3 ).
% Grp_fst_snd
tff(fact_4372_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_4373_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_4374_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.monoid_axioms
tff(fact_4375_Powp__mono,axiom,
! [A: $tType,A3: fun(A,$o),B5: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),A3),B5)
=> aa(fun(set(A),$o),$o,aa(fun(set(A),$o),fun(fun(set(A),$o),$o),ord_less_eq(fun(set(A),$o)),powp(A,A3)),powp(A,B5)) ) ).
% Powp_mono
tff(fact_4376_le__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_unique(A,B,A3)
=> aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,A3),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A3),fequal($o))),ord_less_eq(filter(A))),ord_less_eq(filter(B))) ) ).
% le_filter_parametric
tff(fact_4377_lfp__induct2,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,F2: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),complete_lattice_lfp(set(product_prod(A,B)),F2))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F2)
=> ( ! [A5: A,B2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2),aa(set(product_prod(A,B)),set(product_prod(A,B)),F2,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F2)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))))
=> aa(B,$o,aa(A,fun(B,$o),P,A5),B2) )
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) ) ).
% lfp_induct2
tff(fact_4378_Powp__def,axiom,
! [A: $tType,A3: fun(A,$o),X4: set(A)] :
( aa(set(A),$o,powp(A,A3),X4)
<=> ! [Xa2: A] :
( member(A,Xa2,X4)
=> aa(A,$o,A3,Xa2) ) ) ).
% Powp_def
tff(fact_4379_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_unique(A,B,A3)
=> ( bi_total(A,B,A3)
=> aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A3),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A3),rel_filter(A,B,A3))),inf_inf(filter(A))),inf_inf(filter(B))) ) ) ).
% inf_filter_parametric
tff(fact_4380_relpow__fun__conv,axiom,
! [A: $tType,A4: A,B3: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),compow(set(product_prod(A,A)),N2,R3))
<=> ? [F6: fun(nat,A)] :
( ( aa(nat,A,F6,zero_zero(nat)) = A4 )
& ( aa(nat,A,F6,N2) = B3 )
& ! [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),N2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,I4)),aa(nat,A,F6,aa(nat,nat,suc,I4))),R3) ) ) ) ).
% relpow_fun_conv
tff(fact_4381_prod_Orel__compp__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(C,$o)),R23: fun(B,fun(D,$o))] : basic_rel_prod(A,C,B,D,R12,R23) = relcompp(product_prod(A,B),product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),conversep(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_ta(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),R12),R23)),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)))),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_ta(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),R12),R23)),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).
% prod.rel_compp_Grp
tff(fact_4382_prod_Oin__rel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A4: product_prod(A,C),B3: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),A4),B3)
<=> ? [Z3: product_prod(product_prod(A,B),product_prod(C,D))] :
( member(product_prod(product_prod(A,B),product_prod(C,D)),Z3,aa(fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),set(product_prod(product_prod(A,B),product_prod(C,D))),collect(product_prod(product_prod(A,B),product_prod(C,D))),aa(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_tb(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o)),R12),R23)))
& ( aa(product_prod(product_prod(A,B),product_prod(C,D)),product_prod(A,C),product_map_prod(product_prod(A,B),A,product_prod(C,D),C,product_fst(A,B),product_fst(C,D)),Z3) = A4 )
& ( aa(product_prod(product_prod(A,B),product_prod(C,D)),product_prod(B,D),product_map_prod(product_prod(A,B),B,product_prod(C,D),D,product_snd(A,B),product_snd(C,D)),Z3) = B3 ) ) ) ).
% prod.in_rel
tff(fact_4383_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A4: A,B3: C,C2: B,D3: D] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),B3)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C2),D3))
<=> ( aa(B,$o,aa(A,fun(B,$o),R12,A4),C2)
& aa(D,$o,aa(C,fun(D,$o),R23,B3),D3) ) ) ).
% rel_prod_inject
tff(fact_4384_relpow__Suc__D2_H,axiom,
! [A: $tType,N2: nat,R3: set(product_prod(A,A)),X4: A,Y5: A,Z6: A] :
( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5),compow(set(product_prod(A,A)),N2,R3))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z6),R3) )
=> ? [W2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),W2),R3)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W2),Z6),compow(set(product_prod(A,A)),N2,R3)) ) ) ).
% relpow_Suc_D2'
tff(fact_4385_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),A1),A22)
=> ~ ! [A5: A,B2: B,C3: C] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C3) )
=> ! [D2: D] :
( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D2) )
=> ( aa(B,$o,aa(A,fun(B,$o),R12,A5),B2)
=> ~ aa(D,$o,aa(C,fun(D,$o),R23,C3),D2) ) ) ) ) ).
% rel_prod.cases
tff(fact_4386_rel__prod_Osimps,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),A1),A22)
<=> ? [A7: A,B7: B,C5: C,D5: D] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),C5) )
& ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B7),D5) )
& aa(B,$o,aa(A,fun(B,$o),R12,A7),B7)
& aa(D,$o,aa(C,fun(D,$o),R23,C5),D5) ) ) ).
% rel_prod.simps
tff(fact_4387_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: fun(A,fun(B,$o)),A4: A,B3: B,R23: fun(C,fun(D,$o)),C2: C,D3: D] :
( aa(B,$o,aa(A,fun(B,$o),R12,A4),B3)
=> ( aa(D,$o,aa(C,fun(D,$o),R23,C2),D3)
=> aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),C2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B3),D3)) ) ) ).
% rel_prod.intros
tff(fact_4388_bi__total__rel__filter,axiom,
! [B: $tType,A: $tType,A3: fun(A,fun(B,$o))] :
( bi_total(A,B,A3)
=> bi_total(filter(A),filter(B),rel_filter(A,B,A3)) ) ).
% bi_total_rel_filter
tff(fact_4389_rel__prod__sel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),P2: product_prod(A,C),Q3: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),P2),Q3)
<=> ( aa(B,$o,aa(A,fun(B,$o),R12,aa(product_prod(A,C),A,product_fst(A,C),P2)),aa(product_prod(B,D),B,product_fst(B,D),Q3))
& aa(D,$o,aa(C,fun(D,$o),R23,aa(product_prod(A,C),C,product_snd(A,C),P2)),aa(product_prod(B,D),D,product_snd(B,D),Q3)) ) ) ).
% rel_prod_sel
tff(fact_4390_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: fun(A,fun(C,$o)),B5: fun(B,fun(D,$o))] : aa(fun(product_prod(C,D),D),$o,aa(fun(product_prod(A,B),B),fun(fun(product_prod(C,D),D),$o),bNF_rel_fun(product_prod(A,B),product_prod(C,D),B,D,basic_rel_prod(A,C,B,D,A3,B5),B5),product_snd(A,B)),product_snd(C,D)) ).
% snd_transfer
tff(fact_4391_relpow__0__I,axiom,
! [A: $tType,X: A,R3: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),compow(set(product_prod(A,A)),zero_zero(nat),R3)) ).
% relpow_0_I
tff(fact_4392_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),compow(set(product_prod(A,A)),zero_zero(nat),R3))
=> ( X = Y ) ) ).
% relpow_0_E
tff(fact_4393_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z2: A,N2: nat] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),compow(set(product_prod(A,A)),N2,R3))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N2),R3)) ) ) ).
% relpow_Suc_I2
tff(fact_4394_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N2),R3))
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),N2,R3)) ) ) ).
% relpow_Suc_E2
tff(fact_4395_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N2),R3))
=> ? [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R3)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),N2,R3)) ) ) ).
% relpow_Suc_D2
tff(fact_4396_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N2: nat,R3: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),compow(set(product_prod(A,A)),N2,R3))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R3)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N2),R3)) ) ) ).
% relpow_Suc_I
tff(fact_4397_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N2),R3))
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),compow(set(product_prod(A,A)),N2,R3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),R3) ) ) ).
% relpow_Suc_E
tff(fact_4398_prod__filter__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,$o)),S: fun(C,fun(D,$o))] : aa(fun(filter(B),fun(filter(D),filter(product_prod(B,D)))),$o,aa(fun(filter(A),fun(filter(C),filter(product_prod(A,C)))),fun(fun(filter(B),fun(filter(D),filter(product_prod(B,D)))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(C),filter(product_prod(A,C))),fun(filter(D),filter(product_prod(B,D))),rel_filter(A,B,R3),bNF_rel_fun(filter(C),filter(D),filter(product_prod(A,C)),filter(product_prod(B,D)),rel_filter(C,D,S),rel_filter(product_prod(A,C),product_prod(B,D),basic_rel_prod(A,B,C,D,R3,S)))),prod_filter(A,C)),prod_filter(B,D)) ).
% prod_filter_parametric
tff(fact_4399_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_total(A,B,A3)
=> aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A3),top_top(filter(A))),top_top(filter(B))) ) ).
% top_filter_parametric
tff(fact_4400_relpowp__relpow__eq,axiom,
! [A: $tType,N2: nat,R3: set(product_prod(A,A)),X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N2,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R3)),X4),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3),compow(set(product_prod(A,A)),N2,R3)) ) ).
% relpowp_relpow_eq
tff(fact_4401_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),N2,R3))
=> ( ( ( N2 = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N2 = aa(nat,nat,suc,M3) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),M3,R3)) ) ) ) ) ).
% relpow_E2
tff(fact_4402_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R3: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),N2,R3))
=> ( ( ( N2 = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N2 = aa(nat,nat,suc,M3) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),compow(set(product_prod(A,A)),M3,R3))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),R3) ) ) ) ) ).
% relpow_E
tff(fact_4403_relpow__empty,axiom,
! [A: $tType,N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
=> ( compow(set(product_prod(A,A)),N2,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).
% relpow_empty
tff(fact_4404_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o))] : aa(fun(B,fun(D,product_prod(B,D))),$o,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),$o),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A3,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B5,basic_rel_prod(A,B,C,D,A3,B5))),product_Pair(A,C)),product_Pair(B,D)) ).
% Pair_transfer
tff(fact_4405_curry__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F3: $tType,E: $tType,D: $tType,A3: fun(A,fun(D,$o)),B5: fun(B,fun(E,$o)),C6: fun(C,fun(F3,$o))] : aa(fun(fun(product_prod(D,E),F3),fun(D,fun(E,F3))),$o,aa(fun(fun(product_prod(A,B),C),fun(A,fun(B,C))),fun(fun(fun(product_prod(D,E),F3),fun(D,fun(E,F3))),$o),bNF_rel_fun(fun(product_prod(A,B),C),fun(product_prod(D,E),F3),fun(A,fun(B,C)),fun(D,fun(E,F3)),bNF_rel_fun(product_prod(A,B),product_prod(D,E),C,F3,basic_rel_prod(A,D,B,E,A3,B5),C6),bNF_rel_fun(A,D,fun(B,C),fun(E,F3),A3,bNF_rel_fun(B,E,C,F3,B5,C6))),product_curry(A,B,C)),product_curry(D,E,F3)) ).
% curry_transfer
tff(fact_4406_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X: set(fun(A,nat))] :
( aa(set(fun(A,nat)),$o,aa(set(fun(A,nat)),fun(set(fun(A,nat)),$o),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o))),X),X)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(fun(A,nat)),set(multiset(A)),image2(fun(A,nat),multiset(A),abs_multiset(A)),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat)),X)) ) ) ).
% Inf_multiset.abs_eq
tff(fact_4407_irreflp__irrefl__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( irreflp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R3))
<=> irrefl(A,R3) ) ).
% irreflp_irrefl_eq
tff(fact_4408_equivp__equiv,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( equiv_equiv(A,top_top(set(A)),A3)
<=> equiv_equivp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),A3)) ) ).
% equivp_equiv
tff(fact_4409_is__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_total(A,B,A3)
=> ( bi_unique(A,B,A3)
=> aa(fun(fun(fun(B,$o),$o),$o),$o,aa(fun(fun(fun(A,$o),$o),$o),fun(fun(fun(fun(B,$o),$o),$o),$o),bNF_rel_fun(fun(fun(A,$o),$o),fun(fun(B,$o),$o),$o,$o,bNF_rel_fun(fun(A,$o),fun(B,$o),$o,$o,bNF_rel_fun(A,B,$o,$o,A3,fequal($o)),fequal($o)),fequal($o)),is_filter(A)),is_filter(B)) ) ) ).
% is_filter_parametric
tff(fact_4410_eq__onp__top__eq__eq,axiom,
! [A: $tType] : bNF_eq_onp(A,top_top(fun(A,$o))) = fequal(A) ).
% eq_onp_top_eq_eq
tff(fact_4411_eq__onp__same__args,axiom,
! [A: $tType,P: fun(A,$o),X: A] :
( aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,P),X),X)
<=> aa(A,$o,P,X) ) ).
% eq_onp_same_args
tff(fact_4412_eq__onp__to__eq,axiom,
! [A: $tType,P: fun(A,$o),X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,P),X),Y)
=> ( X = Y ) ) ).
% eq_onp_to_eq
tff(fact_4413_eq__onp__mono0,axiom,
! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
( ! [X3: A] :
( member(A,X3,A3)
=> ( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) ) )
=> ! [X4: A] :
( member(A,X4,A3)
=> ! [Xa3: A] :
( member(A,Xa3,A3)
=> ( aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,P),X4),Xa3)
=> aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,Q),X4),Xa3) ) ) ) ) ).
% eq_onp_mono0
tff(fact_4414_eq__onp__eqD,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,fun(A,$o)),X: A] :
( ( bNF_eq_onp(A,P) = Q )
=> ( aa(A,$o,P,X)
<=> aa(A,$o,aa(A,fun(A,$o),Q,X),X) ) ) ).
% eq_onp_eqD
tff(fact_4415_rel__setI,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B),R3: fun(A,fun(B,$o))] :
( ! [X3: A] :
( member(A,X3,A3)
=> ? [Xa3: B] :
( member(B,Xa3,B5)
& aa(B,$o,aa(A,fun(B,$o),R3,X3),Xa3) ) )
=> ( ! [Y2: B] :
( member(B,Y2,B5)
=> ? [X4: A] :
( member(A,X4,A3)
& aa(B,$o,aa(A,fun(B,$o),R3,X4),Y2) ) )
=> aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,R3),A3),B5) ) ) ).
% rel_setI
tff(fact_4416_irreflp__def,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( irreflp(A,R3)
<=> ! [A7: A] : ~ aa(A,$o,aa(A,fun(A,$o),R3,A7),A7) ) ).
% irreflp_def
tff(fact_4417_irreflpI,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),R3,A5),A5)
=> irreflp(A,R3) ) ).
% irreflpI
tff(fact_4418_is__filter__def,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
<=> ( aa(fun(A,$o),$o,F4,aTP_Lamp_ci(A,$o))
& ! [P4: fun(A,$o)] :
( aa(fun(A,$o),$o,F4,P4)
=> ! [Q4: fun(A,$o)] :
( aa(fun(A,$o),$o,F4,Q4)
=> aa(fun(A,$o),$o,F4,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P4),Q4)) ) )
& ! [P4: fun(A,$o),Q4: fun(A,$o)] :
( ! [X2: A] :
( aa(A,$o,P4,X2)
=> aa(A,$o,Q4,X2) )
=> ( aa(fun(A,$o),$o,F4,P4)
=> aa(fun(A,$o),$o,F4,Q4) ) ) ) ) ).
% is_filter_def
tff(fact_4419_is__filter_Ointro,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o)] :
( aa(fun(A,$o),$o,F4,aTP_Lamp_ci(A,$o))
=> ( ! [P3: fun(A,$o)] :
( aa(fun(A,$o),$o,F4,P3)
=> ! [Q2: fun(A,$o)] :
( aa(fun(A,$o),$o,F4,Q2)
=> aa(fun(A,$o),$o,F4,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P3),Q2)) ) )
=> ( ! [P3: fun(A,$o),Q2: fun(A,$o)] :
( ! [X4: A] :
( aa(A,$o,P3,X4)
=> aa(A,$o,Q2,X4) )
=> ( aa(fun(A,$o),$o,F4,P3)
=> aa(fun(A,$o),$o,F4,Q2) ) )
=> aa(fun(fun(A,$o),$o),$o,is_filter(A),F4) ) ) ) ).
% is_filter.intro
tff(fact_4420_is__filter_Omono,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o),P: fun(A,$o),Q: fun(A,$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> ( aa(fun(A,$o),$o,F4,P)
=> aa(fun(A,$o),$o,F4,Q) ) ) ) ).
% is_filter.mono
tff(fact_4421_is__filter_Oconj,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o),P: fun(A,$o),Q: fun(A,$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
=> ( aa(fun(A,$o),$o,F4,P)
=> ( aa(fun(A,$o),$o,F4,Q)
=> aa(fun(A,$o),$o,F4,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) ) ) ) ).
% is_filter.conj
tff(fact_4422_is__filter_OTrue,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
=> aa(fun(A,$o),$o,F4,aTP_Lamp_ci(A,$o)) ) ).
% is_filter.True
tff(fact_4423_eq__onp__True,axiom,
! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_ci(A,$o)) = fequal(A) ).
% eq_onp_True
tff(fact_4424_eq__onp__def,axiom,
! [A: $tType,R3: fun(A,$o),X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),bNF_eq_onp(A,R3),X4),Xa3)
<=> ( aa(A,$o,R3,X4)
& ( X4 = Xa3 ) ) ) ).
% eq_onp_def
tff(fact_4425_irreflp__less,axiom,
! [A: $tType] :
( preorder(A)
=> irreflp(A,ord_less(A)) ) ).
% irreflp_less
tff(fact_4426_principal__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(set(B),filter(B)),$o,aa(fun(set(A),filter(A)),fun(fun(set(B),filter(B)),$o),bNF_rel_fun(set(A),set(B),filter(A),filter(B),bNF_rel_set(A,B,A3),rel_filter(A,B,A3)),principal(A)),principal(B)) ).
% principal_parametric
tff(fact_4427_rel__set__def,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),X4: set(A),Xa3: set(B)] :
( aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,R3),X4),Xa3)
<=> ( ! [Xb3: A] :
( member(A,Xb3,X4)
=> ? [Xc: B] :
( member(B,Xc,Xa3)
& aa(B,$o,aa(A,fun(B,$o),R3,Xb3),Xc) ) )
& ! [Xb3: B] :
( member(B,Xb3,Xa3)
=> ? [Xc: A] :
( member(A,Xc,X4)
& aa(B,$o,aa(A,fun(B,$o),R3,Xc),Xb3) ) ) ) ) ).
% rel_set_def
tff(fact_4428_eq__onp__Grp,axiom,
! [A: $tType,P: fun(A,$o)] : bNF_eq_onp(A,P) = bNF_Grp(A,A,aa(fun(A,$o),set(A),collect(A),P),id(A)) ).
% eq_onp_Grp
tff(fact_4429_Abs__filter__inject,axiom,
! [A: $tType,X: fun(fun(A,$o),$o),Y: fun(fun(A,$o),$o)] :
( member(fun(fun(A,$o),$o),X,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> ( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> ( ( aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),X) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_filter_inject
tff(fact_4430_Abs__filter__induct,axiom,
! [A: $tType,P: fun(filter(A),$o),X: filter(A)] :
( ! [Y2: fun(fun(A,$o),$o)] :
( member(fun(fun(A,$o),$o),Y2,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> aa(filter(A),$o,P,aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y2)) )
=> aa(filter(A),$o,P,X) ) ).
% Abs_filter_induct
tff(fact_4431_Abs__filter__cases,axiom,
! [A: $tType,X: filter(A)] :
~ ! [Y2: fun(fun(A,$o),$o)] :
( ( X = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y2) )
=> ~ member(fun(fun(A,$o),$o),Y2,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ) ).
% Abs_filter_cases
tff(fact_4432_eq__onp__mono__iff,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),bNF_eq_onp(A,P)),bNF_eq_onp(A,Q))
<=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).
% eq_onp_mono_iff
tff(fact_4433_irreflp__greater,axiom,
! [A: $tType] :
( preorder(A)
=> irreflp(A,aTP_Lamp_tc(A,fun(A,$o))) ) ).
% irreflp_greater
tff(fact_4434_is__filter__map__filter__on,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X5: set(B),F4: filter(B)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),aa(filter(B),fun(fun(A,$o),$o),aa(set(B),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_te(fun(B,A),fun(set(B),fun(filter(B),fun(fun(A,$o),$o))),F2),X5),F4))
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_nz(set(B),fun(B,$o),X5)),F4) ) ).
% is_filter_map_filter_on
tff(fact_4435_UNIV__typedef__to__equivp,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> equiv_equivp(B,fequal(B)) ) ).
% UNIV_typedef_to_equivp
tff(fact_4436_eventually__Abs__filter,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o),P: fun(A,$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),F4))
<=> aa(fun(A,$o),$o,F4,P) ) ) ).
% eventually_Abs_filter
tff(fact_4437_Inf__multiset_Orsp,axiom,
! [A: $tType] : aa(fun(set(fun(A,nat)),fun(A,nat)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(fun(A,nat)),fun(A,nat)),$o),bNF_rel_fun(set(fun(A,nat)),set(fun(A,nat)),fun(A,nat),fun(A,nat),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o))),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o))),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat))) ).
% Inf_multiset.rsp
tff(fact_4438_repeat__mset_Orsp,axiom,
! [A: $tType] : aa(fun(nat,fun(fun(A,nat),fun(A,nat))),$o,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(fun(A,nat),fun(A,nat))),$o),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(nat),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o)),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o)))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))) ).
% repeat_mset.rsp
tff(fact_4439_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X: fun(A,nat),Xa: nat] :
( aa(fun(A,nat),$o,aa(fun(A,nat),fun(fun(A,nat),$o),bNF_eq_onp(fun(A,nat),aTP_Lamp_sc(fun(A,nat),$o)),X),X)
=> ( aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),Xa),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(nat,fun(A,nat),aTP_Lamp_tg(fun(A,nat),fun(nat,fun(A,nat)),X),Xa)) ) ) ).
% repeat_mset.abs_eq
tff(fact_4440_Lifting__Set_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(B,fun(set(B),set(B))),$o,aa(fun(A,fun(set(A),set(A))),fun(fun(B,fun(set(B),set(B))),$o),bNF_rel_fun(A,B,fun(set(A),set(A)),fun(set(B),set(B)),A3,bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A3),bNF_rel_set(A,B,A3))),insert2(A)),insert2(B)) ).
% Lifting_Set.insert_transfer
tff(fact_4441_empty__transfer,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,A3),bot_bot(set(A))),bot_bot(set(B))) ).
% empty_transfer
tff(fact_4442_Abs__filter__inverse,axiom,
! [A: $tType,Y: fun(fun(A,$o),$o)] :
( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),Y)) = Y ) ) ).
% Abs_filter_inverse
tff(fact_4443_repeat__mset__right,axiom,
! [A: $tType,A4: nat,B3: nat,A3: multiset(A)] : aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),A4),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),B3),A3)) = aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),B3)),A3) ).
% repeat_mset_right
tff(fact_4444_Rep__filter__iff__eventually,axiom,
! [A: $tType,F4: filter(A),P: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),F4),P)
<=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F4) ) ).
% Rep_filter_iff_eventually
tff(fact_4445_count__repeat__mset,axiom,
! [A: $tType,I2: nat,A3: multiset(A),A4: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),A3)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A3),A4)) ).
% count_repeat_mset
tff(fact_4446_Rep__filter__inject,axiom,
! [A: $tType,X: filter(A),Y: filter(A)] :
( ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X) = aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),Y) )
<=> ( X = Y ) ) ).
% Rep_filter_inject
tff(fact_4447_Sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] : aa(fun(set(filter(B)),filter(B)),$o,aa(fun(set(filter(A)),filter(A)),fun(fun(set(filter(B)),filter(B)),$o),bNF_rel_fun(set(filter(A)),set(filter(B)),filter(A),filter(B),bNF_rel_set(filter(A),filter(B),rel_filter(A,B,A3)),rel_filter(A,B,A3)),complete_Sup_Sup(filter(A))),complete_Sup_Sup(filter(B))) ).
% Sup_filter_parametric
tff(fact_4448_is__filter__Rep__filter,axiom,
! [A: $tType,F4: filter(A)] : aa(fun(fun(A,$o),$o),$o,is_filter(A),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),F4)) ).
% is_filter_Rep_filter
tff(fact_4449_Rep__filter__inverse,axiom,
! [A: $tType,X: filter(A)] : aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X)) = X ).
% Rep_filter_inverse
tff(fact_4450_repeat__mset_Orep__eq,axiom,
! [A: $tType,X: nat,Xa: multiset(A),X4: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),X),Xa)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Xa),X4)) ).
% repeat_mset.rep_eq
tff(fact_4451_Rep__filter,axiom,
! [A: $tType,X: filter(A)] : member(fun(fun(A,$o),$o),aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X),aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ).
% Rep_filter
tff(fact_4452_Rep__filter__cases,axiom,
! [A: $tType,Y: fun(fun(A,$o),$o)] :
( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> ~ ! [X3: filter(A)] : Y != aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X3) ) ).
% Rep_filter_cases
tff(fact_4453_Rep__filter__induct,axiom,
! [A: $tType,Y: fun(fun(A,$o),$o),P: fun(fun(fun(A,$o),$o),$o)] :
( member(fun(fun(A,$o),$o),Y,aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A)))
=> ( ! [X3: filter(A)] : aa(fun(fun(A,$o),$o),$o,P,aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),X3))
=> aa(fun(fun(A,$o),$o),$o,P,Y) ) ) ).
% Rep_filter_induct
tff(fact_4454_Inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: fun(A,fun(B,$o))] :
( bi_unique(A,B,A3)
=> ( bi_total(A,B,A3)
=> aa(fun(set(filter(B)),filter(B)),$o,aa(fun(set(filter(A)),filter(A)),fun(fun(set(filter(B)),filter(B)),$o),bNF_rel_fun(set(filter(A)),set(filter(B)),filter(A),filter(B),bNF_rel_set(filter(A),filter(B),rel_filter(A,B,A3)),rel_filter(A,B,A3)),complete_Inf_Inf(filter(A))),complete_Inf_Inf(filter(B))) ) ) ).
% Inf_filter_parametric
tff(fact_4455_Abs__filter__inverse_H,axiom,
! [A: $tType,F4: fun(fun(A,$o),$o)] :
( aa(fun(fun(A,$o),$o),$o,is_filter(A),F4)
=> ( aa(filter(A),fun(fun(A,$o),$o),rep_filter(A),aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),F4)) = F4 ) ) ).
% Abs_filter_inverse'
tff(fact_4456_type__definition__filter,axiom,
! [A: $tType] : type_definition(filter(A),fun(fun(A,$o),$o),rep_filter(A),abs_filter(A),aa(fun(fun(fun(A,$o),$o),$o),set(fun(fun(A,$o),$o)),collect(fun(fun(A,$o),$o)),is_filter(A))) ).
% type_definition_filter
tff(fact_4457_Inf__multiset_Otransfer,axiom,
! [A: $tType] : aa(fun(set(multiset(A)),multiset(A)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(multiset(A)),multiset(A)),$o),bNF_rel_fun(set(fun(A,nat)),set(multiset(A)),fun(A,nat),multiset(A),bNF_rel_set(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),pcr_multiset(A,A,fequal(A))),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat))),complete_Inf_Inf(multiset(A))) ).
% Inf_multiset.transfer
tff(fact_4458_in__chain__finite,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A3: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ) ).
% in_chain_finite
tff(fact_4459_cofinite__bot,axiom,
! [A: $tType] :
( ( cofinite(A) = bot_bot(filter(A)) )
<=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).
% cofinite_bot
tff(fact_4460_drop__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N2: nat] : bit_se4197421643247451524op_bit(A,N2,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),N2 = zero_zero(nat)) ) ).
% drop_bit_of_1
tff(fact_4461_chain__empty,axiom,
! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).
% chain_empty
tff(fact_4462_cofinite__eq__sequentially,axiom,
cofinite(nat) = at_top(nat) ).
% cofinite_eq_sequentially
tff(fact_4463_ccpo__Sup__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A3: set(A),B5: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
=> ( comple1602240252501008431_chain(A,ord_less_eq(A),B5)
=> ( ! [X3: A] :
( member(A,X3,A3)
=> ? [Xa3: A] :
( member(A,Xa3,B5)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ).
% ccpo_Sup_mono
tff(fact_4464_INFM__nat__inductI,axiom,
! [P: fun(nat,$o),Q: fun(nat,$o)] :
( aa(nat,$o,P,zero_zero(nat))
=> ( ! [I3: nat] :
( aa(nat,$o,P,I3)
=> ? [J6: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J6)
& aa(nat,$o,P,J6)
& aa(nat,$o,Q,J6) ) )
=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),frequently(nat),Q),cofinite(nat)) ) ) ).
% INFM_nat_inductI
tff(fact_4465_eventually__cofinite,axiom,
! [A: $tType,P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
<=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),P))) ) ).
% eventually_cofinite
tff(fact_4466_frequently__cofinite,axiom,
! [A: $tType,P: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
<=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P)) ) ).
% frequently_cofinite
tff(fact_4467_chain__singleton,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).
% chain_singleton
tff(fact_4468_cofinite__def,axiom,
! [A: $tType] : cofinite(A) = aa(fun(fun(A,$o),$o),filter(A),abs_filter(A),aTP_Lamp_th(fun(A,$o),$o)) ).
% cofinite_def
tff(fact_4469_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A,N2: nat] : divide_divide(A,A4,bit_se4730199178511100633sh_bit(A,N2,one_one(A))) = bit_se4197421643247451524op_bit(A,N2,A4) ) ).
% div_push_bit_of_1_eq_drop_bit
tff(fact_4470_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A4: A,N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),N2)
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,N2,A4)),one_one(A)) = one_one(A) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
tff(fact_4471_repeat__mset_Otransfer,axiom,
! [A: $tType] : aa(fun(nat,fun(multiset(A),multiset(A))),$o,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(multiset(A),multiset(A))),$o),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(nat),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))),repeat_mset(A)) ).
% repeat_mset.transfer
tff(fact_4472_drop__bit__exp__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N2: nat] :
bit_se4197421643247451524op_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N2)) = aa(A,A,
aa(A,fun(A,A),times_times(A),
aa($o,A,zero_neq_one_of_bool(A),
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
& bit_se6407376104438227557le_bit(A,type2(A),N2) ))),
aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N2,M))) ) ).
% drop_bit_exp_eq
tff(fact_4473_wfP__SUP,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,fun(B,$o)))] :
( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,$o)),R2,I3))
=> ( ! [I3: A,J4: A] :
( ( aa(A,fun(B,fun(B,$o)),R2,I3) != aa(A,fun(B,fun(B,$o)),R2,J4) )
=> ( aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),domainp(B,B,aa(A,fun(B,fun(B,$o)),R2,I3))),rangep(B,B,aa(A,fun(B,fun(B,$o)),R2,J4))) = bot_bot(fun(B,$o)) ) )
=> wfP(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Sup_Sup(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),top_top(set(A))))) ) ) ).
% wfP_SUP
tff(fact_4474_bit__double__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A4: A,N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),N2)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A4),minus_minus(nat,N2,one_one(nat)))
& ( N2 != zero_zero(nat) )
& bit_se6407376104438227557le_bit(A,type2(A),N2) ) ) ) ).
% bit_double_iff
tff(fact_4475_override__on__insert,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),X: A,X5: set(A)] : override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),X5)) = fun_upd(A,B,override_on(A,B,F2,G,X5),X,aa(A,B,G,X)) ).
% override_on_insert
tff(fact_4476_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] : override_on(A,B,F2,G,bot_bot(set(A))) = F2 ).
% override_on_emptyset
tff(fact_4477_bit__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N2)
<=> bit_se6407376104438227557le_bit(A,type2(A),N2) ) ) ).
% bit_minus_1_iff
tff(fact_4478_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: A] :
( aa(A,$o,domainp(A,B,R2),A4)
=> ~ ! [B2: B] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A4),B2) ) ).
% Domainp.cases
tff(fact_4479_Domainp_Osimps,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: A] :
( aa(A,$o,domainp(A,B,R2),A4)
<=> ? [A7: A,B7: B] :
( ( A4 = A7 )
& aa(B,$o,aa(A,fun(B,$o),R2,A7),B7) ) ) ).
% Domainp.simps
tff(fact_4480_DomainPI,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),R2,A4),B3)
=> aa(A,$o,domainp(A,B,R2),A4) ) ).
% DomainPI
tff(fact_4481_DomainpE,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A4: A] :
( aa(A,$o,domainp(A,B,R2),A4)
=> ~ ! [B2: B] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A4),B2) ) ).
% DomainpE
tff(fact_4482_bit__minus__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A4: A,N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A4)),N2)
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N2)
& ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,minus_minus(A,A4,one_one(A))),N2) ) ) ) ).
% bit_minus_iff
tff(fact_4483_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X4: A] :
( aa(A,$o,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4)
<=> member(A,X4,aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).
% Domainp_Domain_eq
tff(fact_4484_Domain__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),X4) = aa(fun(A,$o),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),X4))) ).
% Domain_def
tff(fact_4485_wfP__wf__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wfP(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> wf(A,R2) ) ).
% wfP_wf_eq
tff(fact_4486_bit__mask__sub__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N2: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M),one_one(A))),N2)
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N2)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),M) ) ) ) ).
% bit_mask_sub_iff
tff(fact_4487_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),X: A,X5: set(A)] : override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),X5)) = override_on(A,B,fun_upd(A,B,F2,X,aa(A,B,G,X)),G,X5) ).
% override_on_insert'
tff(fact_4488_old_Orec__unit__def,axiom,
! [A: $tType,X4: A,Xa3: product_unit] : product_rec_unit(A,X4,Xa3) = the(A,product_rec_set_unit(A,X4,Xa3)) ).
% old.rec_unit_def
tff(fact_4489_cut__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),R3: set(product_prod(A,A)),X: A,X4: A] :
aa(A,B,cut(A,B,F2,R3,X),X4) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X),R3),aa(A,B,F2,X4),undefined(B)) ).
% cut_def
tff(fact_4490_above__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] : order_above(A,R2,A4) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A4)) ).
% above_def
tff(fact_4491_ID_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),A4: A,B3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R3),A4),B3)
<=> ? [Z3: product_prod(A,B)] :
( member(product_prod(A,B),Z3,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_ti(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),R3)))
& ( aa(product_prod(A,B),A,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),A),bNF_id_bnf(fun(product_prod(A,B),A)),product_fst(A,B)),Z3) = A4 )
& ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),B),bNF_id_bnf(fun(product_prod(A,B),B)),product_snd(A,B)),Z3) = B3 ) ) ) ).
% ID.in_rel
tff(fact_4492_cuts__eq,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),R3: set(product_prod(A,A)),X: A,G: fun(A,B)] :
( ( cut(A,B,F2,R3,X) = cut(A,B,G,R3,X) )
<=> ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R3)
=> ( aa(A,B,F2,Y3) = aa(A,B,G,Y3) ) ) ) ).
% cuts_eq
tff(fact_4493_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A4: A,R3: set(product_prod(A,A)),F2: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A4),R3)
=> ( aa(A,B,cut(A,B,F2,R3,A4),X) = aa(A,B,F2,X) ) ) ).
% cut_apply
tff(fact_4494_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya2: A,Y: B,Xa: B,R3: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
( ( X = Ya2 )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( member(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya2),bot_bot(set(A))))
=> ( member(B,Yb,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B))))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,Z4),Yb)
<=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z4),Yb) ) ) )
=> ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R3),X),Y)
<=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),Ra2),Ya2),Xa) ) ) ) ) ).
% ID.rel_cong
tff(fact_4495_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),X: A,Y: B,Ra2: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),R3),X),Y)
=> ( ! [Z4: A,Yb: B] :
( member(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
=> ( member(B,Yb,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Y),bot_bot(set(B))))
=> ( aa(B,$o,aa(A,fun(B,$o),R3,Z4),Yb)
=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z4),Yb) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),bNF_id_bnf(fun(A,fun(B,$o))),Ra2),X),Y) ) ) ).
% ID.rel_mono_strong
tff(fact_4496_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra2: fun(A,fun(A,$o))] :
( ! [Z4: A] :
( member(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
=> aa(A,$o,aa(A,fun(A,$o),Ra2,Z4),Z4) )
=> aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),bNF_id_bnf(fun(A,fun(A,$o))),Ra2),X),X) ) ).
% ID.rel_refl_strong
tff(fact_4497_BNF__Composition_Otype__definition__id__bnf__UNIV,axiom,
! [A: $tType] : type_definition(A,A,bNF_id_bnf(A),bNF_id_bnf(A),top_top(set(A))) ).
% BNF_Composition.type_definition_id_bnf_UNIV
tff(fact_4498_ctor__rec__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R3: fun(A,fun(C,$o)),S: fun(B,fun(D,$o))] : aa(fun(fun(C,D),fun(C,D)),$o,aa(fun(fun(A,B),fun(A,B)),fun(fun(fun(C,D),fun(C,D)),$o),bNF_rel_fun(fun(A,B),fun(C,D),fun(A,B),fun(C,D),bNF_rel_fun(A,C,B,D,bNF_vimage2p(A,A,C,C,$o,bNF_id_bnf(A),bNF_id_bnf(C),R3),S),bNF_rel_fun(A,C,B,D,R3,S)),basic_BNF_ctor_rec(fun(A,B))),basic_BNF_ctor_rec(fun(C,D))) ).
% ctor_rec_transfer
tff(fact_4499_Basic__BNF__LFPs_Oxtor__rel__induct,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o)),IR: fun(A,fun(B,$o))] :
( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),bNF_vimage2p(A,A,B,B,$o,bNF_id_bnf(A),bNF_id_bnf(B),R3),X3),Y2)
=> aa(B,$o,aa(A,fun(B,$o),IR,aa(A,A,basic_BNF_xtor(A),X3)),aa(B,B,basic_BNF_xtor(B),Y2)) )
=> aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R3),IR) ) ).
% Basic_BNF_LFPs.xtor_rel_induct
tff(fact_4500_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(A,C),fun(A,B),comp(C,B,A,aa(fun(C,B),fun(C,B),basic_BNF_ctor_rec(fun(C,B)),F2)),G) = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),aa(fun(A,C),fun(A,B),comp(C,B,A,F2),aa(fun(A,A),fun(A,C),comp(A,C,A,aa(fun(A,C),fun(A,C),comp(C,C,A,bNF_id_bnf(C)),G)),bNF_id_bnf(A)))) ).
% Basic_BNF_LFPs.ctor_rec_o_map
tff(fact_4501_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : F2 = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),aa(fun(A,A),fun(A,B),comp(A,B,A,F2),bNF_id_bnf(A))) ).
% Basic_BNF_LFPs.ctor_rec_def_alt
tff(fact_4502_Basic__BNF__LFPs_OPair__def__alt,axiom,
! [A: $tType,B: $tType,X4: A,Xa3: B] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3) = aa(product_prod(A,B),product_prod(A,B),basic_BNF_xtor(product_prod(A,B)),aa(product_prod(A,B),product_prod(A,B),bNF_id_bnf(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3))) ).
% Basic_BNF_LFPs.Pair_def_alt
tff(fact_4503_Basic__BNF__LFPs_Oxtor__def,axiom,
! [A: $tType,X: A] : aa(A,A,basic_BNF_xtor(A),X) = X ).
% Basic_BNF_LFPs.xtor_def
tff(fact_4504_Basic__BNF__LFPs_Oxtor__map,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X: B] : aa(B,A,F2,aa(B,B,basic_BNF_xtor(B),X)) = aa(A,A,basic_BNF_xtor(A),aa(B,A,F2,X)) ).
% Basic_BNF_LFPs.xtor_map
tff(fact_4505_Basic__BNF__LFPs_Oxtor__rel,axiom,
! [B: $tType,A: $tType,C: $tType,R3: fun(B,fun(C,A)),X: B,Y: C] : aa(C,A,aa(B,fun(C,A),R3,aa(B,B,basic_BNF_xtor(B),X)),aa(C,C,basic_BNF_xtor(C),Y)) = aa(C,A,aa(B,fun(C,A),R3,X),Y) ).
% Basic_BNF_LFPs.xtor_rel
tff(fact_4506_Basic__BNF__LFPs_Oxtor__set,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X: B] : aa(B,A,F2,aa(B,B,basic_BNF_xtor(B),X)) = aa(B,A,F2,X) ).
% Basic_BNF_LFPs.xtor_set
tff(fact_4507_Basic__BNF__LFPs_Oxtor__xtor,axiom,
! [A: $tType,X: A] : aa(A,A,basic_BNF_xtor(A),aa(A,A,basic_BNF_xtor(A),X)) = X ).
% Basic_BNF_LFPs.xtor_xtor
tff(fact_4508_Basic__BNF__LFPs_Oxtor__induct,axiom,
! [A: $tType,P: fun(A,$o),Z2: A] :
( ! [X3: A] : aa(A,$o,P,aa(A,A,basic_BNF_xtor(A),X3))
=> aa(A,$o,P,Z2) ) ).
% Basic_BNF_LFPs.xtor_induct
tff(fact_4509_Basic__BNF__LFPs_Oxtor__inject,axiom,
! [A: $tType,X: A,Y: A] :
( ( aa(A,A,basic_BNF_xtor(A),X) = aa(A,A,basic_BNF_xtor(A),Y) )
<=> ( X = Y ) ) ).
% Basic_BNF_LFPs.xtor_inject
tff(fact_4510_xtor__iff__xtor,axiom,
! [A: $tType,U: A,W: A] :
( ( U = aa(A,A,basic_BNF_xtor(A),W) )
<=> ( aa(A,A,basic_BNF_xtor(A),U) = W ) ) ).
% xtor_iff_xtor
tff(fact_4511_xtor__map__unique,axiom,
! [B: $tType,A: $tType,U: fun(A,B),F2: fun(A,B)] :
( ( aa(fun(A,A),fun(A,B),comp(A,B,A,U),basic_BNF_xtor(A)) = aa(fun(A,B),fun(A,B),comp(B,B,A,basic_BNF_xtor(B)),F2) )
=> ( U = F2 ) ) ).
% xtor_map_unique
tff(fact_4512_Basic__BNF__LFPs_Octor__rec__def,axiom,
! [A: $tType,X: A] : aa(A,A,basic_BNF_ctor_rec(A),X) = X ).
% Basic_BNF_LFPs.ctor_rec_def
tff(fact_4513_ID_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,$o),X: A,Pa: fun(A,$o)] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X)
=> ( ! [Z4: A] :
( member(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
=> ( aa(A,$o,P,Z4)
=> aa(A,$o,Pa,Z4) ) )
=> aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),Pa),X) ) ) ).
% ID.pred_mono_strong
tff(fact_4514_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya2: A,P: fun(A,$o),Pa: fun(A,$o)] :
( ( X = Ya2 )
=> ( ! [Z4: A] :
( member(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya2),bot_bot(set(A))))
=> ( aa(A,$o,P,Z4)
<=> aa(A,$o,Pa,Z4) ) )
=> ( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X)
<=> aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),Pa),Ya2) ) ) ) ).
% ID.pred_cong
tff(fact_4515_ctor__rec__unique,axiom,
! [B: $tType,A: $tType,G: fun(A,A),F2: fun(A,B),S2: fun(A,B)] :
( ( G = id(A) )
=> ( ( aa(fun(A,A),fun(A,B),comp(A,B,A,F2),basic_BNF_xtor(A)) = aa(fun(A,A),fun(A,B),comp(A,B,A,S2),aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,bNF_id_bnf(A)),G)),bNF_id_bnf(A))) )
=> ( F2 = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),S2) ) ) ) ).
% ctor_rec_unique
tff(fact_4516_Basic__BNF__LFPs_Octor__rec,axiom,
! [B: $tType,A: $tType,G: fun(A,A),F2: fun(A,B),X: A] :
( ( G = id(A) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),F2),aa(A,A,basic_BNF_xtor(A),X)) = aa(A,B,F2,aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,bNF_id_bnf(A)),G)),bNF_id_bnf(A)),X)) ) ) ).
% Basic_BNF_LFPs.ctor_rec
tff(fact_4517_ID_Opred__set,axiom,
! [A: $tType,P: fun(A,$o),X4: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),bNF_id_bnf(fun(A,$o)),P),X4)
<=> ! [Xa2: A] :
( member(A,Xa2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))))
=> aa(A,$o,P,Xa2) ) ) ).
% ID.pred_set
tff(fact_4518_lcm__altdef__int,axiom,
! [A4: int,B3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A4),B3) = divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,A4)),abs_abs(int,B3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A4),B3)) ).
% lcm_altdef_int
tff(fact_4519_times__num__def,axiom,
! [M: num,N2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),N2) = num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat_of_num(M)),nat_of_num(N2))) ).
% times_num_def
tff(fact_4520_is__num_Ocases,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A4: A] :
( neg_numeral_is_num(A,A4)
=> ( ( A4 != one_one(A) )
=> ( ! [X3: A] :
( ( A4 = aa(A,A,uminus_uminus(A),X3) )
=> ~ neg_numeral_is_num(A,X3) )
=> ~ ! [X3: A,Y2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y2) )
=> ( neg_numeral_is_num(A,X3)
=> ~ neg_numeral_is_num(A,Y2) ) ) ) ) ) ) ).
% is_num.cases
tff(fact_4521_is__num_Osimps,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A4: A] :
( neg_numeral_is_num(A,A4)
<=> ( ( A4 = one_one(A) )
| ? [X2: A] :
( ( A4 = aa(A,A,uminus_uminus(A),X2) )
& neg_numeral_is_num(A,X2) )
| ? [X2: A,Y3: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y3) )
& neg_numeral_is_num(A,X2)
& neg_numeral_is_num(A,Y3) ) ) ) ) ).
% is_num.simps
tff(fact_4522_lcm__eq__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3) = one_one(A) )
<=> ( dvd_dvd(A,A4,one_one(A))
& dvd_dvd(A,B3,one_one(A)) ) ) ) ).
% lcm_eq_1_iff
tff(fact_4523_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> neg_numeral_is_num(A,one_one(A)) ) ).
% is_num_normalize(4)
tff(fact_4524_nat__of__num__mult,axiom,
! [X: num,Y: num] : nat_of_num(aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat_of_num(X)),nat_of_num(Y)) ).
% nat_of_num_mult
tff(fact_4525_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).
% lcm_mult_unit1
tff(fact_4526_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).
% lcm_mult_unit2
tff(fact_4527_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),divide_divide(A,C2,A4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).
% lcm_div_unit2
tff(fact_4528_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),divide_divide(A,B3,A4)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),C2) ) ) ) ).
% lcm_div_unit1
tff(fact_4529_prod__gcd__lcm__int,axiom,
! [M: int,N2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,M)),abs_abs(int,N2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N2)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N2)) ).
% prod_gcd_lcm_int
tff(fact_4530_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = $ite(aa(set(A),$o,finite_finite2(A),A3),finite_fold(A,A,gcd_lcm(A),one_one(A),A3),zero_zero(A)) ) ).
% Lcm_fin.eq_fold
tff(fact_4531_Lcm__fin__def,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).
% Lcm_fin_def
tff(fact_4532_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] :
( member(A,A4,A3)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),aa(set(A),A,semiring_gcd_Lcm_fin(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) ) ).
% Lcm_fin.remove
tff(fact_4533_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),aa(set(A),A,semiring_gcd_Lcm_fin(A),minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ).
% Lcm_fin.insert_remove
tff(fact_4534_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_fin.empty
tff(fact_4535_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
( dvd_dvd(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A3),one_one(A))
<=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = one_one(A) ) ) ) ).
% is_unit_Lcm_fin_iff
tff(fact_4536_Lcm__fin_Oinsert,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,A3: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3)) ) ).
% Lcm_fin.insert
tff(fact_4537_prod__gcd__lcm__nat,axiom,
! [M: nat,N2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N2)) ).
% prod_gcd_lcm_nat
tff(fact_4538_lcm__code__integer,axiom,
! [A4: code_integer,B3: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),A4),B3) = divide_divide(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),abs_abs(code_integer,A4)),abs_abs(code_integer,B3)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A4),B3)) ).
% lcm_code_integer
tff(fact_4539_lcm__nat__def,axiom,
! [X: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).
% lcm_nat_def
tff(fact_4540_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A)] :
( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A3) = one_one(A) )
<=> ( ! [X2: A] :
( member(A,X2,A3)
=> dvd_dvd(A,X2,one_one(A)) )
& aa(set(A),$o,finite_finite2(A),A3) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_4541_Lcm__eq__Max__nat,axiom,
! [M2: set(nat)] :
( aa(set(nat),$o,finite_finite2(nat),M2)
=> ( ( M2 != bot_bot(set(nat)) )
=> ( ~ member(nat,zero_zero(nat),M2)
=> ( ! [M3: nat,N3: nat] :
( member(nat,M3,M2)
=> ( member(nat,N3,M2)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N3),M2) ) )
=> ( gcd_Lcm(nat,M2) = lattic643756798349783984er_Max(nat,M2) ) ) ) ) ) ).
% Lcm_eq_Max_nat
tff(fact_4542_gcd__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( ( A4 != zero_zero(A) )
=> ( ( B3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3))) ) ) ) ) ).
% gcd_lcm
tff(fact_4543_Lcm__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A,B3: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3) ) ).
% Lcm_2
tff(fact_4544_nat__of__num__sqr,axiom,
! [X: num] : nat_of_num(sqr(X)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat_of_num(X)),nat_of_num(X)) ).
% nat_of_num_sqr
tff(fact_4545_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A,B3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,normal6383669964737779283malize(A),B3))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% normalize_mult_normalize_right
tff(fact_4546_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A,B3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),B3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% normalize_mult_normalize_left
tff(fact_4547_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% gcd.normalize_bottom
tff(fact_4548_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_4549_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_4550_Lcm__1__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
( ( gcd_Lcm(A,A3) = one_one(A) )
<=> ! [X2: A] :
( member(A,X2,A3)
=> dvd_dvd(A,X2,one_one(A)) ) ) ) ).
% Lcm_1_iff
tff(fact_4551_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A4) = aa(A,A,normal6383669964737779283malize(A),A4) ) ).
% lcm.top_left_normalize
tff(fact_4552_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A4) ) ).
% lcm.top_right_normalize
tff(fact_4553_Lcm__insert,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A,A3: set(A)] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),gcd_Lcm(A,A3)) ) ).
% Lcm_insert
tff(fact_4554_normalize__mult__unit__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,normal6383669964737779283malize(A),B3) ) ) ) ).
% normalize_mult_unit_left
tff(fact_4555_normalize__mult__unit__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,normal6383669964737779283malize(A),A4) ) ) ) ).
% normalize_mult_unit_right
tff(fact_4556_lcm__mult__gcd,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),aa(A,A,normal6383669964737779283malize(A),B3)) ) ).
% lcm_mult_gcd
tff(fact_4557_gcd__mult__lcm,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),aa(A,A,normal6383669964737779283malize(A),B3)) ) ).
% gcd_mult_lcm
tff(fact_4558_Lcm__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A4) ) ).
% Lcm_singleton
tff(fact_4559_Gcd__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A4) ) ).
% Gcd_singleton
tff(fact_4560_normalize__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A4: A,B3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),aa(A,A,normal6383669964737779283malize(A),B3)) ) ).
% normalize_mult
tff(fact_4561_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A),C2: A] :
( ( A3 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A3))) ) ) ) ).
% Lcm_mult
tff(fact_4562_associated__unit,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A4) = aa(A,A,normal6383669964737779283malize(A),B3) )
=> ( dvd_dvd(A,A4,one_one(A))
=> dvd_dvd(A,B3,one_one(A)) ) ) ) ).
% associated_unit
tff(fact_4563_normalize__1__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A4) = one_one(A) )
<=> dvd_dvd(A,A4,one_one(A)) ) ) ).
% normalize_1_iff
tff(fact_4564_is__unit__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( aa(A,A,normal6383669964737779283malize(A),A4) = one_one(A) ) ) ) ).
% is_unit_normalize
tff(fact_4565_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A4) = A4 )
=> ( dvd_dvd(A,A4,one_one(A))
<=> ( A4 = one_one(A) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
tff(fact_4566_gcd__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3))) ) ).
% gcd_mult_left
tff(fact_4567_gcd__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B3),A4)),C2)) ) ).
% gcd_mult_right
tff(fact_4568_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).
% gcd_mult_distrib'
tff(fact_4569_lcm__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3))) ) ).
% lcm_mult_left
tff(fact_4570_lcm__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B3),A4)),C2)) ) ).
% lcm_mult_right
tff(fact_4571_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).
% lcm_mult_distrib'
tff(fact_4572_Lcm__nat__empty,axiom,
gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).
% Lcm_nat_empty
tff(fact_4573_sqr__conv__mult,axiom,
! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).
% sqr_conv_mult
tff(fact_4574_Lcm__in__lcm__closed__set__nat,axiom,
! [M2: set(nat)] :
( aa(set(nat),$o,finite_finite2(nat),M2)
=> ( ( M2 != bot_bot(set(nat)) )
=> ( ! [M3: nat,N3: nat] :
( member(nat,M3,M2)
=> ( member(nat,N3,M2)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N3),M2) ) )
=> member(nat,gcd_Lcm(nat,M2),M2) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
tff(fact_4575_lcm__gcd__prod,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ).
% lcm_gcd_prod
tff(fact_4576_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C2: A,A3: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A3))) ) ).
% Gcd_mult
tff(fact_4577_numeral__sqr,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).
% numeral_sqr
tff(fact_4578_lcm__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3))) ) ).
% lcm_gcd
tff(fact_4579_Lcm__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A),B3: A] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B3)),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(set(A),A,semiring_gcd_Lcm_fin(A),A3))) ) ) ) ).
% Lcm_fin_mult
tff(fact_4580_Gcd__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: set(A),B3: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B3)),A3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A3))) ) ) ) ).
% Gcd_fin_mult
tff(fact_4581_Lcm__no__units,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] : gcd_Lcm(A,A3) = gcd_Lcm(A,minus_minus(set(A),A3,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_tj(A,$o)))) ) ).
% Lcm_no_units
tff(fact_4582_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_4583_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_4584_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] : pow(X,bit1(Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y))),X) ).
% pow.simps(3)
tff(fact_4585_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).
% lcm.bounded_quasi_semilattice_axioms
tff(fact_4586_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).
% gcd.bounded_quasi_semilattice_axioms
tff(fact_4587_Lcm__coprime,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A,B2: A] :
( member(A,A5,A3)
=> ( member(A,B2,A3)
=> ( ( A5 != B2 )
=> algebr8660921524188924756oprime(A,A5,B2) ) ) )
=> ( gcd_Lcm(A,A3) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_tk(A,A)),A3)) ) ) ) ) ) ).
% Lcm_coprime
tff(fact_4588_coprime__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,C2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2)
<=> ( algebr8660921524188924756oprime(A,A4,C2)
& algebr8660921524188924756oprime(A,B3,C2) ) ) ) ).
% coprime_mult_left_iff
tff(fact_4589_coprime__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A4: A,B3: A] :
( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3))
<=> ( algebr8660921524188924756oprime(A,C2,A4)
& algebr8660921524188924756oprime(A,C2,B3) ) ) ) ).
% coprime_mult_right_iff
tff(fact_4590_coprime__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] :
( algebr8660921524188924756oprime(A,A4,A4)
<=> dvd_dvd(A,A4,one_one(A)) ) ) ).
% coprime_self
tff(fact_4591_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,B3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) = one_one(A) ) ) ) ).
% coprime_imp_gcd_eq_1
tff(fact_4592_coprime__0__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] :
( algebr8660921524188924756oprime(A,A4,zero_zero(A))
<=> dvd_dvd(A,A4,one_one(A)) ) ) ).
% coprime_0_right_iff
tff(fact_4593_coprime__0__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] :
( algebr8660921524188924756oprime(A,zero_zero(A),A4)
<=> dvd_dvd(A,A4,one_one(A)) ) ) ).
% coprime_0_left_iff
tff(fact_4594_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,C2: A,B3: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> ( dvd_dvd(A,C2,one_one(A))
& algebr8660921524188924756oprime(A,A4,B3) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_4595_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A4: A,B3: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> ( dvd_dvd(A,C2,one_one(A))
& algebr8660921524188924756oprime(A,A4,B3) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_4596_is__unit__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3),one_one(A))
<=> algebr8660921524188924756oprime(A,A4,B3) ) ) ).
% is_unit_gcd
tff(fact_4597_coprime__crossproduct__nat,axiom,
! [A4: nat,D3: nat,B3: nat,C2: nat] :
( algebr8660921524188924756oprime(nat,A4,D3)
=> ( algebr8660921524188924756oprime(nat,B3,C2)
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),D3) )
<=> ( ( A4 = B3 )
& ( C2 = D3 ) ) ) ) ) ).
% coprime_crossproduct_nat
tff(fact_4598_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,A4,one_one(A)) ) ).
% coprime_1_right
tff(fact_4599_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,one_one(A),A4) ) ).
% coprime_1_left
tff(fact_4600_coprime__add__one__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,A4,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A))) ) ).
% coprime_add_one_right
tff(fact_4601_coprime__add__one__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),one_one(A)),A4) ) ).
% coprime_add_one_left
tff(fact_4602_coprime__doff__one__right,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,A4,minus_minus(A,A4,one_one(A))) ) ).
% coprime_doff_one_right
tff(fact_4603_coprime__diff__one__left,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A4: A] : algebr8660921524188924756oprime(A,minus_minus(A,A4,one_one(A)),A4) ) ).
% coprime_diff_one_left
tff(fact_4604_divides__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] :
( dvd_dvd(A,A4,C2)
=> ( dvd_dvd(A,B3,C2)
=> ( algebr8660921524188924756oprime(A,A4,B3)
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) ) ) ) ) ).
% divides_mult
tff(fact_4605_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,C2)
=> ( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
<=> dvd_dvd(A,A4,B3) ) ) ) ).
% coprime_dvd_mult_left_iff
tff(fact_4606_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,C2)
=> ( dvd_dvd(A,A4,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
<=> dvd_dvd(A,A4,B3) ) ) ) ).
% coprime_dvd_mult_right_iff
tff(fact_4607_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B3: A,A4: A] :
( dvd_dvd(A,B3,one_one(A))
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ).
% is_unit_right_imp_coprime
tff(fact_4608_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ).
% is_unit_left_imp_coprime
tff(fact_4609_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A,C2: A] :
( algebr8660921524188924756oprime(A,A4,B3)
=> ( dvd_dvd(A,C2,A4)
=> ( dvd_dvd(A,C2,B3)
=> dvd_dvd(A,C2,one_one(A)) ) ) ) ) ).
% coprime_common_divisor
tff(fact_4610_coprime__absorb__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Y: A,X: A] :
( dvd_dvd(A,Y,X)
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> dvd_dvd(A,Y,one_one(A)) ) ) ) ).
% coprime_absorb_right
tff(fact_4611_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,D3: A,A4: A,B3: A] :
( algebr8660921524188924756oprime(A,C2,D3)
=> ( ! [E2: A] :
( ~ dvd_dvd(A,E2,one_one(A))
=> ( dvd_dvd(A,E2,A4)
=> ( dvd_dvd(A,E2,B3)
=> dvd_dvd(A,E2,C2) ) ) )
=> ( ! [E2: A] :
( ~ dvd_dvd(A,E2,one_one(A))
=> ( dvd_dvd(A,E2,A4)
=> ( dvd_dvd(A,E2,B3)
=> dvd_dvd(A,E2,D3) ) ) )
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ) ) ).
% coprime_imp_coprime
tff(fact_4612_coprime__absorb__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,Y: A] :
( dvd_dvd(A,X,Y)
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> dvd_dvd(A,X,one_one(A)) ) ) ) ).
% coprime_absorb_left
tff(fact_4613_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A4: A,B3: A] :
( dvd_dvd(A,C2,A4)
=> ( dvd_dvd(A,C2,B3)
=> ( ~ dvd_dvd(A,C2,one_one(A))
=> ~ algebr8660921524188924756oprime(A,A4,B3) ) ) ) ) ).
% not_coprimeI
tff(fact_4614_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( ~ algebr8660921524188924756oprime(A,A4,B3)
=> ~ ! [C3: A] :
( dvd_dvd(A,C3,A4)
=> ( dvd_dvd(A,C3,B3)
=> dvd_dvd(A,C3,one_one(A)) ) ) ) ) ).
% not_coprimeE
tff(fact_4615_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,B3)
<=> ! [C5: A] :
( dvd_dvd(A,C5,A4)
=> ( dvd_dvd(A,C5,B3)
=> dvd_dvd(A,C5,one_one(A)) ) ) ) ) ).
% coprime_def
tff(fact_4616_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: A,B3: A] :
( ! [C3: A] :
( dvd_dvd(A,C3,A4)
=> ( dvd_dvd(A,C3,B3)
=> dvd_dvd(A,C3,one_one(A)) ) )
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ).
% coprimeI
tff(fact_4617_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [N2: A,A4: A,M: A,B3: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N2),A4),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N2),B3),M) )
=> ( algebr8660921524188924756oprime(A,M,N2)
=> ( modulo_modulo(A,A4,M) = modulo_modulo(A,B3,M) ) ) ) ) ).
% mult_mod_cancel_left
tff(fact_4618_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [A4: A,N2: A,M: A,B3: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),N2),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),N2),M) )
=> ( algebr8660921524188924756oprime(A,M,N2)
=> ( modulo_modulo(A,A4,M) = modulo_modulo(A,B3,M) ) ) ) ) ).
% mult_mod_cancel_right
tff(fact_4619_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) ) ) ) ).
% gcd_mult_right_right_cancel
tff(fact_4620_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,C2: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) ) ) ) ).
% gcd_mult_right_left_cancel
tff(fact_4621_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B3: A,C2: A,A4: A] :
( algebr8660921524188924756oprime(A,B3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),B3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) ) ) ) ).
% gcd_mult_left_right_cancel
tff(fact_4622_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B3: A,C2: A,A4: A] :
( algebr8660921524188924756oprime(A,B3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),B3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) ) ) ) ).
% gcd_mult_left_left_cancel
tff(fact_4623_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) = one_one(A) )
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ).
% gcd_eq_1_imp_coprime
tff(fact_4624_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,B3)
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) = one_one(A) ) ) ) ).
% coprime_iff_gcd_eq_1
tff(fact_4625_coprime__crossproduct__int,axiom,
! [A4: int,D3: int,B3: int,C2: int] :
( algebr8660921524188924756oprime(int,A4,D3)
=> ( algebr8660921524188924756oprime(int,B3,C2)
=> ( ( aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,A4)),abs_abs(int,C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,B3)),abs_abs(int,D3)) )
<=> ( ( abs_abs(int,A4) = abs_abs(int,B3) )
& ( abs_abs(int,C2) = abs_abs(int,D3) ) ) ) ) ) ).
% coprime_crossproduct_int
tff(fact_4626_coprime__crossproduct,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,D3: A,B3: A,C2: A] :
( algebr8660921524188924756oprime(A,A4,D3)
=> ( algebr8660921524188924756oprime(A,B3,C2)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),aa(A,A,normal6383669964737779283malize(A),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B3)),aa(A,A,normal6383669964737779283malize(A),D3)) )
<=> ( ( aa(A,A,normal6383669964737779283malize(A),A4) = aa(A,A,normal6383669964737779283malize(A),B3) )
& ( aa(A,A,normal6383669964737779283malize(A),C2) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).
% coprime_crossproduct
tff(fact_4627_invertible__coprime,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C2) = one_one(A) )
=> algebr8660921524188924756oprime(A,A4,C2) ) ) ).
% invertible_coprime
tff(fact_4628_gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A,A6: A,B4: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) != zero_zero(A) )
=> ( ( A4 = aa(A,A,aa(A,fun(A,A),times_times(A),A6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) )
=> ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B4),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) )
=> algebr8660921524188924756oprime(A,A6,B4) ) ) ) ) ).
% gcd_coprime
tff(fact_4629_gcd__coprime__exists,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3) != zero_zero(A) )
=> ? [A16: A,B11: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),times_times(A),A16),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) )
& ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B11),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) )
& algebr8660921524188924756oprime(A,A16,B11) ) ) ) ).
% gcd_coprime_exists
tff(fact_4630_lcm__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
( algebr8660921524188924756oprime(A,A4,B3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) ) ) ) ).
% lcm_coprime
tff(fact_4631_mult__inj__if__coprime__nat,axiom,
! [A: $tType,B: $tType,F2: fun(A,nat),A3: set(A),G: fun(B,nat),B5: set(B)] :
( inj_on(A,nat,F2,A3)
=> ( inj_on(B,nat,G,B5)
=> ( ! [A5: A,B2: B] :
( member(A,A5,A3)
=> ( member(B,B2,B5)
=> algebr8660921524188924756oprime(nat,aa(A,nat,F2,A5),aa(B,nat,G,B2)) ) )
=> inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_tl(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F2),G)),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5))) ) ) ) ).
% mult_inj_if_coprime_nat
tff(fact_4632_normalize__div,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] : divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A4),A4) = divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A4)) ) ).
% normalize_div
tff(fact_4633_unit__factor__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A4)) = one_one(A) ) ) ) ).
% unit_factor_normalize
tff(fact_4634_normalize__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A4)) = one_one(A) ) ) ) ).
% normalize_unit_factor
tff(fact_4635_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_4636_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A4)),unit_f5069060285200089521factor(A,A4)) = A4 ) ).
% normalize_mult_unit_factor
tff(fact_4637_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A4)),aa(A,A,normal6383669964737779283malize(A),A4)) = A4 ) ).
% unit_factor_mult_normalize
tff(fact_4638_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A] :
( ( divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A4)) = zero_zero(A) )
<=> ( A4 = zero_zero(A) ) ) ) ).
% inv_unit_factor_eq_0_iff
tff(fact_4639_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A4),unit_f5069060285200089521factor(A,B3)) ) ) ) ).
% unit_factor_mult_unit_left
tff(fact_4640_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A4: A,B3: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B3)),A4) ) ) ) ).
% unit_factor_mult_unit_right
tff(fact_4641_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A4),divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,B3))) = divide_divide(A,A4,unit_f5069060285200089521factor(A,B3)) ) ).
% mult_one_div_unit_factor
tff(fact_4642_unit__factor__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)) = $ite(
( ( A4 = zero_zero(A) )
| ( B3 = zero_zero(A) ) ),
zero_zero(A),
one_one(A) ) ) ).
% unit_factor_lcm
tff(fact_4643_unit__factor__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A4: A,B3: A] : unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A4)),unit_f5069060285200089521factor(A,B3)) ) ).
% unit_factor_mult
tff(fact_4644_is__unit__unit__factor,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A4: A] :
( dvd_dvd(A,A4,one_one(A))
=> ( unit_f5069060285200089521factor(A,A4) = A4 ) ) ) ).
% is_unit_unit_factor
tff(fact_4645_mult__gcd__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))) ) ).
% mult_gcd_left
tff(fact_4646_mult__gcd__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))),unit_f5069060285200089521factor(A,C2)) ) ).
% mult_gcd_right
tff(fact_4647_gcd__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B3))),unit_f5069060285200089521factor(A,K)) ) ).
% gcd_mult_distrib
tff(fact_4648_mult__lcm__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))) ) ).
% mult_lcm_left
tff(fact_4649_mult__lcm__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A4: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))),unit_f5069060285200089521factor(A,C2)) ) ).
% mult_lcm_right
tff(fact_4650_lcm__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B3))),unit_f5069060285200089521factor(A,K)) ) ).
% lcm_mult_distrib
tff(fact_4651_unit__factor__is__unit,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A4: A] :
( ( A4 != zero_zero(A) )
=> dvd_dvd(A,unit_f5069060285200089521factor(A,A4),one_one(A)) ) ) ).
% unit_factor_is_unit
tff(fact_4652_unit__factor__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: A,B3: A] :
unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A4),B3)) = $ite(
( ( A4 = zero_zero(A) )
& ( B3 = zero_zero(A) ) ),
zero_zero(A),
one_one(A) ) ) ).
% unit_factor_gcd
tff(fact_4653_coprime__crossproduct_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [B3: A,D3: A,A4: A,C2: A] :
( ( B3 != zero_zero(A) )
=> ( ( unit_f5069060285200089521factor(A,B3) = unit_f5069060285200089521factor(A,D3) )
=> ( algebr8660921524188924756oprime(A,A4,B3)
=> ( algebr8660921524188924756oprime(A,C2,D3)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A4),D3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
<=> ( ( A4 = C2 )
& ( B3 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
tff(fact_4654_unit__factor__Lcm,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
unit_f5069060285200089521factor(A,gcd_Lcm(A,A3)) = $ite(gcd_Lcm(A,A3) = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% unit_factor_Lcm
tff(fact_4655_unit__factor__Gcd,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: set(A)] :
unit_f5069060285200089521factor(A,gcd_Gcd(A,A3)) = $ite(gcd_Gcd(A,A3) = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% unit_factor_Gcd
tff(fact_4656_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F4: set(fun(C,set(B))),G: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,bNF_collect(C,B,F4)),G) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),image2(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_tm(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G)),F4)) ).
% collect_comp
tff(fact_4657_subset__mset_OatLeastatMost__empty,axiom,
! [A: $tType,B3: multiset(A),A4: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),B3),A4)
=> ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.atLeastatMost_empty
tff(fact_4658_of__rat__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),field_char_0_of_rat(A,R2)),one_one(A))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).
% of_rat_le_1_iff
tff(fact_4659_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: rat] :
( ( one_one(A) = field_char_0_of_rat(A,A4) )
<=> ( one_one(rat) = A4 ) ) ) ).
% one_eq_of_rat_iff
tff(fact_4660_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: rat] :
( ( field_char_0_of_rat(A,A4) = one_one(A) )
<=> ( A4 = one_one(rat) ) ) ) ).
% of_rat_eq_1_iff
tff(fact_4661_of__rat__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_of_rat(A,one_one(rat)) = one_one(A) ) ) ).
% of_rat_1
tff(fact_4662_of__rat__neg__one,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_of_rat(A,aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% of_rat_neg_one
tff(fact_4663_one__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),field_char_0_of_rat(A,R2))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).
% one_less_of_rat_iff
tff(fact_4664_of__rat__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),one_one(A))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).
% of_rat_less_1_iff
tff(fact_4665_one__le__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R2))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).
% one_le_of_rat_iff
tff(fact_4666_subset__mset_OacyclicI__order,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,multiset(B))] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2),R2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F2,B2)),aa(A,multiset(B),F2,A5)) )
=> transitive_acyclic(A,R2) ) ).
% subset_mset.acyclicI_order
tff(fact_4667_of__rat__mult,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A4: rat,B3: rat] : field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),field_char_0_of_rat(A,A4)),field_char_0_of_rat(A,B3)) ) ).
% of_rat_mult
tff(fact_4668_collect__def,axiom,
! [A: $tType,B: $tType,F4: set(fun(B,set(A))),X: B] : aa(B,set(A),bNF_collect(B,A,F4),X) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(B,set(A))),set(set(A)),image2(fun(B,set(A)),set(A),aTP_Lamp_tn(B,fun(fun(B,set(A)),set(A)),X)),F4)) ).
% collect_def
tff(fact_4669_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),A4,B3) = minus_minus(set(multiset(A)),set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),A4),bot_bot(set(multiset(A))))) ).
% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_4670_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A4,B3) = minus_minus(set(multiset(A)),set_atLeastAtMost(multiset(A),subseteq_mset(A),A4,B3),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),B3),bot_bot(set(multiset(A))))) ).
% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4671_subset__mset_OgreaterThanLessThan__empty,axiom,
! [A: $tType,L: multiset(A),K: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
=> ( set_gr287244882034783167ssThan(multiset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.greaterThanLessThan_empty
tff(fact_4672_subset__mset_OatLeastLessThan__empty,axiom,
! [A: $tType,B3: multiset(A),A4: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B3),A4)
=> ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A4,B3) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.atLeastLessThan_empty
tff(fact_4673_subset__mset_OatLeastLessThan__empty__iff,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] :
( ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A4,B3) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A4),B3) ) ).
% subset_mset.atLeastLessThan_empty_iff
tff(fact_4674_subset__mset_OatLeastLessThan__empty__iff2,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A4,B3) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A4),B3) ) ).
% subset_mset.atLeastLessThan_empty_iff2
tff(fact_4675_subset__mset_OgreaterThanAtMost__empty,axiom,
! [A: $tType,L: multiset(A),K: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
=> ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.greaterThanAtMost_empty
tff(fact_4676_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
! [A: $tType,K: multiset(A),L: multiset(A)] :
( ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).
% subset_mset.greaterThanAtMost_empty_iff
tff(fact_4677_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
! [A: $tType,K: multiset(A),L: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).
% subset_mset.greaterThanAtMost_empty_iff2
tff(fact_4678_subset__mset_OIio__Int__singleton,axiom,
! [A: $tType,K: multiset(A),X: multiset(A)] :
aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_lessThan(multiset(A),subset_mset(A),K)),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A))))) = $ite(aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X),K),aa(set(multiset(A)),set(multiset(A)),aa(multiset(A),fun(set(multiset(A)),set(multiset(A))),insert2(multiset(A)),X),bot_bot(set(multiset(A)))),bot_bot(set(multiset(A)))) ).
% subset_mset.Iio_Int_singleton
tff(fact_4679_subset__mset_Osum__pos,axiom,
! [B: $tType,A: $tType,I: set(A),F2: fun(A,multiset(B))] :
( aa(set(A),$o,finite_finite2(A),I)
=> ( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F2,I3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F2,I)) ) ) ) ).
% subset_mset.sum_pos
tff(fact_4680_subset__mset_Osum__strict__mono,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F2,X3)),aa(A,multiset(B),G,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F2,A3)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,A3)) ) ) ) ).
% subset_mset.sum_strict_mono
tff(fact_4681_typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: set(B)] :
( type_definition(A,B,Rep,Abs,S)
=> equiv_part_equivp(B,bNF_eq_onp(B,aTP_Lamp_nz(set(B),fun(B,$o),S))) ) ).
% typedef_to_part_equivp
tff(fact_4682_open__typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,$o)] :
( type_definition(A,B,Rep,Abs,aa(fun(B,$o),set(B),collect(B),P))
=> equiv_part_equivp(B,bNF_eq_onp(B,P)) ) ).
% open_typedef_to_part_equivp
tff(fact_4683_in__range_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ! [X3: nat] :
( member(nat,X3,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),lim(product_unit,H)) ) ) ) ).
% in_range.elims(3)
tff(fact_4684_in__range_Osimps,axiom,
! [Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ! [X2: nat] :
( member(nat,X2,As)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,Ha)) ) ) ).
% in_range.simps
tff(fact_4685_in__range_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ~ ! [X2: nat] :
( member(nat,X2,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,H)) ) ) ) ) ).
% in_range.elims(1)
tff(fact_4686_in__range_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ~ ! [X4: nat] :
( member(nat,X4,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H)) ) ) ) ).
% in_range.elims(2)
tff(fact_4687_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A,Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ( ( ref_get(A,Ha,R2) = X )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,R2)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,R2)),lim(product_unit,Ha)) ) ) ) ).
% sngr_assn_raw.simps
tff(fact_4688_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ~ ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
tff(fact_4689_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ~ ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) ) ) ) ).
% sngr_assn_raw.elims(2)
tff(fact_4690_relH__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit),R2: ref(A)] :
( relH(As,Ha,H2)
=> ( member(nat,addr_of_ref(A,R2),As)
=> ( ref_get(A,Ha,R2) = ref_get(A,H2,R2) ) ) ) ) ).
% relH_ref
tff(fact_4691_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) ) ) ) ).
% sngr_assn_raw.elims(3)
tff(fact_4692_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
tff(fact_4693_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ~ ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
tff(fact_4694_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ( ( ref_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H)) ) )
=> ~ aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
tff(fact_4695_relH__set__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),As: set(nat),Ha: heap_ext(product_unit),X: A] :
( ~ member(nat,addr_of_ref(A,R2),As)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> relH(As,Ha,ref_set(A,R2,X,Ha)) ) ) ) ).
% relH_set_ref
tff(fact_4696_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A),Ha: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
<=> ( ( array_get(A,Ha,R2) = X )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,R2)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,R2)),lim(product_unit,Ha)) ) ) ) ).
% snga_assn_raw.simps
tff(fact_4697_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( (Y)
<=> ~ ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
tff(fact_4698_relH__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [As: set(nat),Ha: heap_ext(product_unit),H2: heap_ext(product_unit),R2: array(A)] :
( relH(As,Ha,H2)
=> ( member(nat,addr_of_array(A,R2),As)
=> ( array_get(A,Ha,R2) = array_get(A,H2,R2) ) ) ) ) ).
% relH_array
tff(fact_4699_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) ) ) ) ).
% snga_assn_raw.elims(3)
tff(fact_4700_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ~ ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) ) ) ) ).
% snga_assn_raw.elims(2)
tff(fact_4701_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
tff(fact_4702_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))))
=> ~ ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
tff(fact_4703_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ( ( array_get(A,H,X) = Xa )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H)) ) )
=> ~ aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
tff(fact_4704_relH__set__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),As: set(nat),Ha: heap_ext(product_unit),X: list(A)] :
( ~ member(nat,addr_of_array(A,R2),As)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Ha),As))
=> relH(As,Ha,array_set(A,R2,X,Ha)) ) ) ) ).
% relH_set_array
tff(fact_4705_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [R2: set(product_prod(A,A)),As8: fun(A,B)] :
( bNF_Ca3754400796208372196lChain(A,B,R2,As8)
<=> ! [I4: A,J3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3),R2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As8,I4)),aa(A,B,As8,J3)) ) ) ) ).
% relChain_def
tff(fact_4706_prod_H__def,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ( groups1962203154675924110t_prod(A,B) = groups_comm_monoid_G(B,A,times_times(B),one_one(B)) ) ) ).
% prod'_def
tff(fact_4707_cofinal__def,axiom,
! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
( bNF_Ca7293521722713021262ofinal(A,A3,R2)
<=> ! [X2: A] :
( member(A,X2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ? [Xa2: A] :
( member(A,Xa2,A3)
& ( X2 != Xa2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R2) ) ) ) ).
% cofinal_def
tff(fact_4708_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B5: set(A),A3: set(B)] :
( ( B5 != bot_bot(set(A)) )
=> ( ~ ? [F6: fun(B,A)] : aa(set(B),set(A),image2(B,A,F6),A3) = B5
<=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A3)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_We4044943003108391690rdLess(B,A)) ) ) ).
% card_of_ordLess2
tff(fact_4709_Gr__def,axiom,
! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] : bNF_Gr(A,B,A3,F2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),A3),F2)) ).
% Gr_def
tff(fact_4710_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A3: set(A),F2: fun(A,B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx),bNF_Gr(A,B,A3,F2))
=> member(A,X,A3) ) ).
% GrD1
tff(fact_4711_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A3: set(A),F2: fun(A,B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx),bNF_Gr(A,B,A3,F2))
=> ( aa(A,B,F2,X) = Fx ) ) ).
% GrD2
tff(fact_4712_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A3: set(A),B3: B] :
( ( A3 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A)) ) ).
% card_of_singl_ordLeq
tff(fact_4713_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( ( A3 != bot_bot(set(A)) )
=> ( ? [G4: fun(B,A)] : aa(set(B),set(A),image2(B,A,G4),B5) = A3
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)) ) ) ).
% card_of_ordLeq2
tff(fact_4714_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.comm_monoid_axioms
tff(fact_4715_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: A] :
( comm_monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A4),Z2) = A4 ) ) ).
% comm_monoid.comm_neutral
tff(fact_4716_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.comm_monoid_axioms
tff(fact_4717_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> comm_monoid(A,times_times(A),one_one(A)) ) ).
% mult.comm_monoid_axioms
tff(fact_4718_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> comm_monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.comm_monoid_axioms
tff(fact_4719_card__of__empty3,axiom,
! [B: $tType,A: $tType,A3: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B))
=> ( A3 = bot_bot(set(A)) ) ) ).
% card_of_empty3
tff(fact_4720_card__of__empty,axiom,
! [B: $tType,A: $tType,A3: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A3)),bNF_Wellorder_ordLeq(A,B)) ).
% card_of_empty
tff(fact_4721_exists__minim__Well__order,axiom,
! [A: $tType,R3: set(set(product_prod(A,A)))] :
( ( R3 != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
=> order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
& ! [Xa3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa3,R3)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).
% exists_minim_Well_order
tff(fact_4722_card__of__Times2,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( A3 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ).
% card_of_Times2
tff(fact_4723_card__of__Times1,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( A3 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),A3)))),bNF_Wellorder_ordLeq(B,product_prod(B,A))) ) ).
% card_of_Times1
tff(fact_4724_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set(A),B5: set(B)] :
( ( ( A1 != A22 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A3) )
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A3,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).
% card_of_Plus_Times_aux
tff(fact_4725_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordIso(A,product_prod(A,B))) ) ) ) ).
% card_of_Times_infinite_simps(2)
tff(fact_4726_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),A3)))),bNF_Wellorder_ordIso(A,product_prod(B,A))) ) ) ) ).
% card_of_Times_infinite_simps(4)
tff(fact_4727_card__of__empty2,axiom,
! [B: $tType,A: $tType,A3: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B))
=> ( A3 = bot_bot(set(A)) ) ) ).
% card_of_empty2
tff(fact_4728_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B)) ).
% card_of_empty_ordIso
tff(fact_4729_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set(A),B1: B,B22: B,B5: set(B)] :
( ( ( A1 != A22 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A3) )
=> ( ( ( B1 != B22 )
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B1),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B22),bot_bot(set(B))))),B5) )
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A3,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).
% card_of_Plus_Times
tff(fact_4730_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ( sum_Plus(A,B,A3,B5) = bot_bot(set(sum_sum(A,B))) )
<=> ( ( A3 = bot_bot(set(A)) )
& ( B5 = bot_bot(set(B)) ) ) ) ).
% Plus_eq_empty_conv
tff(fact_4731_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),A3)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ).
% card_of_Times_infinite_simps(3)
tff(fact_4732_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A3: set(A),B5: set(B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).
% card_of_Times_infinite_simps(1)
tff(fact_4733_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of($o,top_top(set($o)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A)))))),bNF_Wellorder_ordIso($o,A)) ) ).
% card_of_bool
tff(fact_4734_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A3: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A3,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B))) ).
% card_of_Plus_empty1
tff(fact_4735_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A3: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A3))),bNF_Wellorder_ordIso(A,sum_sum(B,A))) ).
% card_of_Plus_empty2
tff(fact_4736_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A3: set(A),B5: set(B)] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordIso(product_prod(A,B),A))
& member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_au(set(A),fun(B,set(A)),A3)))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ).
% card_of_Times_infinite
tff(fact_4737_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( aa(set(product_prod(B,B)),set(B),field2(B),P2) != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P2),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_tp(set(product_prod(B,B)),fun(A,set(B)),P2)))),R2),bNF_Wellorder_ordIso(product_prod(A,B),A))
& member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,aa(set(product_prod(B,B)),set(B),field2(B),P2),aTP_Lamp_tq(set(product_prod(A,A)),fun(B,set(A)),R2)))),R2),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ) ).
% Card_order_Times_infinite
tff(fact_4738_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A3: set(B)] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( A3 != bot_bot(set(B)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A3,aTP_Lamp_tq(set(product_prod(A,A)),fun(B,set(A)),R2)))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ) ) ).
% Card_order_Times2
tff(fact_4739_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B5: set(B)] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( B5 != bot_bot(set(B)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_aw(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(A,product_prod(A,B))) ) ) ).
% Card_order_Times1
tff(fact_4740_UNIV__bool,axiom,
top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).
% UNIV_bool
tff(fact_4741_Card__order__trans,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( X != Y )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( ( Y != Z2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> ( ( X != Z2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ) ) ) ).
% Card_order_trans
tff(fact_4742_Card__order__wo__rel,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> bNF_Wellorder_wo_rel(A,R2) ) ).
% Card_order_wo_rel
tff(fact_4743_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: A] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,A4,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ? [X3: A] :
( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ( A4 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),X3),R2) ) ) ) ) ).
% infinite_Card_order_limit
tff(fact_4744_exists__minim__Card__order,axiom,
! [A: $tType,R3: set(set(product_prod(A,A)))] :
( ( R3 != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
=> bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R3)
& ! [Xa3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa3,R3)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).
% exists_minim_Card_order
tff(fact_4745_Card__order__empty,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).
% Card_order_empty
tff(fact_4746_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B3: B] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B))))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).
% Card_order_singl_ordLeq
tff(fact_4747_card__of__empty1,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
( ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
| bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).
% card_of_empty1
tff(fact_4748_Cinfinite__limit__finite,axiom,
! [A: $tType,X5: set(A),R2: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite2(A),X5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
=> ? [X3: A] :
( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ! [Xa3: A] :
( member(A,Xa3,X5)
=> ( ( Xa3 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3),R2) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
tff(fact_4749_czeroI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
=> ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)) ) ) ).
% czeroI
tff(fact_4750_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
=> ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) ) ) ).
% Cnotzero_imp_not_empty
tff(fact_4751_czero__def,axiom,
! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).
% czero_def
tff(fact_4752_Cinfinite__limit,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] :
( member(A,X,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
=> ? [X3: A] :
( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ( X != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X3),R2) ) ) ) ).
% Cinfinite_limit
tff(fact_4753_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R2: set(product_prod(A,A)),X22: A] :
( member(A,X1,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(A,X22,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
=> ? [X3: A] :
( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
& ( X1 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3),R2)
& ( X22 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X3),R2) ) ) ) ) ).
% Cinfinite_limit2
tff(fact_4754_czeroE,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
=> ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) ) ) ).
% czeroE
tff(fact_4755_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A3: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
<=> ( A3 = bot_bot(set(A)) ) ) ).
% card_of_ordIso_czero_iff_empty
tff(fact_4756_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q3),Q3)
=> ( ( ( aa(set(product_prod(A,A)),set(A),field2(A),P23) = bot_bot(set(A)) )
=> ( aa(set(product_prod(B,B)),set(B),field2(B),R22) = bot_bot(set(B)) ) )
=> member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P23)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).
% cexp_mono2'
tff(fact_4757_Rep__unit__induct,axiom,
! [Y: $o,P: fun($o,$o)] :
( member($o,(Y),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> ( ! [X3: product_unit] : aa($o,$o,P,aa(product_unit,$o,product_Rep_unit,X3))
=> aa($o,$o,P,(Y)) ) ) ).
% Rep_unit_induct
tff(fact_4758_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( member($o,(X),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> ( member($o,(Y),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> ( ( aa($o,product_unit,product_Abs_unit,(X)) = aa($o,product_unit,product_Abs_unit,(Y)) )
<=> ( (X)
<=> (Y) ) ) ) ) ).
% Abs_unit_inject
tff(fact_4759_Rep__unit__inject,axiom,
! [X: product_unit,Y: product_unit] :
( ( aa(product_unit,$o,product_Rep_unit,X)
<=> aa(product_unit,$o,product_Rep_unit,Y) )
<=> ( X = Y ) ) ).
% Rep_unit_inject
tff(fact_4760_Rep__unit__inverse,axiom,
! [X: product_unit] : aa($o,product_unit,product_Abs_unit,aa(product_unit,$o,product_Rep_unit,X)) = X ).
% Rep_unit_inverse
tff(fact_4761_Abs__unit__inverse,axiom,
! [Y: $o] :
( member($o,(Y),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> ( aa(product_unit,$o,product_Rep_unit,aa($o,product_unit,product_Abs_unit,(Y)))
<=> (Y) ) ) ).
% Abs_unit_inverse
tff(fact_4762_type__definition__unit,axiom,
type_definition(product_unit,$o,product_Rep_unit,product_Abs_unit,aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).
% type_definition_unit
tff(fact_4763_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P13),R1),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordLeq(C,D))
=> ( ( ( aa(set(product_prod(C,C)),set(C),field2(C),P23) = bot_bot(set(C)) )
=> ( aa(set(product_prod(D,D)),set(D),field2(D),R22) = bot_bot(set(D)) ) )
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P13,P23)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ).
% cexp_mono'
tff(fact_4764_Rep__unit,axiom,
! [X: product_unit] : member($o,aa(product_unit,$o,product_Rep_unit,X),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).
% Rep_unit
tff(fact_4765_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X = aa($o,product_unit,product_Abs_unit,(Y2)) )
=> ~ member($o,(Y2),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ) ).
% Abs_unit_cases
tff(fact_4766_Rep__unit__cases,axiom,
! [Y: $o] :
( member($o,(Y),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> ~ ! [X3: product_unit] :
( (Y)
<=> ~ aa(product_unit,$o,product_Rep_unit,X3) ) ) ).
% Rep_unit_cases
tff(fact_4767_Abs__unit__induct,axiom,
! [P: fun(product_unit,$o),X: product_unit] :
( ! [Y2: $o] :
( member($o,(Y2),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o))))
=> aa(product_unit,$o,P,aa($o,product_unit,product_Abs_unit,(Y2))) )
=> aa(product_unit,$o,P,X) ) ).
% Abs_unit_induct
tff(fact_4768_prod_Oreindex__bij__betw__not__neutral,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [S4: set(A),T6: set(B),Ha: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
( aa(set(A),$o,finite_finite2(A),S4)
=> ( aa(set(B),$o,finite_finite2(B),T6)
=> ( bij_betw(A,B,Ha,minus_minus(set(A),S,S4),minus_minus(set(B),T3,T6))
=> ( ! [A5: A] :
( member(A,A5,S4)
=> ( aa(B,C,G,aa(A,B,Ha,A5)) = one_one(C) ) )
=> ( ! [B2: B] :
( member(B,B2,T6)
=> ( aa(B,C,G,B2) = one_one(C) ) )
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_tr(fun(A,B),fun(fun(B,C),fun(A,C)),Ha),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T3) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
tff(fact_4769_prod__mset__def,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ( comm_m9189036328036947845d_mset(A) = comm_monoid_F(A,times_times(A),one_one(A)) ) ) ).
% prod_mset_def
tff(fact_4770_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite_folding_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_folding_F(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_folding_F(A,B,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% folding_on.insert_remove
tff(fact_4771_bijI_H,axiom,
! [A: $tType,B: $tType,F2: fun(A,B)] :
( ! [X3: A,Y2: A] :
( ( aa(A,B,F2,X3) = aa(A,B,F2,Y2) )
<=> ( X3 = Y2 ) )
=> ( ! [Y2: B] :
? [X4: A] : Y2 = aa(A,B,F2,X4)
=> bij_betw(A,B,F2,top_top(set(A)),top_top(set(B))) ) ) ).
% bijI'
tff(fact_4772_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X5: set(A),F2: fun(A,B),Y4: set(B)] :
( ! [X3: A] :
( member(A,X3,X5)
=> ! [Y2: A] :
( member(A,Y2,X5)
=> ( ( aa(A,B,F2,X3) = aa(A,B,F2,Y2) )
<=> ( X3 = Y2 ) ) ) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> member(B,aa(A,B,F2,X3),Y4) )
=> ( ! [Y2: B] :
( member(B,Y2,Y4)
=> ? [X4: A] :
( member(A,X4,X5)
& ( Y2 = aa(A,B,F2,X4) ) ) )
=> bij_betw(A,B,F2,X5,Y4) ) ) ) ).
% bij_betwI'
tff(fact_4773_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Z2: B] :
( finite_folding_on(A,B,S,F2)
=> ( finite_folding_F(A,B,F2,Z2,bot_bot(set(A))) = Z2 ) ) ).
% folding_on.empty
tff(fact_4774_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
( bij_betw(A,B,F2,A3,bot_bot(set(B)))
=> ( A3 = bot_bot(set(A)) ) ) ).
% bij_betw_empty2
tff(fact_4775_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
( bij_betw(A,B,F2,bot_bot(set(A)),A3)
=> ( A3 = bot_bot(set(B)) ) ) ).
% bij_betw_empty1
tff(fact_4776_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite_folding_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( finite_folding_F(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_folding_F(A,B,F2,Z2,A3)) ) ) ) ) ) ).
% folding_on.insert
tff(fact_4777_bij__swap,axiom,
! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).
% bij_swap
tff(fact_4778_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A3: set(A),F2: fun(A,B),A9: set(B)] :
( ~ member(A,B3,A3)
=> ( ~ member(B,aa(A,B,F2,B3),A9)
=> ( bij_betw(A,B,F2,A3,A9)
<=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A9),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw3
tff(fact_4779_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A3: set(A),F2: fun(A,B),A9: set(B)] :
( ~ member(A,B3,A3)
=> ( ~ member(B,aa(A,B,F2,B3),A9)
=> ( bij_betw(A,B,F2,A3,A9)
=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A9),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw
tff(fact_4780_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B5: set(B),C6: set(A),D4: set(B)] :
( bij_betw(A,B,F2,A3,B5)
=> ( bij_betw(A,B,F2,C6,D4)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D4)) ) ) ) ).
% bij_betw_combine
tff(fact_4781_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C6: set(A),B5: set(B),D4: set(B)] :
( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D4))
=> ( bij_betw(A,B,F2,C6,D4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C6) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F2,A3,B5) ) ) ) ) ).
% bij_betw_partition
tff(fact_4782_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A3)),R2),bNF_Wellorder_ordLeq(A,B))
=> ? [F: fun(B,A),B6: set(B)] : bij_betw(B,A,F,B6,A3) ) ).
% ex_bij_betw
tff(fact_4783_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C6: set(B),G: fun(A,B),B5: set(A),D4: set(B)] :
( bij_betw(A,B,F2,A3,C6)
=> ( bij_betw(A,B,G,B5,D4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B5) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C6),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_fa(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C6),D4)) ) ) ) ) ).
% bij_betw_disjoint_Un
tff(fact_4784_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set(A),A4: A] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ? [H: fun(A,A)] : bij_betw(A,A,H,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw2
tff(fact_4785_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set(A),A4: A] :
( ~ aa(set(A),$o,finite_finite2(A),A3)
=> ? [H: fun(A,A)] : bij_betw(A,A,H,A3,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw
tff(fact_4786_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A3: set(A),X: A,Z2: B] :
( finite_folding_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( finite_folding_F(A,B,F2,Z2,A3) = aa(B,B,aa(A,fun(B,B),F2,X),finite_folding_F(A,B,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% folding_on.remove
tff(fact_4787_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z2: B] :
( finite1890593828518410140dem_on(A,B,S,F2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( finite_folding_F(A,B,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_folding_F(A,B,F2,Z2,A3)) ) ) ) ) ).
% folding_idem_on.insert_idem
tff(fact_4788_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B22)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B1)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).
% integer_of_char_code
tff(fact_4789_trancl__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : transitive_trancl(A,X4) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_tranclp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),X4)))) ).
% trancl_def
tff(fact_4790_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( ! [A5: A,B2: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2))
=> aa(B,$o,aa(A,fun(B,$o),P,A5),B2) )
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
=> ( aa(B,$o,aa(A,fun(B,$o),P,A5),B2)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% tranclp_induct2
tff(fact_4791_tranclp__trancl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3),transitive_trancl(A,R2)) ) ).
% tranclp_trancl_eq
tff(fact_4792_Nitpick_Otranclp__unfold,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,R2),A4),B3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),transitive_trancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% Nitpick.tranclp_unfold
tff(fact_4793_reflp__refl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( reflp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> refl_on(A,top_top(set(A)),R2) ) ).
% reflp_refl_eq
tff(fact_4794_fun__of__rel__def,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R3,X) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),R3),X)) ).
% fun_of_rel_def
tff(fact_4795_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
% Eps_case_prod_eq
tff(fact_4796_some__insert__self,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S))),S) = S ) ) ).
% some_insert_self
tff(fact_4797_reflp__inf,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( reflp(A,R2)
=> ( reflp(A,S2)
=> reflp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S2)) ) ) ).
% reflp_inf
tff(fact_4798_reflpD,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A] :
( reflp(A,R2)
=> aa(A,$o,aa(A,fun(A,$o),R2,X),X) ) ).
% reflpD
tff(fact_4799_reflpE,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A] :
( reflp(A,R2)
=> aa(A,$o,aa(A,fun(A,$o),R2,X),X) ) ).
% reflpE
tff(fact_4800_reflpI,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( ! [X3: A] : aa(A,$o,aa(A,fun(A,$o),R2,X3),X3)
=> reflp(A,R2) ) ).
% reflpI
tff(fact_4801_reflp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( reflp(A,R2)
<=> ! [X2: A] : aa(A,$o,aa(A,fun(A,$o),R2,X2),X2) ) ).
% reflp_def
tff(fact_4802_reflp__mono,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
( reflp(A,R3)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),Q,X3),Y2) )
=> reflp(A,Q) ) ) ).
% reflp_mono
tff(fact_4803_reflp__equality,axiom,
! [A: $tType] : reflp(A,fequal(A)) ).
% reflp_equality
tff(fact_4804_reflp__sup,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( reflp(A,R2)
=> ( reflp(A,S2)
=> reflp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),R2),S2)) ) ) ).
% reflp_sup
tff(fact_4805_reflp__eq,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( reflp(A,R3)
<=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R3) ) ).
% reflp_eq
tff(fact_4806_some__elem,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),S) ) ).
% some_elem
tff(fact_4807_some__in__eq,axiom,
! [A: $tType,A3: set(A)] :
( member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),A3)
<=> ( A3 != bot_bot(set(A)) ) ) ).
% some_in_eq
tff(fact_4808_inv__on__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),X: B] : aa(B,A,inv_on(A,B,F2,A3),X) = fChoice(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_lb(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),A3),X)) ).
% inv_on_def
tff(fact_4809_pick__middlep__def,axiom,
! [B: $tType,A: $tType,C: $tType,P: fun(B,fun(A,$o)),Q: fun(A,fun(C,$o)),A4: B,C2: C] : bNF_pick_middlep(B,A,C,P,Q,A4,C2) = fChoice(A,aa(C,fun(A,$o),aa(B,fun(C,fun(A,$o)),aa(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o))),aTP_Lamp_ts(fun(B,fun(A,$o)),fun(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o)))),P),Q),A4),C2)) ).
% pick_middlep_def
tff(fact_4810_split__paired__Eps,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_tt(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).
% split_paired_Eps
tff(fact_4811_some__theI,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
( ? [A12: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),P,A12),X_13)
=> ( ! [B17: B,B24: B] :
( ? [A5: A] : aa(B,$o,aa(A,fun(B,$o),P,A5),B17)
=> ( ? [A5: A] : aa(B,$o,aa(A,fun(B,$o),P,A5),B24)
=> ( B17 = B24 ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,fChoice(A,aTP_Lamp_mf(fun(A,fun(B,$o)),fun(A,$o),P))),the(B,aTP_Lamp_tu(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).
% some_theI
tff(fact_4812_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = fChoice(product_prod(A,B),aTP_Lamp_le(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).
% Eps_case_prod
tff(fact_4813_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M)) ).
% inj_on_fst_map_to_set
tff(fact_4814_old_Orec__bool__def,axiom,
! [A: $tType,X4: A,Xa3: A,Xb2: $o] : product_rec_bool(A,X4,Xa3,(Xb2)) = the(A,product_rec_set_bool(A,X4,Xa3,(Xb2))) ).
% old.rec_bool_def
tff(fact_4815_sum__encode__def,axiom,
! [X: sum_sum(nat,nat)] : nat_sum_encode(X) = aa(sum_sum(nat,nat),nat,sum_case_sum(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aTP_Lamp_tv(nat,nat)),X) ).
% sum_encode_def
tff(fact_4816_old_Obool_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F22: A] : product_rec_bool(A,F1,F22,$false) = F22 ).
% old.bool.simps(6)
tff(fact_4817_old_Obool_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: A] : product_rec_bool(A,F1,F22,$true) = F1 ).
% old.bool.simps(5)
tff(fact_4818_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V: B] : map_to_set(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),minus_minus(set(product_prod(A,B)),map_to_set(A,B,M),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_tw(A,fun(product_prod(A,B),$o),K)))) ).
% map_to_set_upd
tff(fact_4819_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( inj_on(product_prod(A,B),A,product_fst(A,B),S)
=> ( map_to_set(A,B,set_to_map(A,B,S)) = S ) ) ).
% set_to_map_inverse
tff(fact_4820_map__to__set__ran,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M)) ).
% map_to_set_ran
tff(fact_4821_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K2: set(A),M: fun(A,option(B)),V: B] :
( member(A,K,K2)
=> ( aa(A,option(B),map_mmupd(A,B,M,K2,V),K) = aa(B,option(B),some(B),V) ) ) ).
% mmupd_in_upd
tff(fact_4822_map__mmupd__def,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),K2: set(B),V: A,K: B] :
aa(B,option(A),map_mmupd(B,A,M,K2,V),K) = $ite(member(B,K,K2),aa(A,option(A),some(A),V),aa(B,option(A),M,K)) ).
% map_mmupd_def
tff(fact_4823_map__mmupdE,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),K2: set(B),V: A,K: B,X: A] :
( ( aa(B,option(A),map_mmupd(B,A,M,K2,V),K) = aa(A,option(A),some(A),X) )
=> ( ( ~ member(B,K,K2)
=> ( aa(B,option(A),M,K) != aa(A,option(A),some(A),X) ) )
=> ~ ( member(B,K,K2)
=> ( X != V ) ) ) ) ).
% map_mmupdE
tff(fact_4824_le__some__optE,axiom,
! [A: $tType] :
( preorder(A)
=> ! [M: A,X: option(A)] :
( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),X)
=> ~ ! [M6: A] :
( ( X = aa(A,option(A),some(A),M6) )
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),M6) ) ) ) ).
% le_some_optE
tff(fact_4825_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B)),K: A,V: B] :
( inj_on(product_prod(A,B),A,product_fst(A,B),S)
=> ( ( aa(A,option(B),set_to_map(A,B,S),K) = aa(B,option(B),some(B),V) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),S) ) ) ).
% set_to_map_simp
tff(fact_4826_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S)) ).
% set_to_map_ran
tff(fact_4827_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M)) = M ).
% map_to_set_inverse
tff(fact_4828_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod(A,B),S: set(product_prod(A,B))] :
( ~ member(A,aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
=> ( set_to_map(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),Kv),S)) = fun_upd(A,option(B),set_to_map(A,B,S),aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),Kv))) ) ) ).
% set_to_map_insert
tff(fact_4829_map__to__set__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_tx(fun(A,option(B)),fun(A,fun(B,$o)),M))) ).
% map_to_set_def
tff(fact_4830_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [A3: set(A),F2: fun(A,B)] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_ty(fun(A,B),fun(A,option(B)),F2)),A3)) ) ) ) ).
% Some_SUP
tff(fact_4831_rel__of__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_tz(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).
% rel_of_def
tff(fact_4832_Some__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(A)] :
( ( A3 != bot_bot(set(A)) )
=> ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A3)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A3)) ) ) ) ).
% Some_Sup
tff(fact_4833_the__dflt__None__nonempty,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ).
% the_dflt_None_nonempty
tff(fact_4834_set__to__map__def,axiom,
! [A: $tType,B: $tType,S: set(product_prod(B,A)),K: B] : aa(B,option(A),set_to_map(B,A,S),K) = eps_Opt(A,aa(B,fun(A,$o),aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),S),K)) ).
% set_to_map_def
tff(fact_4835_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu: A,X: A] : the_default(A,Uu,aa(A,option(A),some(A),X)) = X ).
% the_default.simps(1)
tff(fact_4836_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] : eps_Opt(A,aa(A,fun(A,$o),fequal(A),X)) = aa(A,option(A),some(A),X) ).
% some_opt_sym_eq_trivial
tff(fact_4837_the__dflt__None__set,axiom,
! [A: $tType,X: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X)) = X ).
% the_dflt_None_set
tff(fact_4838_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_ai(A,fun(A,$o),X)) = aa(A,option(A),some(A),X) ).
% some_opt_eq_trivial
tff(fact_4839_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: fun(A,$o),X: A] :
( ! [X8: A] :
( aa(A,$o,P,X)
=> ( aa(A,$o,P,X8)
=> ( X8 = X ) ) )
=> ( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
<=> aa(A,$o,P,X) ) ) ).
% Eps_Opt_eq_Some
tff(fact_4840_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: fun(A,$o),X: A] :
( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
=> aa(A,$o,P,X) ) ).
% Eps_Opt_eq_Some_implies
tff(fact_4841_Eps__Opt__def,axiom,
! [A: $tType,P: fun(A,$o)] :
eps_Opt(A,P) = $ite(
? [X9: A] : aa(A,$o,P,X9),
aa(A,option(A),some(A),fChoice(A,P)),
none(A) ) ).
% Eps_Opt_def
tff(fact_4842_dflt__None__set__def,axiom,
! [A: $tType,S: set(A)] :
dflt_None_set(A,S) = $ite(S = bot_bot(set(A)),none(set(A)),aa(set(A),option(set(A)),some(set(A)),S)) ).
% dflt_None_set_def
tff(fact_4843_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y: A,Z2: A] :
( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) )
=> ( inj_on(B,option(A),M,dom(B,A,M))
=> ( ~ member(A,Z2,ran(B,A,M))
=> ( ran(B,A,fun_upd(B,option(A),M,X,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),ran(B,A,M),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z2),bot_bot(set(A)))) ) ) ) ) ).
% ran_map_upd_Some
tff(fact_4844_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
( ! [X2: A,Y3: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3))
<=> ( V = none(product_prod(A,B)) ) ) ).
% not_Some_eq2
tff(fact_4845_empty__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).
% empty_Sup
tff(fact_4846_singleton__None__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).
% singleton_None_Sup
tff(fact_4847_some__opt__false__trivial,axiom,
! [A: $tType] : eps_Opt(A,aTP_Lamp_cy(A,$o)) = none(A) ).
% some_opt_false_trivial
tff(fact_4848_dom__eq__empty__conv,axiom,
! [A: $tType,B: $tType,F2: fun(A,option(B))] :
( ( dom(A,B,F2) = bot_bot(set(A)) )
<=> ! [X2: A] : aa(A,option(B),F2,X2) = none(B) ) ).
% dom_eq_empty_conv
tff(fact_4849_Eps__Opt__eq__None,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( eps_Opt(A,P) = none(A) )
<=> ~ ? [X_12: A] : aa(A,$o,P,X_12) ) ).
% Eps_Opt_eq_None
tff(fact_4850_dom__empty,axiom,
! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_ua(A,option(B))) = bot_bot(set(A)) ).
% dom_empty
tff(fact_4851_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_ub(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K),V),M)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),dom(A,B,M)) ).
% map_update_eta_repair(1)
tff(fact_4852_dom__const_H,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] : dom(A,B,aTP_Lamp_uc(fun(A,B),fun(A,option(B)),F2)) = top_top(set(A)) ).
% dom_const'
tff(fact_4853_set__to__map__empty,axiom,
! [A: $tType,B: $tType,X4: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X4) = none(B) ).
% set_to_map_empty
tff(fact_4854_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K2: set(A),V: B] : dom(A,B,map_mmupd(A,B,M,K2,V)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M)),K2) ).
% dom_mmupd
tff(fact_4855_ran__empty,axiom,
! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_ud(B,option(A))) = bot_bot(set(A)) ).
% ran_empty
tff(fact_4856_map__to__set__empty,axiom,
! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_ua(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% map_to_set_empty
tff(fact_4857_map__update__eta__repair_I2_J,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
( ( aa(B,option(A),M,K) = none(A) )
=> ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_ue(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),ran(B,A,M)) ) ) ).
% map_update_eta_repair(2)
tff(fact_4858_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_ua(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).
% rel_of_empty
tff(fact_4859_the__dflt__None__empty,axiom,
! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).
% the_dflt_None_empty
tff(fact_4860_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A4: B,B3: A] :
( ( aa(B,option(A),M,A4) = none(A) )
=> ( ran(B,A,fun_upd(B,option(A),M,A4,aa(A,option(A),some(A),B3))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),ran(B,A,M)) ) ) ).
% ran_map_upd
tff(fact_4861_dom__fun__upd,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),X: A,Y: option(B)] :
dom(A,B,fun_upd(A,option(B),F2,X,Y)) = $ite(Y = none(B),minus_minus(set(A),dom(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),dom(A,B,F2))) ).
% dom_fun_upd
tff(fact_4862_insert__dom,axiom,
! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,Y: A] :
( ( aa(B,option(A),F2,X) = aa(A,option(A),some(A),Y) )
=> ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),dom(B,A,F2)) = dom(B,A,F2) ) ) ).
% insert_dom
tff(fact_4863_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] :
( ( dom(A,B,F2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
<=> ? [V4: B] : F2 = fun_upd(A,option(B),aTP_Lamp_ua(A,option(B)),X,aa(B,option(B),some(B),V4)) ) ).
% dom_eq_singleton_conv
tff(fact_4864_dom__minus,axiom,
! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,A3: set(B)] :
( ( aa(B,option(A),F2,X) = none(A) )
=> ( minus_minus(set(B),dom(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = minus_minus(set(B),dom(B,A,F2),A3) ) ) ).
% dom_minus
tff(fact_4865_nempty__dom,axiom,
! [B: $tType,A: $tType,E4: fun(A,option(B))] :
( ~ ! [X4: A] : aa(A,option(B),E4,X4) = none(B)
=> ~ ! [M3: A] : ~ member(A,M3,dom(A,B,E4)) ) ).
% nempty_dom
tff(fact_4866_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B))] :
( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M),M7)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),dom(A,B,M)),dom(A,B,M7)) ) ) ).
% le_map_dom_mono
tff(fact_4867_bot__option__def,axiom,
! [A: $tType] :
( order(A)
=> ( bot_bot(option(A)) = none(A) ) ) ).
% bot_option_def
tff(fact_4868_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] :
( aa(set(A),$o,finite_finite2(A),dom(A,B,M2))
=> aa(set(B),$o,finite_finite2(B),ran(A,B,M2)) ) ).
% map_dom_ran_finite
tff(fact_4869_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] : the_default(A,X,none(A)) = X ).
% the_default.simps(2)
tff(fact_4870_map__to__set__empty__iff_I2_J,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B))] :
( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
<=> ! [X2: A] : aa(A,option(B),M,X2) = none(B) ) ).
% map_to_set_empty_iff(2)
tff(fact_4871_map__to__set__empty__iff_I1_J,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B))] :
( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
<=> ! [X2: A] : aa(A,option(B),M,X2) = none(B) ) ).
% map_to_set_empty_iff(1)
tff(fact_4872_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ! [X2: A] : aa(A,option(B),set_to_map(A,B,S),X2) = none(B)
<=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(1)
tff(fact_4873_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y: A] :
( aa(set(A),$o,finite_finite2(A),dom(A,B,M2))
=> ( ( finite_card(A,dom(A,B,M2)) = finite_card(B,ran(A,B,M2)) )
=> ( member(A,X,dom(A,B,M2))
=> ( ( aa(A,option(B),M2,X) = aa(A,option(B),M2,Y) )
<=> ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
tff(fact_4874_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ( aTP_Lamp_ua(A,option(B)) = set_to_map(A,B,S) )
<=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(2)
tff(fact_4875_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] :
( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),map_to_set(A,B,M))
<=> aa(set(A),$o,finite_finite2(A),dom(A,B,M)) ) ).
% finite_map_to_set
tff(fact_4876_map__to__set__dom,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M)) ).
% map_to_set_dom
tff(fact_4877_set__to__map__dom,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] : dom(A,B,set_to_map(A,B,S)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S) ).
% set_to_map_dom
tff(fact_4878_card__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : finite_card(product_prod(A,B),map_to_set(A,B,M)) = finite_card(A,dom(A,B,M)) ).
% card_map_to_set
tff(fact_4879_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).
% graph_map_upd
tff(fact_4880_Sup__option__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: set(option(A))] :
aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A3) = $ite(
( ( A3 = bot_bot(set(option(A))) )
| ( A3 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ),
none(A),
aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A3))) ) ) ).
% Sup_option_def
tff(fact_4881_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ).
% restrict_upd_same
tff(fact_4882_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B))] : restrict_map(A,B,F2,top_top(set(A))) = F2 ).
% restrict_map_UNIV
tff(fact_4883_restrict__map__self,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : restrict_map(A,B,M,dom(A,B,M)) = M ).
% restrict_map_self
tff(fact_4884_restrict__map__to__empty,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),X4: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X4) = none(B) ).
% restrict_map_to_empty
tff(fact_4885_graph__empty,axiom,
! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_ua(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% graph_empty
tff(fact_4886_restrict__map__inv,axiom,
! [A: $tType,B: $tType,F2: fun(A,option(B)),X4: A] : aa(A,option(B),restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F2))),X4) = none(B) ).
% restrict_map_inv
tff(fact_4887_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set(A),M: fun(A,option(B)),Y: option(B)] :
( member(A,X,D4)
=> ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,minus_minus(set(A),D4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ) ) ).
% fun_upd_restrict_conv
tff(fact_4888_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: option(B),D4: set(A)] :
restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D4) = $ite(member(A,X,D4),fun_upd(A,option(B),restrict_map(A,B,M,minus_minus(set(A),D4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y),restrict_map(A,B,M,D4)) ).
% restrict_fun_upd
tff(fact_4889_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),D4: set(A),X: A] :
fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = $ite(member(A,X,D4),restrict_map(A,B,M,minus_minus(set(A),D4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),restrict_map(A,B,M,D4)) ).
% fun_upd_None_restrict
tff(fact_4890_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B)),A3: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M,A3)))
=> ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).
% graph_restrictD(2)
tff(fact_4891_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B)),A3: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M,A3)))
=> member(A,K,A3) ) ).
% graph_restrictD(1)
tff(fact_4892_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M: fun(A,option(B)),X5: set(A)] : aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X5)),M) ) ).
% le_map_restrict
tff(fact_4893_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),R3: set(A),M7: fun(A,option(B)),R4: set(A)] :
( ( restrict_map(A,B,M,R3) = M7 )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R4),R3)
=> ( restrict_map(A,B,M,R4) = restrict_map(A,B,M7,R4) ) ) ) ).
% restrict_map_subset_eq
tff(fact_4894_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A3: set(B),K: B,V: A] :
( ( aa(B,option(A),restrict_map(B,A,M,A3),K) = aa(A,option(A),some(A),V) )
<=> ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
& member(B,K,A3) ) ) ).
% restrict_map_eq(2)
tff(fact_4895_restrict__map__insert,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),A4: A,A3: set(A)] : restrict_map(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A3)) = fun_upd(A,option(B),restrict_map(A,B,F2,A3),A4,aa(A,option(B),F2,A4)) ).
% restrict_map_insert
tff(fact_4896_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,M))
=> ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).
% in_graphD
tff(fact_4897_in__graphI,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V),graph(B,A,M)) ) ).
% in_graphI
tff(fact_4898_graph__ranD,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),M: fun(A,option(B))] :
( member(product_prod(A,B),X,graph(A,B,M))
=> member(B,aa(product_prod(A,B),B,product_snd(A,B),X),ran(A,B,M)) ) ).
% graph_ranD
tff(fact_4899_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),X: B,S2: set(B)] :
( ( aa(B,option(A),M,X) = none(A) )
=> ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),S2))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) ) ).
% map_restrict_insert_none_simp
tff(fact_4900_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A3: set(B),K: B] :
( ( aa(B,option(A),restrict_map(B,A,M,A3),K) = none(A) )
<=> ~ member(B,K,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M)),A3)) ) ).
% restrict_map_eq(1)
tff(fact_4901_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),S: set(A),K: A,V: B] : fun_upd(A,option(B),restrict_map(A,B,F2,S),K,aa(B,option(B),some(B),V)) = restrict_map(A,B,fun_upd(A,option(B),F2,K,aa(B,option(B),some(B),V)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),S)) ).
% restrict_map_upd
tff(fact_4902_graph__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_uf(fun(A,option(B)),fun(product_prod(A,B),$o),M)) ).
% graph_def
tff(fact_4903_snd__graph__ran,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),graph(B,A,M)) = ran(B,A,M) ).
% snd_graph_ran
tff(fact_4904_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),D4: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,minus_minus(set(A),D4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ).
% fun_upd_restrict
tff(fact_4905_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X: A] : fun_upd(A,option(B),M,X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ).
% map_upd_eq_restrict
tff(fact_4906_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] : restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = fun_upd(A,option(B),F2,X,none(B)) ).
% restrict_complement_singleton_eq
tff(fact_4907_these__insert__Some,axiom,
! [A: $tType,X: A,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),aa(A,option(A),some(A),X)),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),these(A,A3)) ).
% these_insert_Some
tff(fact_4908_these__empty,axiom,
! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).
% these_empty
tff(fact_4909_these__not__empty__eq,axiom,
! [A: $tType,B5: set(option(A))] :
( ( these(A,B5) != bot_bot(set(A)) )
<=> ( ( B5 != bot_bot(set(option(A))) )
& ( B5 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_not_empty_eq
tff(fact_4910_these__empty__eq,axiom,
! [A: $tType,B5: set(option(A))] :
( ( these(A,B5) = bot_bot(set(A)) )
<=> ( ( B5 = bot_bot(set(option(A))) )
| ( B5 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_empty_eq
tff(fact_4911_ran__map__add,axiom,
! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,M12,M23)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M12)),ran(A,B,M23)) ) ) ).
% ran_map_add
tff(fact_4912_ran__add,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F2)),dom(A,B,G)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,F2,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F2)),ran(A,B,G)) ) ) ).
% ran_add
tff(fact_4913_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S2: set(A),F2: fun(fun(A,option(B)),fun(A,option(B))),A3: fun(A,option(B))] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3)))
=> ( ( A3 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3),aa(set(A),set(A),uminus_uminus(set(A)),S2)) )
<=> ( map_le(A,B,A3,aa(fun(A,option(B)),fun(A,option(B)),F2,A3))
& ( S2 = minus_minus(set(A),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3)),dom(A,B,A3)) ) ) ) ) ).
% eq_f_restr_ss_eq
tff(fact_4914_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),K2: set(A),V: B] : map_le(A,B,M,map_mmupd(A,B,M,minus_minus(set(A),K2,dom(A,B,M)),V)) ).
% le_map_mmupd_not_dom
tff(fact_4915_map__leI,axiom,
! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
( ! [X3: A,V3: B] :
( ( aa(A,option(B),M12,X3) = aa(B,option(B),some(B),V3) )
=> ( aa(A,option(B),M23,X3) = aa(B,option(B),some(B),V3) ) )
=> map_le(A,B,M12,M23) ) ).
% map_leI
tff(fact_4916_map__leD,axiom,
! [A: $tType,B: $tType,M12: fun(A,option(B)),M23: fun(A,option(B)),K: A,V: B] :
( map_le(A,B,M12,M23)
=> ( ( aa(A,option(B),M12,K) = aa(B,option(B),some(B),V) )
=> ( aa(A,option(B),M23,K) = aa(B,option(B),some(B),V) ) ) ) ).
% map_leD
tff(fact_4917_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B)),N2: fun(A,option(B))] :
( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M),M7)
=> aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N2)),map_add(A,B,M7,N2)) ) ) ).
% map_add_first_le
tff(fact_4918_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: fun(B,option(A)),K: B,F2: fun(B,option(A))] :
( ( aa(B,option(A),G,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F2,G),K) = aa(B,option(A),F2,K) ) ) ).
% map_add_find_left
tff(fact_4919_map__add__left__None,axiom,
! [A: $tType,B: $tType,F2: fun(B,option(A)),K: B,G: fun(B,option(A))] :
( ( aa(B,option(A),F2,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F2,G),K) = aa(B,option(A),G,K) ) ) ).
% map_add_left_None
tff(fact_4920_map__add__comm,axiom,
! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
=> ( map_add(A,B,M12,M23) = map_add(A,B,M23,M12) ) ) ).
% map_add_comm
tff(fact_4921_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A3: fun(A,option(B)),B5: fun(A,option(B)),C6: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,A3)),dom(A,B,B5)) = bot_bot(set(A)) )
=> ( map_add(A,B,A3,map_add(A,B,B5,C6)) = map_add(A,B,B5,map_add(A,B,A3,C6)) ) ) ).
% map_add_left_comm
tff(fact_4922_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B)),N2: fun(A,option(B)),N8: fun(A,option(B))] :
( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M),M7)
=> ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),N2),N8)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M7)),dom(A,B,N8)) = bot_bot(set(A)) )
=> aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N2)),map_add(A,B,M7,N8)) ) ) ) ) ).
% map_add_distinct_le
tff(fact_4923_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K2: set(A),K7: set(A),M: fun(A,option(B)),V: B] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K2),K7)
=> map_le(A,B,map_mmupd(A,B,M,minus_minus(set(A),K2,dom(A,B,M)),V),map_mmupd(A,B,M,minus_minus(set(A),K7,dom(A,B,M)),V)) ) ).
% map_mmupd_update_less
tff(fact_4924_graph__map__add,axiom,
! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M12)),dom(A,B,M23)) = bot_bot(set(A)) )
=> ( graph(A,B,map_add(A,B,M12,M23)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M12)),graph(A,B,M23)) ) ) ).
% graph_map_add
tff(fact_4925_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S2: set(A),F2: fun(fun(A,option(B)),fun(A,option(B))),A3: fun(A,option(B))] :
( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3)))
& ( A3 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3),aa(set(A),set(A),uminus_uminus(set(A)),S2)) ) )
<=> ( map_le(A,B,A3,aa(fun(A,option(B)),fun(A,option(B)),F2,A3))
& ( S2 = minus_minus(set(A),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F2,A3)),dom(A,B,A3)) ) ) ) ).
% eq_f_restr_conv
tff(fact_4926_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_ug(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).
% graph_eq_to_snd_dom
tff(fact_4927_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D4: set(A),M: fun(A,option(B))] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),dom(A,B,M))
=> ( inj_on(A,option(B),M,D4)
=> inj_on(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),M),D4) ) ) ).
% inj_on_map_the
tff(fact_4928_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] : set_option(A,aa(A,option(A),some(A),X22)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X22),bot_bot(set(A))) ).
% option.simps(15)
tff(fact_4929_set__empty__eq,axiom,
! [A: $tType,Xo: option(A)] :
( ( set_option(A,Xo) = bot_bot(set(A)) )
<=> ( Xo = none(A) ) ) ).
% set_empty_eq
tff(fact_4930_option_Osimps_I14_J,axiom,
! [A: $tType] : set_option(A,none(A)) = bot_bot(set(A)) ).
% option.simps(14)
tff(fact_4931_ran__is__image,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A))] : ran(B,A,M2) = aa(set(B),set(A),image2(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),M2)),dom(B,A,M2)) ).
% ran_is_image
tff(fact_4932_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top.axioms(2)
tff(fact_4933_group_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> group_axioms(A,F2,Z2,Inverse) ) ).
% group.axioms(2)
tff(fact_4934_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),bit0(one2))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
& ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs2)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).
% mergesort_by_rel_split_length
tff(fact_4935_group__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A5)),A5) = Z2
=> group_axioms(A,F2,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_4936_group__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F2,Z2,Inverse)
<=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A7) = A7
& ! [A7: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A7)),A7) = Z2 ) ) ).
% group_axioms_def
tff(fact_4937_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top_axioms.intro
tff(fact_4938_ordering__top__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
( ordering_top_axioms(A,Less_eq,Top)
<=> ! [A7: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),Top) ) ).
% ordering_top_axioms_def
tff(fact_4939_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = minus_minus(nat,To,From) ) ) ) ).
% slice_len
tff(fact_4940_nth__step__trancl,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),N2: nat,M: nat] :
( ! [N3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N3))),aa(nat,A,nth(A,Xs),N3)),R3) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N2)),aa(nat,A,nth(A,Xs),M)),transitive_trancl(A,R3)) ) ) ) ).
% nth_step_trancl
tff(fact_4941_slice__eq__bounds__empty,axiom,
! [A: $tType,I2: nat,Xs: list(A)] : slice(A,I2,I2,Xs) = nil(A) ).
% slice_eq_bounds_empty
tff(fact_4942_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).
% slice_Nil
tff(fact_4943_slice__complete,axiom,
! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).
% slice_complete
tff(fact_4944_length__ge__1__conv,axiom,
! [A: $tType,L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
<=> ( L != nil(A) ) ) ).
% length_ge_1_conv
tff(fact_4945_obtain__list__from__elements,axiom,
! [A: $tType,N2: nat,P: fun(A,fun(nat,$o))] :
( ! [I3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),N2)
=> ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I3) )
=> ~ ! [L3: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L3) = N2 )
=> ~ ! [I6: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I6),N2)
=> aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L3),I6)),I6) ) ) ) ).
% obtain_list_from_elements
tff(fact_4946_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),L))
=> ( L != nil(A) ) ) ).
% len_greater_imp_nonempty
tff(fact_4947_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2) ).
% mergesort_by_rel_split.simps(1)
tff(fact_4948_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A),I2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),minus_minus(nat,To,From))
=> ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I2)) ) ) ) ) ).
% slice_nth
tff(fact_4949_product__nth,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list(A),Ys: list(B)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),divide_divide(nat,N2,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N2,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).
% product_nth
tff(fact_4950_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F2: fun(B,A),A4: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_uh(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A4),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum_funpow
tff(fact_4951_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F2: fun(B,A),A4: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ui(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A4),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum
tff(fact_4952_length__product,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).
% length_product
tff(fact_4953_sorted__in__between,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I2: nat,J2: nat,L: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L))
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),J2))
=> ~ ! [K3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),K3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),J2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K3)),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_4954_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),$o))] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),A0),A1))
=> ( ! [R8: fun(A,fun(A,$o)),Xs3: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R8),Xs3))
=> ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R8),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
=> ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R8),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R8),Xs3) ) ) )
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,A0),A1) ) ) ).
% mergesort_by_rel.pinduct
tff(fact_4955_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
=> ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa,merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))))) ) ) ).
% mergesort_by_rel.elims
tff(fact_4956_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] : aa(list(A),list(A),mergesort_by_rel(A,R3),nil(A)) = nil(A) ).
% mergesort_by_rel_simps(1)
tff(fact_4957_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Ys: list(A)] : merges9089515139780605204_merge(A,R3,nil(A),Ys) = Ys ).
% mergesort_by_rel_merge_simps(3)
tff(fact_4958_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
| aa(A,$o,aa(A,fun(A,$o),R3,Y2),X3) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R3,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),R3,X3),Z4) ) )
=> ( sorted_wrt(A,R3,merges9089515139780605204_merge(A,R3,Xs,Ys))
<=> ( sorted_wrt(A,R3,Xs)
& sorted_wrt(A,R3,Ys) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
tff(fact_4959_sorted__mergesort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xs)) ) ).
% sorted_mergesort_by_rel
tff(fact_4960_sorted__wrt__mergesort__by__rel,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
| aa(A,$o,aa(A,fun(A,$o),R3,Y2),X3) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R3,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),R3,X3),Z4) ) )
=> sorted_wrt(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),Xs)) ) ) ).
% sorted_wrt_mergesort_by_rel
tff(fact_4961_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] : merges9089515139780605204_merge(A,R3,Xs,nil(A)) = Xs ).
% mergesort_by_rel_merge_simps(2)
tff(fact_4962_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R3),Xs))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R3),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ) ) ).
% mergesort_by_rel.psimps
tff(fact_4963_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa,merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))))) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa)) ) ) ) ).
% mergesort_by_rel.pelims
tff(fact_4964_mergesort__by__rel_Osimps,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
aa(list(A),list(A),mergesort_by_rel(A,R3),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ).
% mergesort_by_rel.simps
tff(fact_4965_mergesort__def,axiom,
! [A: $tType] :
( ord(A)
=> ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% mergesort_def
tff(fact_4966_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),X1: A,X22: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_uj(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),R3)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),nil(A))),Xs)) ).
% mergesort_by_rel_simps(3)
tff(fact_4967_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))),mset(A,aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Xs1))),mset(A,Xs2)) ).
% mset_mergesort_by_rel_split
tff(fact_4968_mergesort__by__rel__permutes,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] : mset(A,aa(list(A),list(A),mergesort_by_rel(A,R3),Xs)) = mset(A,Xs) ).
% mergesort_by_rel_permutes
tff(fact_4969_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),X: A] : aa(list(A),list(A),mergesort_by_rel(A,R3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).
% mergesort_by_rel_simps(2)
tff(fact_4970_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] : mset(A,merges9089515139780605204_merge(A,R3,Xs,Ys)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys)) ).
% mset_mergesort_by_rel_merge
tff(fact_4971_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F2: fun(B,A),A4: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Xs))) ) ).
% horner_sum_simps(2)
tff(fact_4972_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),X: A,Xs: list(A),Y: A,Ys: list(A)] :
merges9089515139780605204_merge(A,R3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(A,$o,aa(A,fun(A,$o),R3,X),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,R3,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).
% mergesort_by_rel_merge_simps(1)
tff(fact_4973_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),R3: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
( ! [Xs3: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs3),nil(B))
=> ( ! [Ys2: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),Ys2)
=> ( ! [X3: A,Xs3: list(A),Y2: B,Ys2: list(B)] :
( aa(B,$o,aa(A,fun(B,$o),R3,X3),Y2)
=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)) ) )
=> ( ! [X3: A,Xs3: list(A),Y2: B,Ys2: list(B)] :
( ~ aa(B,$o,aa(A,fun(B,$o),R3,X3),Y2)
=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),Ys2)
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)) ) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_4974_list__induct__first2,axiom,
! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
( aa(list(A),$o,P,nil(A))
=> ( ! [X3: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
=> ( ! [X12: A,X23: A,Xs3: list(A)] :
( aa(list(A),$o,P,Xs3)
=> aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3))) )
=> aa(list(A),$o,P,Xs) ) ) ) ).
% list_induct_first2
tff(fact_4975_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),W1: list(A),W22: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
=> ( ! [E2: A,W12: list(A),W23: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W12),W23)
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),W12)),W23) )
=> ( ! [E2: B,W13: list(A),W24: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),W24)
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24)) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W1),W22) ) ) ) ).
% list_2pre_induct
tff(fact_4976_neq__NilE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [X3: A,Xs3: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) ) ).
% neq_NilE
tff(fact_4977_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list(A),N22: A,R22: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N1),R1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N22),R22) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
tff(fact_4978_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
( ! [F: fun(A,fun(B,C))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
=> ( ! [F: fun(A,fun(B,C)),A5: A,As2: list(A),B2: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2)))
=> ( ! [A5: fun(A,fun(B,C)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
=> ~ ! [A5: fun(A,fun(B,C)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).
% zipf.cases
tff(fact_4979_merge_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: product_prod(list(A),list(A))] :
( ! [L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
=> ( ! [V3: A,Va: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A))
=> ~ ! [X12: A,L1: list(A),X23: A,L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)) ) ) ) ).
% merge.cases
tff(fact_4980_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
( ! [P3: fun(A,fun(B,$o))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
=> ( ! [P3: fun(A,fun(B,$o)),A5: A,As2: list(A),B2: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2)))
=> ( ! [P3: fun(A,fun(B,$o)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
=> ~ ! [P3: fun(A,fun(B,$o)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).
% list_all_zip.cases
tff(fact_4981_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
( ! [P3: fun(A,$o),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P3),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
=> ~ ! [P3: fun(A,$o),Yes: list(A),No: list(A),X3: A,Xs3: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P3),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3))) ) ).
% partition_rev.cases
tff(fact_4982_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
( ! [R8: fun(A,fun(A,$o)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
=> ~ ! [R8: fun(A,fun(A,$o)),Sl: list(A),X3: A,Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3))) ) ).
% quicksort_by_rel.cases
tff(fact_4983_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
( ! [R8: fun(A,fun(A,$o)),X3: A,Xs3: list(A),Y2: A,Ys2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
=> ( ! [R8: fun(A,fun(A,$o)),Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A)))
=> ~ ! [R8: fun(A,fun(A,$o)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va))) ) ) ).
% mergesort_by_rel_merge.cases
tff(fact_4984_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
( ! [Xs12: list(A),Xs22: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),nil(A))
=> ( ! [Xs12: list(A),Xs22: list(A),X3: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
=> ~ ! [Xs12: list(A),Xs22: list(A),X12: A,X23: A,Xs3: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3))) ) ) ).
% mergesort_by_rel_split.cases
tff(fact_4985_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
=> ( ! [X3: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( Y != $ite(aa(A,$o,aa(A,fun(A,$o),X,X3),Y2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3),Ys2))) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V3: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
tff(fact_4986_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),V: A,Va2: list(A)] : merges9089515139780605204_merge(A,R3,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2) ).
% mergesort_by_rel_merge.simps(3)
tff(fact_4987_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),X1: A,X22: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2)),Xs) ).
% mergesort_by_rel_split.simps(3)
tff(fact_4988_length__compl__induct,axiom,
! [A: $tType,P: fun(list(A),$o),L: list(A)] :
( aa(list(A),$o,P,nil(A))
=> ( ! [E2: A,L3: list(A)] :
( ! [Ll: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3))
=> aa(list(A),$o,P,Ll) )
=> aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L3)) )
=> aa(list(A),$o,P,L) ) ) ).
% length_compl_induct
tff(fact_4989_list__decomp__1,axiom,
! [A: $tType,L: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
=> ? [A5: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),nil(A)) ) ).
% list_decomp_1
tff(fact_4990_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Xs2) ).
% mergesort_by_rel_split.simps(2)
tff(fact_4991_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X,Xa) = Y )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) ) ) )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs22) ) ) )
=> ~ ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X12: A,X23: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)) )
=> ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs3) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
tff(fact_4992_list__decomp__2,axiom,
! [A: $tType,L: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
=> ? [A5: A,B2: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),nil(A))) ) ).
% list_decomp_2
tff(fact_4993_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(
( ( Begin = zero_zero(nat) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),slice(A,Begin,minus_minus(nat,End,one_one(nat)),Xs)),
slice(A,minus_minus(nat,Begin,one_one(nat)),minus_minus(nat,End,one_one(nat)),Xs) ) ).
% slice_Cons
tff(fact_4994_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ! [X3: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),X,X3),Y2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3),Ys2))) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V3: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
tff(fact_4995_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set(A)] : set_Cons(A,A3,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_uk(A,list(A))),A3) ).
% set_Cons_sing_Nil
tff(fact_4996_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( linord4507533701916653071of_set(A,A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),lattic643756798350308766er_Min(A,A3)),linord4507533701916653071of_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),lattic643756798350308766er_Min(A,A3)),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set_nonempty
tff(fact_4997_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_4998_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( linord4507533701916653071of_set(A,A3) = nil(A) )
<=> ( A3 = bot_bot(set(A)) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4999_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B,Xs: list(B)] : image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),bot_bot(set(list(B))))) = set_Cons(A,image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))),image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),Xs),bot_bot(set(list(B)))))) ).
% listrel_Cons
tff(fact_5000_listset_Osimps_I1_J,axiom,
! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% listset.simps(1)
tff(fact_5001_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list(A),N2: A,Ns: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N2),Ns)),lenlex(A,R2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
| ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N2),R2) )
| ( ( M = N2 )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2)) ) ) ) ).
% Cons_lenlex_iff
tff(fact_5002_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),nil(B)),bot_bot(set(list(B))))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% listrel_Nil
tff(fact_5003_lenlex__irreflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R2)) ) ).
% lenlex_irreflexive
tff(fact_5004_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R2))
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys),listrel(A,B,R2)) ) ) ) ).
% listrel_Cons2
tff(fact_5005_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs),listrel(A,B,R2))
=> ~ ! [Y2: B,Ys2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y2),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys2),listrel(A,B,R2)) ) ) ) ).
% listrel_Cons1
tff(fact_5006_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R2)
=> ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R2)) ) ) ).
% listrel.Cons
tff(fact_5007_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X3: A,Y2: B,Xs3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ! [Ys2: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys2),listrel(A,B,R2)) ) ) ) ) ) ).
% listrel.cases
tff(fact_5008_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X2: A,Y3: B,Xs4: list(A),Ys3: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3),R2)
& member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs4),Ys3),listrel(A,B,R2)) ) ) ) ).
% listrel.simps
tff(fact_5009_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N4)),aa(nat,B,nth(B,Ys),N4)),R2) ) ) ) ).
% listrel_iff_nth
tff(fact_5010_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(A,R2))
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
& ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
| ( ( X = Y )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2)) ) ) ) ).
% Cons_in_lex
tff(fact_5011_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = linorder_insort_key(A,A,aTP_Lamp_ul(A,A),X,linord4507533701916653071of_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5012_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_um(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).
% ran_nth_set_encoding_conv
tff(fact_5013_set__mergesort__by__rel,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R3),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_mergesort_by_rel
tff(fact_5014_set__empty,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
<=> ( Xs = nil(A) ) ) ).
% set_empty
tff(fact_5015_set__empty2,axiom,
! [A: $tType,Xs: list(A)] :
( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
<=> ( Xs = nil(A) ) ) ).
% set_empty2
tff(fact_5016_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X222)) ).
% list.simps(15)
tff(fact_5017_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R3,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).
% set_mergesort_by_rel_merge
tff(fact_5018_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = linorder_insort_key(A,A,aTP_Lamp_ul(A,A),X,linord4507533701916653071of_set(A,A3)) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5019_nth__image__indices,axiom,
! [A: $tType,L: list(A)] : aa(set(nat),set(A),image2(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).
% nth_image_indices
tff(fact_5020_set__insort__key,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linorder_insort_key(A,B,F2,X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).
% set_insort_key
tff(fact_5021_empty__set,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).
% empty_set
tff(fact_5022_all__set__conv__nth,axiom,
! [A: $tType,L: list(A),P: fun(A,$o)] :
( ! [X2: A] :
( member(A,X2,aa(list(A),set(A),set2(A),L))
=> aa(A,$o,P,X2) )
<=> ! [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),L))
=> aa(A,$o,P,aa(nat,A,nth(A,L),I4)) ) ) ).
% all_set_conv_nth
tff(fact_5023_the__elem__set,axiom,
! [A: $tType,X: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).
% the_elem_set
tff(fact_5024_in__set__image__conv__nth,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),X: B,L: list(B)] :
( member(A,aa(B,A,F2,X),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),L)))
<=> ? [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(B),nat,size_size(list(B)),L))
& ( aa(B,A,F2,aa(nat,B,nth(B,L),I4)) = aa(B,A,F2,X) ) ) ) ).
% in_set_image_conv_nth
tff(fact_5025_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list(A),L4: list(A),F2: fun(A,B)] :
( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L4) )
=> ( ! [I3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(A,B,F2,aa(nat,A,nth(A,L),I3)) = aa(A,B,F2,aa(nat,A,nth(A,L4),I3)) ) )
=> ( aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),L4)) ) ) ) ).
% set_image_eq_pointwiseI
tff(fact_5026_Pow__set_I1_J,axiom,
! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_set(1)
tff(fact_5027_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( linord4507533701916653071of_set(A,A3) = linorder_insort_key(A,A,aTP_Lamp_ul(A,A),X,linord4507533701916653071of_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
tff(fact_5028_Pow__set_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] :
pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = $let(
a: set(set(A)),
a:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),a)) ) ).
% Pow_set(2)
tff(fact_5029_is__empty__set,axiom,
! [A: $tType,Xs: list(A)] :
( is_empty(A,aa(list(A),set(A),set2(A),Xs))
<=> null(A,Xs) ) ).
% is_empty_set
tff(fact_5030_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),removeAll(A,X,Xs)) = minus_minus(set(A),aa(list(A),set(A),set2(A),Xs),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% set_removeAll
tff(fact_5031_set__map__filter,axiom,
! [A: $tType,B: $tType,G: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G,Xs)) = aa(fun(A,$o),set(A),collect(A),aa(list(B),fun(A,$o),aTP_Lamp_un(fun(B,option(A)),fun(list(B),fun(A,$o)),G),Xs)) ).
% set_map_filter
tff(fact_5032_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( finite_card(list(A),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).
% card_disjoint_shuffles
tff(fact_5033_Un__set__drop__extend,axiom,
! [A: $tType,J2: nat,L: list(set(A))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(set(A)),nat,size_size(list(set(A))),L))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),minus_minus(nat,J2,aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J2,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),minus_minus(nat,J2,aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).
% Un_set_drop_extend
tff(fact_5034_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys: list(B)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).
% nth_zip
tff(fact_5035_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A4: list(A),B3: list(B),A6: list(A),B4: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
=> ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B4) )
=> ( ( zip(A,B,A4,B3) = zip(A,B,A6,B4) )
<=> ( ( A4 = A6 )
& ( B3 = B4 ) ) ) ) ) ).
% zip_eq_zip_same_len
tff(fact_5036_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),zip(A,B,Xs,Ys)) ).
% zip_Cons_Cons
tff(fact_5037_zip__inj,axiom,
! [A: $tType,B: $tType,A4: list(A),B3: list(B),A6: list(A),B4: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
=> ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B4) )
=> ( ( zip(A,B,A4,B3) = zip(A,B,A6,B4) )
=> ( ( A4 = A6 )
& ( B3 = B4 ) ) ) ) ) ).
% zip_inj
tff(fact_5038_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
=> ~ ! [X3: A,Xs5: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5) )
=> ! [Y2: B,Ys4: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys4) )
=> ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ( Xys != zip(A,B,Xs5,Ys4) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_5039_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
=> member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ).
% set_zip_rightD
tff(fact_5040_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
=> member(A,X,aa(list(A),set(A),set2(A),Xs)) ) ).
% set_zip_leftD
tff(fact_5041_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
=> ~ ( member(A,X,aa(list(A),set(A),set2(A),Xs))
=> ~ member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ) ).
% in_set_zipE
tff(fact_5042_zip__same,axiom,
! [A: $tType,A4: A,B3: A,Xs: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
<=> ( member(A,A4,aa(list(A),set(A),set2(A),Xs))
& ( A4 = B3 ) ) ) ).
% zip_same
tff(fact_5043_drop__eq__ConsD,axiom,
! [A: $tType,N2: nat,Xs: list(A),X: A,Xs6: list(A)] :
( ( drop(A,N2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs6) )
=> ( drop(A,aa(nat,nat,suc,N2),Xs) = Xs6 ) ) ).
% drop_eq_ConsD
tff(fact_5044_set__zip__cart,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L4: list(B)] :
( member(product_prod(A,B),X,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,L,L4)))
=> member(product_prod(A,B),X,product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_uo(list(B),fun(A,set(B)),L4))) ) ).
% set_zip_cart
tff(fact_5045_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
~ ! [L1: list(A),L22: list(B)] :
( ( L = zip(A,B,L1,L22) )
=> ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
=> ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L22) ) ) ) ).
% pair_list_split
tff(fact_5046_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( member(B,Y,aa(list(B),set(B),set2(B),Ys))
=> ~ ! [X3: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).
% in_set_impl_in_set_zip2
tff(fact_5047_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( member(A,X,aa(list(A),set(A),set2(A),Xs))
=> ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).
% in_set_impl_in_set_zip1
tff(fact_5048_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ).
% shuffles.simps(1)
tff(fact_5049_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ).
% shuffles.simps(2)
tff(fact_5050_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N2: nat,L: list(A)] :
( member(A,X,aa(list(A),set(A),set2(A),drop(A,N2,L)))
<=> ? [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),I4)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),L))
& ( X = aa(nat,A,nth(A,L),I4) ) ) ) ).
% in_set_drop_conv_nth
tff(fact_5051_in__set__zip,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B),Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),P2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
<=> ? [N4: nat] :
( ( aa(nat,A,nth(A,Xs),N4) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
& ( aa(nat,B,nth(B,Ys),N4) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).
% in_set_zip
tff(fact_5052_shuffles_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( shuffles(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa),bot_bot(set(list(A)))) ) )
=> ( ( ( Xa = nil(A) )
=> ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3),Ys2))) ) ) ) ) ) ) ).
% shuffles.elims
tff(fact_5053_set__drop__conv,axiom,
! [A: $tType,N2: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,N2,L)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_up(nat,fun(list(A),fun(A,$o)),N2),L)) ).
% set_drop_conv
tff(fact_5054_set__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_uq(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).
% set_zip
tff(fact_5055_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X2: product_prod(A,B)] :
( member(product_prod(A,B),X2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X2) ) ) ) ).
% listrel_iff_zip
tff(fact_5056_drop__last__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ( drop(A,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,L)),nil(A)) ) ) ).
% drop_last_conv
tff(fact_5057_list__collect__set__alt,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_ur(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F2),L))) ).
% list_collect_set_alt
tff(fact_5058_lists__empty,axiom,
! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% lists_empty
tff(fact_5059_Misc_Olast__in__set,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> member(A,last(A,L),aa(list(A),set(A),set2(A),L)) ) ).
% Misc.last_in_set
tff(fact_5060_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),A4: B] : list_collect_set(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),nil(B))) = aa(B,set(A),F2,A4) ).
% list_collect_set_simps(2)
tff(fact_5061_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,set(A))] : list_collect_set(B,A,F2,nil(B)) = bot_bot(set(A)) ).
% list_collect_set_simps(1)
tff(fact_5062_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),A4: B,L: list(B)] : list_collect_set(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),L)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,A4)),list_collect_set(B,A,F2,L)) ).
% list_collect_set_simps(3)
tff(fact_5063_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).
% last_zip
tff(fact_5064_lists__of__len__fin2,axiom,
! [A: $tType,P: set(A),N2: nat] :
( aa(set(A),$o,finite_finite2(A),P)
=> aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_us(nat,fun(list(A),$o),N2)))) ) ).
% lists_of_len_fin2
tff(fact_5065_lists__of__len__fin1,axiom,
! [A: $tType,P: set(A),N2: nat] :
( aa(set(A),$o,finite_finite2(A),P)
=> aa(set(list(A)),$o,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_ut(nat,fun(list(A),$o),N2)))) ) ).
% lists_of_len_fin1
tff(fact_5066_list__collect__set__def,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_uu(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F2),L))) ).
% list_collect_set_def
tff(fact_5067_last__take__nth__conv,axiom,
! [A: $tType,N2: nat,L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),L))
=> ( ( N2 != zero_zero(nat) )
=> ( last(A,take(A,N2,L)) = aa(nat,A,nth(A,L),minus_minus(nat,N2,one_one(nat))) ) ) ) ).
% last_take_nth_conv
tff(fact_5068_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))) = fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2),Xs,bot_bot(A)) ) ).
% SUP_set_fold
tff(fact_5069_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: set(A),X: A] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( linord4507533701916653071of_set(A,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,A3)) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5070_union__set__fold,axiom,
! [A: $tType,Xs: list(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A3) = fold(A,set(A),insert2(A),Xs,A3) ).
% union_set_fold
tff(fact_5071_slice__def,axiom,
! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = take(A,minus_minus(nat,To,From),drop(A,From,List)) ).
% slice_def
tff(fact_5072_Sup__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,sup_sup(A),Xs,bot_bot(A)) ) ).
% Sup_set_fold
tff(fact_5073_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).
% Lcm_set_eq_fold
tff(fact_5074_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).
% Lcm_fin.set_eq_fold
tff(fact_5075_Union__take__drop__id,axiom,
! [A: $tType,N2: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),N2,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),N2,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).
% Union_take_drop_id
tff(fact_5076_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
=> ~ ! [I3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys))
=> ( ( take(A,I3,Xs) = take(A,I3,Ys) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys),I3)),R2) ) ) ) ) ).
% lex_take_index
tff(fact_5077_lexord__take__index__conv,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2))
<=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))
& ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
| ? [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
& ( take(A,I4,X) = take(A,I4,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I4)),aa(nat,A,nth(A,Y),I4)),R2) ) ) ) ).
% lexord_take_index_conv
tff(fact_5078_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),J2: nat,X: A,Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),Xs))
=> ( mset(product_prod(A,B),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),drop(A,J2,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J2,Ys)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),drop(B,J2,Ys))))) = add_mset(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),mset(product_prod(A,B),zip(A,B,Xs,Ys))) ) ) ) ).
% mset_zip_take_Cons_drop_twice
tff(fact_5079_take__butlast__conv,axiom,
! [A: $tType,L: list(A)] : take(A,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,L) ).
% take_butlast_conv
tff(fact_5080_empty__append__eq__id,axiom,
! [A: $tType,X4: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X4) = X4 ).
% empty_append_eq_id
tff(fact_5081_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list(A),E12: A,E23: A,L23: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) )
<=> ( ( L12 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23 = nil(A) ) ) ) ).
% list_ee_eq_leel(1)
tff(fact_5082_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list(A),E12: A,E23: A,L23: list(A),E1: A,E22: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
<=> ( ( L12 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23 = nil(A) ) ) ) ).
% list_ee_eq_leel(2)
tff(fact_5083_list__se__match_I1_J,axiom,
! [A: $tType,L12: list(A),L23: list(A),A4: A] :
( ( L12 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) )
<=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) )
& ( L23 = nil(A) ) ) ) ) ).
% list_se_match(1)
tff(fact_5084_list__se__match_I2_J,axiom,
! [A: $tType,L23: list(A),L12: list(A),A4: A] :
( ( L23 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) ) ) ) ) ).
% list_se_match(2)
tff(fact_5085_list__se__match_I3_J,axiom,
! [A: $tType,L12: list(A),A4: A,L23: list(A)] :
( ( L12 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
<=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) )
& ( L23 = nil(A) ) ) ) ) ).
% list_se_match(3)
tff(fact_5086_list__se__match_I4_J,axiom,
! [A: $tType,L23: list(A),A4: A,L12: list(A)] :
( ( L23 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
<=> ( ( L12 = nil(A) )
& ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) ) ) ) ) ).
% list_se_match(4)
tff(fact_5087_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E4: A,L12: list(A),E5: A,L23: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) )
<=> ( ( L12 = nil(A) )
& ( E5 = E4 )
& ( L23 = nil(A) ) ) ) ).
% list_e_eq_lel(1)
tff(fact_5088_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list(A),E5: A,L23: list(A),E4: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( E5 = E4 )
& ( L23 = nil(A) ) ) ) ).
% list_e_eq_lel(2)
tff(fact_5089_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B),L4: list(B)] : list_collect_set(B,A,F2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F2,L)),list_collect_set(B,A,F2,L4)) ).
% list_collect_set_simps(4)
tff(fact_5090_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set(list(A))] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L5) = L5 ).
% op_conc_empty_img_id
tff(fact_5091_nth__append__first,axiom,
! [A: $tType,I2: nat,L: list(A),L4: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L4)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).
% nth_append_first
tff(fact_5092_lexord__cons__cons,axiom,
! [A: $tType,A4: A,X: list(A),B3: A,Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Y)),lexord(A,R2))
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
| ( ( A4 = B3 )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2)) ) ) ) ).
% lexord_cons_cons
tff(fact_5093_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),X: A] :
( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
<=> ( ( Ys != nil(A) )
& ( butlast(A,Ys) = Xs )
& ( last(A,Ys) = X ) ) ) ).
% snoc_eq_iff_butlast'
tff(fact_5094_lexord__append__leftD,axiom,
! [A: $tType,X: list(A),U: list(A),V: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V)),lexord(A,R2))
=> ( ! [A5: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R2)) ) ) ).
% lexord_append_leftD
tff(fact_5095_butlast__eq__consE,axiom,
! [A: $tType,L: list(A),X: A,Xs: list(A)] :
( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
=> ~ ! [Xl: A] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ).
% butlast_eq_consE
tff(fact_5096_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list(A),X: A,Xs: list(A)] :
( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
<=> ? [Xl2: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ).
% butlast_eq_cons_conv
tff(fact_5097_list__match__lel__lel,axiom,
! [A: $tType,C12: list(A),Qs: A,C23: list(A),C13: list(A),Qs2: A,C24: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C23)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C24)) )
=> ( ! [C21: list(A)] :
( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
=> ( C24 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C23)) ) )
=> ( ( ( C13 = C12 )
=> ( ( Qs2 = Qs )
=> ( C24 != C23 ) ) )
=> ~ ! [C212: list(A)] :
( ( C13 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
=> ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C24)) ) ) ) ) ) ).
% list_match_lel_lel
tff(fact_5098_lexord__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_uv(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% lexord_def
tff(fact_5099_lexord__append__left__rightI,axiom,
! [A: $tType,A4: A,B3: A,R2: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),Y))),lexord(A,R2)) ) ).
% lexord_append_left_rightI
tff(fact_5100_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lexord(A,R2))
<=> ( ? [X2: A] :
( member(A,X2,aa(list(A),set(A),set2(A),Xs))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2) )
| member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lexord(A,R2)) ) ) ).
% lexord_same_pref_iff
tff(fact_5101_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list(A),Zs2: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
=> ( ( ( Ys = nil(A) )
=> ( Zs2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
=> ~ ! [Ys4: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys4) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),Zs2) != Xs ) ) ) ) ).
% list_append_eq_Cons_cases
tff(fact_5102_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs2) )
=> ( ( ( Ys = nil(A) )
=> ( Zs2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
=> ~ ! [Ys4: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys4) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),Zs2) != Xs ) ) ) ) ).
% list_Cons_eq_append_cases
tff(fact_5103_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),$o))] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ! [X3: A,Y2: B] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))
=> ( ! [X3: A,Xs3: list(A),Y2: B] :
( ( Xs3 != nil(A) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))) )
=> ( ! [X3: A,Y2: B,Ys2: list(B)] :
( ( Ys2 != nil(B) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) )
=> ( ! [X3: A,Xs3: list(A),Y2: B,Ys2: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs3),Ys2)
=> ( ( Xs3 != nil(A) )
=> ( ( Ys2 != nil(B) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) ) ) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
tff(fact_5104_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
<=> ? [Xs4: list(A),X2: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) ) ).
% neq_Nil_rev_conv
tff(fact_5105_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
=> ( ! [X3: A,Xs3: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),nil(B))
=> ( ! [Y2: B,Ys2: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))))
=> ( ! [X3: A,Xs3: list(A),Y2: B,Ys2: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs3),Ys2)
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).
% rev_induct2'
tff(fact_5106_neq__Nil__revE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [Ll2: list(A),E2: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))) ) ).
% neq_Nil_revE
tff(fact_5107_in__set__list__format,axiom,
! [A: $tType,E4: A,L: list(A)] :
( member(A,E4,aa(list(A),set(A),set2(A),L))
=> ~ ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),L22)) ) ).
% in_set_list_format
tff(fact_5108_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list(A),Y: A] :
( member(A,X,aa(list(A),set(A),set2(A),L))
=> ( member(A,Y,aa(list(A),set(A),set2(A),L))
=> ( ( ( X = Y )
=> ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) )
=> ( ( ( X != Y )
=> ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
=> ~ ( ( X != Y )
=> ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L32)))) ) ) ) ) ) ).
% xy_in_set_cases
tff(fact_5109_list__rest__coinc,axiom,
! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R22: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R22) )
=> ? [R1p: list(A)] : R22 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R1) ) ) ).
% list_rest_coinc
tff(fact_5110_set__union__code,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).
% set_union_code
tff(fact_5111_lexord__irreflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lexord(A,R2)) ) ).
% lexord_irreflexive
tff(fact_5112_lexord__linear,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: list(A),Y: list(A)] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2),R2)
| ( A5 = B2 )
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A5),R2) )
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2))
| ( X = Y )
| member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X),lexord(A,R2)) ) ) ).
% lexord_linear
tff(fact_5113_drop__take__drop__unsplit,axiom,
! [A: $tType,I2: nat,J2: nat,L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,I2,take(A,J2,L))),drop(A,J2,L)) = drop(A,I2,L) ) ) ).
% drop_take_drop_unsplit
tff(fact_5114_lex__append__leftD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lex(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lex(A,R2)) ) ) ).
% lex_append_leftD
tff(fact_5115_lex__append__left__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lex(A,R2))
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lex(A,R2)) ) ) ).
% lex_append_left_iff
tff(fact_5116_butlast__subset,axiom,
! [A: $tType,Xs: list(A),A3: set(A)] :
( ( Xs != nil(A) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A3) ) ) ).
% butlast_subset
tff(fact_5117_lexord__partial__trans,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs2: list(A)] :
( ! [X3: A,Y2: A,Z4: A] :
( member(A,X3,aa(list(A),set(A),set2(A),Xs))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z4),R2) ) ) )
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lexord(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),lexord(A,R2)) ) ) ) ).
% lexord_partial_trans
tff(fact_5118_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list(A),N2: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N2) )
<=> ? [Ys3: list(A),Y3: A] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N2 ) ) ) ).
% length_Suc_rev_conv
tff(fact_5119_length__compl__rev__induct,axiom,
! [A: $tType,P: fun(list(A),$o),L: list(A)] :
( aa(list(A),$o,P,nil(A))
=> ( ! [L3: list(A),E2: A] :
( ! [Ll: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3))
=> aa(list(A),$o,P,Ll) )
=> aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)))) )
=> aa(list(A),$o,P,L) ) ) ).
% length_compl_rev_induct
tff(fact_5120_slice__prepend,axiom,
! [A: $tType,I2: nat,K: nat,Xs: list(A),Ys: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),K)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
=> $let(
p: nat,
p:= aa(list(A),nat,size_size(list(A)),Ys),
slice(A,I2,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).
% slice_prepend
tff(fact_5121_sorted__append__bigger,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Y: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ! [X3: A] :
( member(A,X3,aa(list(A),set(A),set2(A),Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y) )
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).
% sorted_append_bigger
tff(fact_5122_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F2: fun(B,A),A4: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A4),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Ys))) ) ).
% horner_sum_append
tff(fact_5123_lex__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_uw(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% lex_conv
tff(fact_5124_take__minus__one__conv__butlast,axiom,
! [A: $tType,N2: nat,L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(list(A),nat,size_size(list(A)),L))
=> ( take(A,minus_minus(nat,N2,aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,N2,L)) ) ) ).
% take_minus_one_conv_butlast
tff(fact_5125_butlast__upd__last__eq,axiom,
! [A: $tType,L: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),L))
=> ( list_update(A,butlast(A,L),minus_minus(nat,aa(list(A),nat,size_size(list(A)),L),aa(num,nat,numeral_numeral(nat),bit0(one2))),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,minus_minus(nat,aa(list(A),nat,size_size(list(A)),L),aa(num,nat,numeral_numeral(nat),bit0(one2))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).
% butlast_upd_last_eq
tff(fact_5126_lexn__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N2: nat] : lexn(A,R2,N2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ux(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),N2))) ).
% lexn_conv
tff(fact_5127_take__update__last,axiom,
! [A: $tType,N2: nat,List: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),List))
=> ( list_update(A,take(A,aa(nat,nat,suc,N2),List),N2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N2,List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).
% take_update_last
tff(fact_5128_take__update,axiom,
! [A: $tType,N2: nat,L: list(A),I2: nat,X: A] : take(A,N2,list_update(A,L,I2,X)) = list_update(A,take(A,N2,L),I2,X) ).
% take_update
tff(fact_5129_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N2: nat,L: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
=> ( drop(A,N2,list_update(A,L,M,X)) = drop(A,N2,L) ) ) ).
% drop_upd_irrelevant
tff(fact_5130_nth__update__invalid,axiom,
! [A: $tType,I2: nat,L: list(A),J2: nat,X: A] :
( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(nat,A,nth(A,list_update(A,L,J2,X)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).
% nth_update_invalid
tff(fact_5131_butlast__update_H,axiom,
! [A: $tType,L: list(A),I2: nat,X: A] : list_update(A,butlast(A,L),I2,X) = butlast(A,list_update(A,L,I2,X)) ).
% butlast_update'
tff(fact_5132_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I2: nat,X: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I2,X),list_update(B,Ys,I2,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_update
tff(fact_5133_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I2: nat,Xy: product_prod(A,B)] : list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,Xy) = zip(A,B,list_update(A,Xs,I2,aa(product_prod(A,B),A,product_fst(A,B),Xy)),list_update(B,Ys,I2,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).
% update_zip
tff(fact_5134_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).
% lexn.simps(1)
tff(fact_5135_set__update__subset__insert,axiom,
! [A: $tType,Xs: list(A),I2: nat,X: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs))) ).
% set_update_subset_insert
tff(fact_5136_in__set__upd__eq__aux,axiom,
! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I2,Y)))
<=> ( ( X = Y )
| ! [Y3: A] : member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I2,Y3))) ) ) ) ).
% in_set_upd_eq_aux
tff(fact_5137_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list(A),I2: nat,Y: A] :
( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I2,Y)))
=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( X != Y ) )
=> member(A,X,aa(list(A),set(A),set2(A),L)) ) ) ).
% in_set_upd_cases
tff(fact_5138_in__set__upd__eq,axiom,
! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I2,Y)))
<=> ( ( X = Y )
| ( member(A,X,aa(list(A),set(A),set2(A),L))
& ! [Y3: A] : member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I2,Y3))) ) ) ) ) ).
% in_set_upd_eq
tff(fact_5139_nth__list__update_H,axiom,
! [A: $tType,L: list(A),I2: nat,X: A,J2: nat] :
aa(nat,A,nth(A,list_update(A,L,I2,X)),J2) = $ite(
( ( I2 = J2 )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)) ),
X,
aa(nat,A,nth(A,L),J2) ) ).
% nth_list_update'
tff(fact_5140_insert__swap__set__eq,axiom,
! [A: $tType,I2: nat,L: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,L),I2)),aa(list(A),set(A),set2(A),list_update(A,L,I2,X))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),L)) ) ) ).
% insert_swap_set_eq
tff(fact_5141_set__update__distinct,axiom,
! [A: $tType,Xs: list(A),N2: nat,X: A] :
( distinct(A,Xs)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(list(A),set(A),set2(A),list_update(A,Xs,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),N2)),bot_bot(set(A))))) ) ) ) ).
% set_update_distinct
tff(fact_5142_listrel1__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_uy(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% listrel1_def
tff(fact_5143_foldl__list__update,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list(A),F2: fun(B,fun(A,B)),A4: B,X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F2),A4),list_update(A,Xs,N2,X)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F2),aa(A,B,aa(B,fun(A,B),F2,aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F2),A4),take(A,N2,Xs))),X)),drop(A,aa(nat,nat,suc,N2),Xs)) ) ) ).
% foldl_list_update
tff(fact_5144_foldl__length,axiom,
! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_uz(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).
% foldl_length
tff(fact_5145_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
& ( Xs = Ys ) )
| ( ( X = Y )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_5146_distinct__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( distinct(A,Xs)
& distinct(A,Ys)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).
% distinct_append
tff(fact_5147_set__remove1__eq,axiom,
! [A: $tType,Xs: list(A),X: A] :
( distinct(A,Xs)
=> ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = minus_minus(set(A),aa(list(A),set(A),set2(A),Xs),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ) ).
% set_remove1_eq
tff(fact_5148_foldl__A1__eq,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),N2: A,I2: A,Ww: list(A)] :
( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F2,N2),E2) = E2
=> ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F2,E2),N2) = E2
=> ( ! [A5: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B2),C3)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B2)),C3)
=> ( aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F2),I2),Ww) = aa(A,A,aa(A,fun(A,A),F2,I2),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F2),N2),Ww)) ) ) ) ) ).
% foldl_A1_eq
tff(fact_5149_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),A4: A,B3: B,Xs: list(A)] :
( finite6289374366891150609ommute(A,B,F2)
=> ( aa(B,B,aa(A,fun(B,B),F2,A4),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_va(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),B3),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_va(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),aa(B,B,aa(A,fun(B,B),F2,A4),B3)),Xs) ) ) ).
% comp_fun_commute.foldl_f_commute
tff(fact_5150_distinct__match,axiom,
! [A: $tType,Al: list(A),E4: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl)))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl2)) )
<=> ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
tff(fact_5151_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(A,fun(C,A)),G: fun(A,fun(B,fun(C,B))),A4: A,B3: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_vb(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F2),G))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,F2),A4),Xs) ).
% fst_foldl
tff(fact_5152_foldl__conc__empty__eq,axiom,
! [A: $tType,I2: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),I2),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I2),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),nil(A)),Ww)) ).
% foldl_conc_empty_eq
tff(fact_5153_foldl__absorb1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Zs2: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),one_one(A)),Zs2)) = aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),X),Zs2) ) ).
% foldl_absorb1
tff(fact_5154_foldl__un__empty__eq,axiom,
! [A: $tType,I2: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),I2),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I2),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),Ww)) ).
% foldl_un_empty_eq
tff(fact_5155_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F2: fun(C,fun(A,C)),B3: C] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
=> ( aa(list(product_prod(B,A)),C,aa(C,fun(list(product_prod(B,A)),C),foldl(C,product_prod(B,A),aTP_Lamp_vd(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F2)),B3),zip(B,A,Xs,Ys)) = aa(list(A),C,aa(C,fun(list(A),C),foldl(C,A,F2),B3),Ys) ) ) ).
% foldl_snd_zip
tff(fact_5156_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list(A),I: fun(set(A),fun(B,$o)),Sigma_0: B,F2: fun(B,fun(A,B))] :
( distinct(A,S)
=> ( aa(B,$o,aa(set(A),fun(B,$o),I,aa(list(A),set(A),set2(A),S)),Sigma_0)
=> ( ! [X3: A,It: set(A),Sigma: B] :
( member(A,X3,It)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S))
=> ( aa(B,$o,aa(set(A),fun(B,$o),I,It),Sigma)
=> aa(B,$o,aa(set(A),fun(B,$o),I,minus_minus(set(A),It,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F2,Sigma),X3)) ) ) )
=> aa(B,$o,aa(set(A),fun(B,$o),I,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F2),Sigma_0),S)) ) ) ) ).
% distinct_foldl_invar
tff(fact_5157_distinct__finite__set,axiom,
! [A: $tType,X: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_ve(set(A),fun(list(A),$o),X))) ).
% distinct_finite_set
tff(fact_5158_distinct__length__le,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( distinct(A,Ys)
=> ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).
% distinct_length_le
tff(fact_5159_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list(A),I2: nat] :
( distinct(A,Pq)
=> distinct(A,butlast(A,list_update(A,Pq,I2,last(A,Pq)))) ) ).
% distinct_butlast_swap
tff(fact_5160_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
=> ( ! [X3: A] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2) )
=> ~ ! [Zs: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs) )
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs),Ys),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1E2
tff(fact_5161_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys),listrel1(A,R2))
=> ( ! [Y2: A] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2) )
=> ~ ! [Zs: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs) )
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1E1
tff(fact_5162_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R2)) ) ).
% listrel1I1
tff(fact_5163_foldl__rule,axiom,
! [A: $tType,B: $tType,I: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A))] :
( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,Sigma_0),nil(B)),L0)
=> ( ! [L1: list(B),L22: list(B),X3: B,Sigma: A] :
( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22)) )
=> ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22))
=> aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,aa(B,A,aa(A,fun(B,A),F2,Sigma),X3)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),nil(B)))),L22) ) )
=> aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F2),Sigma_0),L0)),L0),nil(B)) ) ) ).
% foldl_rule
tff(fact_5164_foldl__rule__P,axiom,
! [A: $tType,B: $tType,I: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A)),P: fun(A,$o)] :
( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,Sigma_0),nil(B)),L0)
=> ( ! [L1: list(B),L22: list(B),X3: B,Sigma: A] :
( ( L0 = 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),X3),L22)) )
=> ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22))
=> aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,aa(B,A,aa(A,fun(B,A),F2,Sigma),X3)),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),X3),nil(B)))),L22) ) )
=> ( ! [Sigma: A] :
( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I,Sigma),L0),nil(B))
=> aa(A,$o,P,Sigma) )
=> aa(A,$o,P,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F2),Sigma_0),L0)) ) ) ) ).
% foldl_rule_P
tff(fact_5165_foldl__rule__aux,axiom,
! [A: $tType,B: $tType,I: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A))] :
( aa(list(B),$o,aa(A,fun(list(B),$o),I,Sigma_0),L0)
=> ( ! [L1: list(B),L22: list(B),X3: B,Sigma: A] :
( ( L0 = 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),X3),L22)) )
=> ( aa(list(B),$o,aa(A,fun(list(B),$o),I,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22))
=> aa(list(B),$o,aa(A,fun(list(B),$o),I,aa(B,A,aa(A,fun(B,A),F2,Sigma),X3)),L22) ) )
=> aa(list(B),$o,aa(A,fun(list(B),$o),I,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F2),Sigma_0),L0)),nil(B)) ) ) ).
% foldl_rule_aux
tff(fact_5166_foldl__rule__aux__P,axiom,
! [A: $tType,B: $tType,I: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F2: fun(A,fun(B,A)),P: fun(A,$o)] :
( aa(list(B),$o,aa(A,fun(list(B),$o),I,Sigma_0),L0)
=> ( ! [L1: list(B),L22: list(B),X3: B,Sigma: A] :
( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22)) )
=> ( aa(list(B),$o,aa(A,fun(list(B),$o),I,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X3),L22))
=> aa(list(B),$o,aa(A,fun(list(B),$o),I,aa(B,A,aa(A,fun(B,A),F2,Sigma),X3)),L22) ) )
=> ( ! [Sigma: A] :
( aa(list(B),$o,aa(A,fun(list(B),$o),I,Sigma),nil(B))
=> aa(A,$o,P,Sigma) )
=> aa(A,$o,P,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F2),Sigma_0),L0)) ) ) ) ).
% foldl_rule_aux_P
tff(fact_5167_finite__set__image,axiom,
! [A: $tType,A3: set(list(A))] :
( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A3))
=> ( ! [Xs3: list(A)] :
( member(list(A),Xs3,A3)
=> distinct(A,Xs3) )
=> aa(set(list(A)),$o,finite_finite2(list(A)),A3) ) ) ).
% finite_set_image
tff(fact_5168_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ~ distinct(A,Xs)
=> ~ ! [Y2: A,Ys2: list(A)] :
( distinct(A,Ys2)
=> ( member(A,Y2,aa(list(A),set(A),set2(A),Ys2))
=> ! [Zs: list(A)] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),Zs)) ) ) ) ).
% not_distinct_split_distinct
tff(fact_5169_foldl__length__aux,axiom,
! [A: $tType,A4: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_uz(nat,fun(A,nat))),A4),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),aa(list(A),nat,size_size(list(A)),L)) ).
% foldl_length_aux
tff(fact_5170_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( distinct(A,Xs)
=> ( distinct(A,Ys)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> distinct(A,Zs2) ) ) ) ) ).
% distinct_disjoint_shuffles
tff(fact_5171_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
=> ~ ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ! [Us: list(A),Vs: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs)) )
=> ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Vs)) ) ) ) ) ).
% listrel1E
tff(fact_5172_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us2: list(A),Vs2: list(A),Ys: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs2)) )
=> ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs2)) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).
% listrel1I
tff(fact_5173_distinct__finite__subset,axiom,
! [A: $tType,X: set(A)] :
( aa(set(A),$o,finite_finite2(A),X)
=> aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_vf(set(A),fun(list(A),$o),X))) ) ).
% distinct_finite_subset
tff(fact_5174_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2)))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_5175_foldl__set,axiom,
! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_vg(list(set(A)),fun(set(A),$o),L))) ).
% foldl_set
tff(fact_5176_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))),listrel1(A,R2))
<=> ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
& ( X = Y ) )
| ( ( Xs = Ys )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_5177_distinct__sorted__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I2: nat,J2: nat] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( distinct(A,L)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2)) ) ) ) ) ) ).
% distinct_sorted_mono
tff(fact_5178_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I2: nat,J2: nat] :
( distinct(A,L)
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
tff(fact_5179_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I2: nat,J2: nat] :
( distinct(A,L)
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
tff(fact_5180_distinct__list__update,axiom,
! [A: $tType,Xs: list(A),A4: A,I2: nat] :
( distinct(A,Xs)
=> ( ~ member(A,A4,minus_minus(set(A),aa(list(A),set(A),set2(A),Xs),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),I2)),bot_bot(set(A)))))
=> distinct(A,list_update(A,Xs,I2,A4)) ) ) ).
% distinct_list_update
tff(fact_5181_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs2: list(A),I2: nat,J2: nat] :
( distinct(A,Vs2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I2,Vs2))),aa(list(A),set(A),set2(A),drop(A,J2,Vs2))) = bot_bot(set(A)) ) ) ) ).
% set_take_disj_set_drop_if_distinct
tff(fact_5182_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
<=> ? [Y3: A,N4: nat] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N4)),Y3),R2)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
& ( Ys = list_update(A,Xs,N4,Y3) ) ) ) ).
% listrel1_iff_update
tff(fact_5183_mergesort__remdups__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( distinct(A,aa(list(A),list(A),mergesort_remdups(A),L))
& sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_remdups(A),L))
& ( aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_remdups(A),L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).
% mergesort_remdups_correct
tff(fact_5184_inv__image__partition,axiom,
! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
( ! [X3: A] :
( member(A,X3,aa(list(A),set(A),set2(A),Xs))
=> aa(A,$o,P,X3) )
=> ( ! [Y2: A] :
( member(A,Y2,aa(list(A),set(A),set2(A),Ys))
=> ~ aa(A,$o,P,Y2) )
=> ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert2(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).
% inv_image_partition
tff(fact_5185_merge__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L12: list(A),L23: list(A)] :
( ( distinct(A,L12)
& sorted_wrt(A,ord_less_eq(A),L12) )
=> ( ( distinct(A,L23)
& sorted_wrt(A,ord_less_eq(A),L23) )
=> ( distinct(A,merge(A,L12,L23))
& sorted_wrt(A,ord_less_eq(A),merge(A,L12,L23))
& ( aa(list(A),set(A),set2(A),merge(A,L12,L23)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L12)),aa(list(A),set(A),set2(A),L23)) ) ) ) ) ) ).
% merge_correct
tff(fact_5186_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L23: list(A)] : merge(A,nil(A),L23) = L23 ) ).
% merge.simps(1)
tff(fact_5187_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,L12: list(A),X22: A,L23: list(A)] :
merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23)) = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),X1),X22),
aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L23))),
$ite(X1 = X22,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,L23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),L23))) ) ) ).
% merge.simps(3)
tff(fact_5188_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [V: A,Va2: list(A)] : merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2) ) ).
% merge.simps(2)
tff(fact_5189_merge_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != Xa ) )
=> ( ! [V3: A,Va: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) )
=> ~ ! [X12: A,L1: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ( Y != $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),X12),X23),
aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))),
$ite(X12 = X23,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,L22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1),L22))) ) ) ) ) ) ) ) ) ).
% merge.elims
tff(fact_5190_listrel__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : listrel(A,B,X4) = aa(fun(product_prod(list(A),list(B)),$o),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),$o)),fun(product_prod(list(A),list(B)),$o),product_case_prod(list(A),list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),X4)))) ).
% listrel_def
tff(fact_5191_shuffles_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( shuffles(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa),bot_bot(set(list(A)))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ( ( ( Xa = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3),Ys2))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_5192_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X4: list(A),Xa3: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4),Xa3)
<=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X4),Xa3),listrel(A,B,R2)) ) ).
% listrelp_listrel_eq
tff(fact_5193_merge__list_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: product_prod(list(list(A)),list(list(A)))] :
( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
=> ( ! [L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))
=> ( ! [La: list(A),Acc2: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A)))
=> ( ! [La: list(A),Acc2: list(list(A)),L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))
=> ~ ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) ) ) ) ) ).
% merge_list.cases
tff(fact_5194_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)))
=> ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(2)
tff(fact_5195_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
=> ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(1)
tff(fact_5196_merge_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ( ! [V3: A,Va: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A))) ) ) )
=> ~ ! [X12: A,L1: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ( ( Y = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),X12),X23),
aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))),
$ite(X12 = X23,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,L22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1),L22))) ) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) ) ) ) ) ) ) ) ).
% merge.pelims
tff(fact_5197_zipf__zip,axiom,
! [A: $tType,B: $tType,L12: list(A),L23: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L23) )
=> ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L12,L23) = zip(A,B,L12,L23) ) ) ).
% zipf_zip
tff(fact_5198_set__rec,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_vh(A,fun(list(A),fun(set(A),set(A)))),Xs) ).
% set_rec
tff(fact_5199_zipf_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A4: B,As: list(B),B3: C,Bs: list(C)] : zipf(B,C,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),As),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B3),Bs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F2,A4),B3)),zipf(B,C,A,F2,As,Bs)) ).
% zipf.simps(2)
tff(fact_5200_zipf_Osimps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(B,fun(C,A))] : zipf(B,C,A,F2,nil(B),nil(C)) = nil(A) ).
% zipf.simps(1)
tff(fact_5201_zipf_Oelims,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(B,fun(C,A)),Xa: list(B),Xb: list(C),Y: list(A)] :
( ( zipf(B,C,A,X,Xa,Xb) = Y )
=> ( ( ( Xa = nil(B) )
=> ( ( Xb = nil(C) )
=> ( Y != nil(A) ) ) )
=> ( ! [A5: B,As2: list(B)] :
( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),As2) )
=> ! [B2: C,Bs2: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B2),Bs2) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),X,A5),B2)),zipf(B,C,A,X,As2,Bs2)) ) ) )
=> ( ( ? [V3: B,Va: list(B)] : Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)
=> ( ( Xb = nil(C) )
=> ( Y != undefined(list(A)) ) ) )
=> ~ ( ( Xa = nil(B) )
=> ( ? [V3: C,Va: list(C)] : Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V3),Va)
=> ( Y != undefined(list(A)) ) ) ) ) ) ) ) ).
% zipf.elims
tff(fact_5202_zipf_Opelims,axiom,
! [A: $tType,B: $tType,C: $tType,X: fun(B,fun(C,A)),Xa: list(B),Xb: list(C),Y: list(A)] :
( ( zipf(B,C,A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),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)),Xa),Xb)))
=> ( ( ( Xa = nil(B) )
=> ( ( Xb = nil(C) )
=> ( ( Y = nil(A) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),nil(C)))) ) ) )
=> ( ! [A5: B,As2: list(B)] :
( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),As2) )
=> ! [B2: C,Bs2: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B2),Bs2) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),X,A5),B2)),zipf(B,C,A,X,As2,Bs2)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),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),A5),As2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),B2),Bs2)))) ) ) )
=> ( ! [V3: B,Va: list(B)] :
( ( Xa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
=> ( ( Xb = nil(C) )
=> ( ( Y = undefined(list(A)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)),nil(C)))) ) ) )
=> ~ ( ( Xa = nil(B) )
=> ! [V3: C,Va: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V3),Va) )
=> ( ( Y = undefined(list(A)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),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),V3),Va)))) ) ) ) ) ) ) ) ) ).
% zipf.pelims
tff(fact_5203_merge__list_Opinduct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A0: list(list(A)),A1: list(list(A)),P: fun(list(list(A)),fun(list(list(A)),$o))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1))
=> ( ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),nil(list(A))) )
=> ( ! [L3: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A))) ) )
=> ( ! [La: list(A),Acc2: list(list(A)),L3: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))) ) )
=> ( ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc2)),Ls)
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,Acc2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) )
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,A0),A1) ) ) ) ) ) ) ) ).
% merge_list.pinduct
tff(fact_5204_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I2: nat,L: list(A),F2: fun(A,B),X: B] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( distinct(A,L)
=> ( list_update(B,aa(list(A),list(B),map(A,B,F2),L),I2,X) = aa(list(A),list(B),map(A,B,fun_upd(A,B,F2,aa(nat,A,nth(A,L),I2),X)),L) ) ) ) ).
% map_distinct_upd_conv
tff(fact_5205_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))),K)),L)) = L ).
% map_fst_mk_snd
tff(fact_5206_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).
% map_snd_mk_fst
tff(fact_5207_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).
% map_snd_zip
tff(fact_5208_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs2: list(product_prod(A,B))] : zip(A,B,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs2),aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs2)) = Zs2 ).
% zip_map_fst_snd
tff(fact_5209_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [L: list(product_prod(A,B))] :
( sorted_wrt(product_prod(A,B),aTP_Lamp_vi(product_prod(A,B),fun(product_prod(A,B),$o)),L)
<=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).
% sorted_wrt_map_linord
tff(fact_5210_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(B,set(A)),X: fun(C,B),A4: C] : list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A4),nil(C)))) = aa(B,set(A),F2,aa(C,B,X,A4)) ).
% list_collect_set_map_simps(2)
tff(fact_5211_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),nil(C))) = bot_bot(set(A)) ).
% list_collect_set_map_simps(1)
tff(fact_5212_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(B,set(A)),X: fun(C,B),A4: C,L: list(C)] : list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A4),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,aa(C,B,X,A4))),list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),L))) ).
% list_collect_set_map_simps(3)
tff(fact_5213_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(B,set(A)),X: fun(C,B),L: list(C),L4: list(C)] : list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),L))),list_collect_set(B,A,F2,aa(list(C),list(B),map(C,B,X),L4))) ).
% list_collect_set_map_simps(4)
tff(fact_5214_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list(A),A3: set(A),F2: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A3)
=> ( inj_on(A,B,F2,A3)
=> ( aa(list(B),list(A),map(B,A,inv_on(A,B,F2,A3)),aa(list(A),list(B),map(A,B,F2),L)) = L ) ) ) ).
% inj_on_map_inv_f
tff(fact_5215_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F2),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_vj(fun(C,A),fun(C,fun(B,product_prod(A,B))),F2))),zip(C,B,Xs,Ys)) ).
% zip_map1
tff(fact_5216_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F2: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F2),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_vk(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),zip(A,C,Xs,Ys)) ).
% zip_map2
tff(fact_5217_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F2),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_vl(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F2),G))),zip(D,C,Xs,Ys)) ).
% map_zip_map
tff(fact_5218_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F2),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_vm(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F2),G))),zip(B,D,Xs,Ys)) ).
% map_zip_map2
tff(fact_5219_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_iu(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),zip(C,D,Xs,Ys)) = zip(A,B,aa(list(C),list(A),map(C,A,F2),Xs),aa(list(D),list(B),map(D,B,G),Ys)) ).
% map_prod_fun_zip
tff(fact_5220_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),X4: list(A)] : aa(list(A),set(B),aa(fun(list(A),list(B)),fun(list(A),set(B)),comp(list(B),set(B),list(A),set2(B)),map(A,B,F2)),X4) = aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),X4)) ).
% set_oo_map_alt
tff(fact_5221_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),L4: list(B),I2: nat] :
( ( aa(list(B),list(A),map(B,A,F2),L) = aa(list(B),list(A),map(B,A,F2),L4) )
=> ( aa(B,A,F2,aa(nat,B,nth(B,L),I2)) = aa(B,A,F2,aa(nat,B,nth(B,L4),I2)) ) ) ).
% map_eq_nth_eq
tff(fact_5222_Misc_Omap__eq__append__conv,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),map(B,A,F2),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
<=> ? [L2: list(B),L6: list(B)] :
( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
& ( aa(list(B),list(A),map(B,A,F2),L2) = Fl )
& ( aa(list(B),list(A),map(B,A,F2),L6) = Fl2 ) ) ) ).
% Misc.map_eq_append_conv
tff(fact_5223_Misc_Oappend__eq__map__conv,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F2: fun(B,A),Ls2: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F2),Ls2) )
<=> ? [L2: list(B),L6: list(B)] :
( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
& ( aa(list(B),list(A),map(B,A,F2),L2) = Fl )
& ( aa(list(B),list(A),map(B,A,F2),L6) = Fl2 ) ) ) ).
% Misc.append_eq_map_conv
tff(fact_5224_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),map(B,A,F2),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
=> ~ ! [L3: list(B),L7: list(B)] :
( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
=> ( ( aa(list(B),list(A),map(B,A,F2),L3) = Fl )
=> ( aa(list(B),list(A),map(B,A,F2),L7) != Fl2 ) ) ) ) ).
% map_eq_appendE
tff(fact_5225_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F2: fun(B,A),Ls2: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F2),Ls2) )
=> ~ ! [L3: list(B),L7: list(B)] :
( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
=> ( ( aa(list(B),list(A),map(B,A,F2),L3) = Fl )
=> ( aa(list(B),list(A),map(B,A,F2),L7) != Fl2 ) ) ) ) ).
% append_eq_mapE
tff(fact_5226_distinct__mapI,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),L: list(B)] :
( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
=> distinct(B,L) ) ).
% distinct_mapI
tff(fact_5227_map__eq__consE,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),Ls2: list(B),Fa: A,Fl: list(A)] :
( ( aa(list(B),list(A),map(B,A,F2),Ls2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
=> ~ ! [A5: B,L3: list(B)] :
( ( Ls2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),L3) )
=> ( ( aa(B,A,F2,A5) = Fa )
=> ( aa(list(B),list(A),map(B,A,F2),L3) != Fl ) ) ) ) ).
% map_eq_consE
tff(fact_5228_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list(A),F2: fun(B,A),Ww: list(B),A4: B] :
( ( W = aa(list(B),list(A),map(B,A,F2),Ww) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,A4)),W) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),Ww)) ) ) ).
% map_consI(1)
tff(fact_5229_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list(A),L: list(A),F2: fun(B,A),Ww: list(B),A4: B] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F2),Ww)),L) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,A4)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),L)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),Ww))),L) ) ) ).
% map_consI(2)
tff(fact_5230_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),X: B,Y: B] :
( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
=> ( ( aa(B,A,F2,X) = aa(B,A,F2,Y) )
=> ( member(B,X,aa(list(B),set(B),set2(B),L))
=> ( member(B,Y,aa(list(B),set(B),set2(B),L))
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
tff(fact_5231_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys) )
=> ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ys) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
tff(fact_5232_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list(A),F2: fun(B,A),Q: set(B)] :
( member(list(A),X,lists(A,aa(set(B),set(A),image2(B,A,F2),Q)))
=> ~ ! [Xo2: list(B)] :
( member(list(B),Xo2,lists(B,Q))
=> ( X != aa(list(B),list(A),map(B,A,F2),Xo2) ) ) ) ).
% lists_image_witness
tff(fact_5233_zip__same__conv__map,axiom,
! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_ex(A,product_prod(A,A))),Xs) ).
% zip_same_conv_map
tff(fact_5234_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),Ys)),product(A,B,Xs,Ys)) ).
% product.simps(2)
tff(fact_5235_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs2: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs2)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_vn(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs2)) ).
% zip_assoc
tff(fact_5236_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs2: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs2)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_vp(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs2))) ).
% zip_left_commute
tff(fact_5237_zip__commute,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).
% zip_commute
tff(fact_5238_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F2,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(B),list(set(A)),map(B,set(A),F2),L))) ).
% list_collect_set_as_map
tff(fact_5239_distinct__idx,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),L: list(B),I2: nat,J2: nat] :
( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),aa(list(B),nat,size_size(list(B)),L))
=> ( ( aa(B,A,F2,aa(nat,B,nth(B,L),I2)) = aa(B,A,F2,aa(nat,B,nth(B,L),J2)) )
=> ( I2 = J2 ) ) ) ) ) ).
% distinct_idx
tff(fact_5240_map__upd__eq,axiom,
! [B: $tType,A: $tType,I2: nat,L: list(A),F2: fun(A,B),X: A] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(A,B,F2,aa(nat,A,nth(A,L),I2)) = aa(A,B,F2,X) ) )
=> ( aa(list(A),list(B),map(A,B,F2),list_update(A,L,I2,X)) = aa(list(A),list(B),map(A,B,F2),L) ) ) ).
% map_upd_eq
tff(fact_5241_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs2: list(product_prod(A,B))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( zip(A,B,Xs,Ys) = Zs2 )
<=> ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs2) = Xs )
& ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs2) = Ys ) ) ) ) ).
% zip_eq_conv
tff(fact_5242_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
tff(fact_5243_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z2: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
tff(fact_5244_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),X: A,Xs: list(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(B),map(A,B,F2),removeAll(A,X,Xs)) = removeAll(B,aa(A,B,F2,X),aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ).
% map_removeAll_inj_on
tff(fact_5245_Id__on__set,axiom,
! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_ex(A,product_prod(A,A))),Xs)) ).
% Id_on_set
tff(fact_5246_map__by__foldl,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),L: list(B)] : aa(list(B),list(A),aa(list(A),fun(list(B),list(A)),foldl(list(A),B,aTP_Lamp_vq(fun(B,A),fun(list(A),fun(B,list(A))),F2)),nil(A)),L) = aa(list(B),list(A),map(B,A,F2),L) ).
% map_by_foldl
tff(fact_5247_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),zip(B,A,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).
% map_snd_zip_take
tff(fact_5248_merge__list_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X,Xa) = Y )
=> ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),Xa))
=> ( ( ( X = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = nil(A) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))) ) ) )
=> ( ( ( X = nil(list(A)) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( ( Y = L3 )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A)))) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) ) )
=> ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)) )
=> ( ( Y = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),X),Ls) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)))) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
tff(fact_5249_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls2: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))))
=> ( merge_list(A,Acc22,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L23)),Acc22),Ls2) ) ) ) ).
% merge_list.psimps(5)
tff(fact_5250_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))))
=> ( merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ) ) ).
% merge_list.psimps(2)
tff(fact_5251_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ).
% merge_list.simps(1)
tff(fact_5252_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] : merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ).
% merge_list.simps(2)
tff(fact_5253_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A))] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22)) ) ).
% merge_list.simps(3)
tff(fact_5254_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A)),L: list(A)] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22))) ) ).
% merge_list.simps(4)
tff(fact_5255_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls2: list(list(A))] : merge_list(A,Acc22,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L23)),Acc22),Ls2) ) ).
% merge_list.simps(5)
tff(fact_5256_merge__list_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X,Xa) = Y )
=> ( ( ( X = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != nil(A) ) ) )
=> ( ( ( X = nil(list(A)) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( Y != L3 ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) ) ) )
=> ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)) )
=> ( Y != merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),X),Ls) ) ) ) ) ) ) ) ) ).
% merge_list.elims
tff(fact_5257_mergesort__remdups__def,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),mergesort_remdups(A),Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_vr(A,list(A))),Xs)) ) ).
% mergesort_remdups_def
tff(fact_5258_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
=> ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ) ).
% merge_list.psimps(1)
tff(fact_5259_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A)),L: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22))) ) ) ) ).
% merge_list.psimps(4)
tff(fact_5260_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22)),nil(list(A))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc22)) ) ) ) ).
% merge_list.psimps(3)
tff(fact_5261_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
=> ( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))),X,none(B)) ) ) ) ).
% map_of_distinct_upd4
tff(fact_5262_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y7: B] :
( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
=> ( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),Ys))),X,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd3
tff(fact_5263_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
=> ( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)),X,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd2
tff(fact_5264_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))
=> ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_is_SomeI
tff(fact_5265_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).
% Some_eq_map_of_iff
tff(fact_5266_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).
% map_of_eq_Some_iff
tff(fact_5267_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
=> ? [X3: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X3) ) ).
% weak_map_of_SomeI
tff(fact_5268_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).
% map_of_SomeD
tff(fact_5269_map__of__Cons__code_I2_J,axiom,
! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K: B] :
aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V)),Ps)),K) = $ite(L = K,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K)) ).
% map_of_Cons_code(2)
tff(fact_5270_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ).
% ran_map_of
tff(fact_5271_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P2),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P2),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P2))) ).
% map_of.simps(2)
tff(fact_5272_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ? [Ys2: list(product_prod(B,A)),Zs: list(product_prod(B,A))] :
( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys2),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),Zs)) )
& ( aa(B,option(A),map_of(B,A,Ys2),K) = none(A) ) ) ) ).
% map_of_Some_split
tff(fact_5273_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
=> ( map_to_set(A,B,map_of(A,B,L)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) ) ) ).
% map_to_set_map_of
tff(fact_5274_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B)),M: fun(A,option(B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
=> ( ( map_of(A,B,L) = M )
<=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M) ) ) ) ).
% map_of_map_to_set
tff(fact_5275_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_db(fun(A,B),fun(A,product_prod(A,B)),F2)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F2),aa(list(A),set(A),set2(A),Ks)) ).
% map_of_map_restrict
tff(fact_5276_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
=> ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_ug(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).
% map_of_map_keys
tff(fact_5277_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),T5: list(product_prod(A,C)),K: A,X: C] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( ( aa(A,option(C),map_of(A,C,T5),K) = aa(C,option(C),some(C),X) )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_vs(fun(A,B),fun(A,fun(C,product_prod(B,C))),F2))),T5)),aa(A,B,F2,K)) = aa(C,option(C),some(C),X) ) ) ) ).
% map_of_mapk_SomeI
tff(fact_5278_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
=> ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).
% Misc.ran_distinct
tff(fact_5279_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
=> ( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
=> ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))),X) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_distinct_lookup
tff(fact_5280_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
=> ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_ua(A,option(B)),X,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),X,aa(B,option(B),some(B),Y)) ) ) ).
% map_of_distinct_upd
tff(fact_5281_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_vt(list(product_prod(A,B)),fun(A,B),Xs)),mset(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))) = mset(B,aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs)) ) ) ).
% image_mset_map_of
tff(fact_5282_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [L: list(product_prod(A,B))] :
( sorted_wrt(product_prod(A,B),aTP_Lamp_vu(product_prod(A,B),fun(product_prod(A,B),$o)),L)
<=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).
% sorted_wrt_map_rev_linord
tff(fact_5283_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_vw(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).
% set_relcomp
tff(fact_5284_product__code,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_vx(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).
% product_code
tff(fact_5285_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,aTP_Lamp_nr(A,fun(A,$o)),L)
<=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).
% sorted_wrt_rev_linord
tff(fact_5286_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),M))
=> ( map_of(A,B,rev(product_prod(A,B),M)) = map_of(A,B,M) ) ) ).
% map_of_rev_distinct
tff(fact_5287_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A4: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),foldl(A,list(B),foldl(A,B,F2)),A4),Xs) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F2),A4),concat(B,Xs)) ).
% foldl_foldl_conv_concat
tff(fact_5288_product__concat__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_vx(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).
% product_concat_map
tff(fact_5289_distinct__concat,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(list(A),Xs)
=> ( ! [Ys2: list(A)] :
( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> distinct(A,Ys2) )
=> ( ! [Ys2: list(A),Zs: list(A)] :
( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> ( member(list(A),Zs,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> ( ( Ys2 != Zs )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys2)),aa(list(A),set(A),set2(A),Zs)) = bot_bot(set(A)) ) ) ) )
=> distinct(A,concat(A,Xs)) ) ) ) ).
% distinct_concat
tff(fact_5290_distinct__concat__iff,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(A,concat(A,Xs))
<=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
& ! [Ys3: list(A)] :
( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> distinct(A,Ys3) )
& ! [Ys3: list(A),Zs3: list(A)] :
( ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
& member(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
& ( Ys3 != Zs3 ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).
% distinct_concat_iff
tff(fact_5291_merge__list__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ls2: list(list(A)),As: list(list(A))] :
( ! [L3: list(A)] :
( member(list(A),L3,aa(list(list(A)),set(list(A)),set2(list(A)),Ls2))
=> ( distinct(A,L3)
& sorted_wrt(A,ord_less_eq(A),L3) ) )
=> ( ! [L3: list(A)] :
( member(list(A),L3,aa(list(list(A)),set(list(A)),set2(list(A)),As))
=> ( distinct(A,L3)
& sorted_wrt(A,ord_less_eq(A),L3) ) )
=> ( distinct(A,merge_list(A,As,Ls2))
& sorted_wrt(A,ord_less_eq(A),merge_list(A,As,Ls2))
& ( aa(list(A),set(A),set2(A),merge_list(A,As,Ls2)) = aa(list(A),set(A),set2(A),concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As),Ls2))) ) ) ) ) ) ).
% merge_list_correct
tff(fact_5292_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_vy(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs),Ys) ).
% zip_Cons1
tff(fact_5293_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_vz(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys),Xs) ).
% zip_Cons
tff(fact_5294_revg__fun,axiom,
! [A: $tType,A4: list(A),B3: list(A)] : revg(A,A4,B3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,A4)),B3) ).
% revg_fun
tff(fact_5295_revg_Osimps_I2_J,axiom,
! [A: $tType,A4: A,As: list(A),B3: list(A)] : revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As),B3) = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),B3)) ).
% revg.simps(2)
tff(fact_5296_revg_Osimps_I1_J,axiom,
! [A: $tType,B3: list(A)] : revg(A,nil(A),B3) = B3 ).
% revg.simps(1)
tff(fact_5297_revg_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [A5: A,As2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ( Y != revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) ) ) ) ) ).
% revg.elims
tff(fact_5298_revg_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ~ ! [A5: A,As2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ( ( Y = revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),Xa)) ) ) ) ) ) ).
% revg.pelims
tff(fact_5299_distinct__concat_H,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(list(A),filter2(list(A),aTP_Lamp_wa(list(A),$o),Xs))
=> ( ! [Ys2: list(A)] :
( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> distinct(A,Ys2) )
=> ( ! [Ys2: list(A),Zs: list(A)] :
( member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> ( member(list(A),Zs,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
=> ( ( Ys2 != Zs )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys2)),aa(list(A),set(A),set2(A),Zs)) = bot_bot(set(A)) ) ) ) )
=> distinct(A,concat(A,Xs)) ) ) ) ).
% distinct_concat'
tff(fact_5300_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_wb(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss,aTP_Lamp_ua(A,option(B))) ).
% map_of_concat
tff(fact_5301_sorted__filter_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),P: fun(A,$o)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> sorted_wrt(A,ord_less_eq(A),filter2(A,P,L)) ) ) ).
% sorted_filter'
tff(fact_5302_last__filter,axiom,
! [A: $tType,Xs: list(A),P: fun(A,$o)] :
( ( Xs != nil(A) )
=> ( aa(A,$o,P,last(A,Xs))
=> ( last(A,filter2(A,P,Xs)) = last(A,Xs) ) ) ) ).
% last_filter
tff(fact_5303_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list(list(A))] : concat(A,filter2(list(A),aTP_Lamp_wa(list(A),$o),Xs)) = concat(A,Xs) ).
% concat_filter_neq_Nil
tff(fact_5304_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F2: fun(A,fun(C,C)),B3: C] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
=> ( foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_wc(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F2)),zip(B,A,Xs,Ys),B3) = foldr(A,C,F2,Ys,B3) ) ) ).
% foldr_snd_zip
tff(fact_5305_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
=> ( filter2(A,aa(A,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(A,fun(A,$o)),F2),Y),Xs) = filter2(A,aa(A,fun(A,$o),fequal(A),Y),Xs) ) ) ).
% inj_on_filter_key_eq
tff(fact_5306_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Xs: list(A),A4: B] :
( finite6289374366891150609ommute(A,B,F2)
=> ( foldr(A,B,F2,Xs,A4) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_va(fun(A,fun(B,B)),fun(B,fun(A,B)),F2)),A4),Xs) ) ) ).
% comp_fun_commute.foldr_conv_foldl
tff(fact_5307_filter__eq__snocD,axiom,
! [A: $tType,P: fun(A,$o),L: list(A),L4: list(A),X: A] :
( ( filter2(A,P,L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
=> ( member(A,X,aa(list(A),set(A),set2(A),L))
& aa(A,$o,P,X) ) ) ).
% filter_eq_snocD
tff(fact_5308_set__minus__filter__out,axiom,
! [A: $tType,Xs: list(A),Y: A] : minus_minus(set(A),aa(list(A),set(A),set2(A),Xs),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_ce(A,fun(A,$o)),Y),Xs)) ).
% set_minus_filter_out
tff(fact_5309_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_we(list(A),fun(A,$o),Xs),Zs2) = Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
tff(fact_5310_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_wf(list(A),fun(A,$o),Xs),Zs2) = Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
tff(fact_5311_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_we(list(A),fun(A,$o),Ys),Zs2) = Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
tff(fact_5312_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_wf(list(A),fun(A,$o),Ys),Zs2) = Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
tff(fact_5313_filter__nth__ex__nth,axiom,
! [A: $tType,N2: nat,P: fun(A,$o),Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs)))
=> ? [M3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M3)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),aa(list(A),nat,size_size(list(A)),Xs))
& ( aa(nat,A,nth(A,filter2(A,P,Xs)),N2) = aa(nat,A,nth(A,Xs),M3) )
& ( filter2(A,P,take(A,M3,Xs)) = take(A,N2,filter2(A,P,Xs)) ) ) ) ).
% filter_nth_ex_nth
tff(fact_5314_horner__sum__foldr,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F2: fun(B,A),A4: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A4),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_wg(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A4),Xs,zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_5315_remove__rev__alt__def,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remove_rev(A,X),Xs) = filter2(A,aa(A,fun(A,$o),aTP_Lamp_ce(A,fun(A,$o)),X),rev(A,Xs)) ).
% remove_rev_alt_def
tff(fact_5316_filter__rev__alt,axiom,
! [A: $tType,P: fun(A,$o),L: list(A)] : aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev(A),P),L) = filter2(A,P,rev(A,L)) ).
% filter_rev_alt
tff(fact_5317_filter__rev__aux__alt,axiom,
! [A: $tType,A4: list(A),P: fun(A,$o),L: list(A)] : aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A4),P),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter2(A,P,rev(A,L))),A4) ).
% filter_rev_aux_alt
tff(fact_5318_foldr__length,axiom,
! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_wh(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).
% foldr_length
tff(fact_5319_filter__rev__def,axiom,
! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).
% filter_rev_def
tff(fact_5320_remove__rev__def,axiom,
! [A: $tType,X: A] : remove_rev(A,X) = aa(fun(A,$o),fun(list(A),list(A)),filter_rev(A),aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),aa(A,fun(A,$o),fequal(A),X))) ).
% remove_rev_def
tff(fact_5321_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),X: B,P: fun(product_prod(B,A),$o)] :
( ( aa(B,option(A),map_of(B,A,Xs),X) = none(A) )
=> ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P,Xs)),X) = none(A) ) ) ).
% map_of_None_filterD
tff(fact_5322_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list(A)] : foldr(A,list(A),cons(A),Xs,nil(A)) = Xs ).
% Misc.foldr_Cons
tff(fact_5323_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,A4: list(A),P: fun(A,$o),X: A,Xs: list(A)] :
aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A4),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = $ite(aa(A,$o,P,X),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A4)),P),Xs),aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A4),P),Xs)) ).
% filter_rev_aux.simps(2)
tff(fact_5324_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A4: list(A),P: fun(A,$o)] : aa(list(A),list(A),aa(fun(A,$o),fun(list(A),list(A)),filter_rev_aux(A,A4),P),nil(A)) = A4 ).
% filter_rev_aux.simps(1)
tff(fact_5325_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P: fun(product_prod(A,B),$o)] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),filter2(product_prod(A,B),P,Xs))) ) ).
% distinct_map_fst_filterI
tff(fact_5326_filter__conv__foldr,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter2(A,P,Xs) = foldr(A,list(A),aTP_Lamp_wi(fun(A,$o),fun(A,fun(list(A),list(A))),P),Xs,nil(A)) ).
% filter_conv_foldr
tff(fact_5327_foldr__length__aux,axiom,
! [A: $tType,L: list(A),A4: nat] : foldr(A,nat,aTP_Lamp_wh(A,fun(nat,nat)),L,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),aa(list(A),nat,size_size(list(A)),L)) ).
% foldr_length_aux
tff(fact_5328_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A,P: fun(product_prod(B,A),$o)] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ( ~ aa(product_prod(B,A),$o,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V))
=> ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
=> ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P,Xs)),K) = none(A) ) ) ) ) ).
% map_of_Some_filter_not_in
tff(fact_5329_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_is(A,fun(B,product_prod(B,A))),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_snd_mk_snd
tff(fact_5330_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_fst_mk_fst
tff(fact_5331_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : list_all2(A,B,R3) = relcompp(list(A),list(product_prod(A,B)),list(B),conversep(list(product_prod(A,B)),list(A),bNF_Grp(list(product_prod(A,B)),list(A),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_wj(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R3)),map(product_prod(A,B),A,product_fst(A,B)))),bNF_Grp(list(product_prod(A,B)),list(B),aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_wj(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R3)),map(product_prod(A,B),B,product_snd(A,B)))) ).
% list.rel_compp_Grp
tff(fact_5332_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: nat,X: A,J2: nat,Y: B] : zip(A,B,replicate(A,I2,X),replicate(B,J2,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_replicate
tff(fact_5333_set__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2 != zero_zero(nat) )
=> ( aa(list(A),set(A),set2(A),replicate(A,N2,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ).
% set_replicate
tff(fact_5334_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),L: list(A),L4: list(B),Q: fun(list(A),fun(list(B),$o))] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),L),L4)
=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,nil(A)),nil(B))
=> ( ! [X3: A,X8: B,Ls: list(A),Ls3: list(B)] :
( aa(B,$o,aa(A,fun(B,$o),P,X3),X8)
=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Ls),Ls3)
=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,Ls),Ls3)
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ls)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X8),Ls3)) ) ) )
=> aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,L),L4) ) ) ) ).
% Misc.list_all2_induct
tff(fact_5335_set__replicate__conv__if,axiom,
! [A: $tType,N2: nat,X: A] :
aa(list(A),set(A),set2(A),replicate(A,N2,X)) = $ite(N2 = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% set_replicate_conv_if
tff(fact_5336_set__replicate__Suc,axiom,
! [A: $tType,N2: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N2),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).
% set_replicate_Suc
tff(fact_5337_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N2: nat,X: A] : replicate(A,aa(nat,nat,suc,N2),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N2,X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
% replicate_Suc_conv_snoc
tff(fact_5338_zip__replicate1,axiom,
! [A: $tType,B: $tType,N2: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N2,X),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),take(B,N2,Ys)) ).
% zip_replicate1
tff(fact_5339_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))),K)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).
% map_zip1
tff(fact_5340_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).
% map_zip2
tff(fact_5341_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list(A),N2: nat,Y: B] : zip(A,B,Xs,replicate(B,N2,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))),Y)),take(A,N2,Xs)) ).
% zip_replicate2
tff(fact_5342_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( comm_semiring_0(B)
& comm_semiring_0(A) )
=> ! [A3: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o))] :
( aa(B,$o,aa(A,fun(B,$o),A3,zero_zero(A)),zero_zero(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),plus_plus(A)),plus_plus(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),times_times(A)),times_times(B))
=> aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B5,A3),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A3,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B5),A3))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).
% horner_sum_transfer
tff(fact_5343_list_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),A4: list(A),B3: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R3),A4),B3)
<=> ? [Z3: list(product_prod(A,B))] :
( member(list(product_prod(A,B)),Z3,aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_wj(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R3)))
& ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z3) = A4 )
& ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z3) = B3 ) ) ) ).
% list.in_rel
tff(fact_5344_partition__rev__filter__conv,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,P,Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs))),No2)) ).
% partition_rev_filter_conv
tff(fact_5345_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_mult(B)
& monoid_mult(A) )
=> ! [A3: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),A3,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),times_times(A)),times_times(B))
=> aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A3),A3),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).
% prod_list_transfer
tff(fact_5346_length__product__lists,axiom,
! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss),one_one(nat)) ).
% length_product_lists
tff(fact_5347_prod__list_OCons,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).
% prod_list.Cons
tff(fact_5348_prod__list_ONil,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).
% prod_list.Nil
tff(fact_5349_prod__list_Oappend,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ).
% prod_list.append
tff(fact_5350_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),X: A,Xs: list(A)] :
partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = partition_rev(A,P,
$ite(aa(A,$o,P,X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Yes2)),No2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),No2))),
Xs) ).
% partition_rev.simps(2)
tff(fact_5351_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) ).
% partition_rev.simps(1)
tff(fact_5352_partition__rev_Oelims,axiom,
! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X,Xa,Xb) = Y )
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ( Y != partition_rev(A,X,
$ite(aa(A,$o,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),
Xs3) ) ) ) ) ) ).
% partition_rev.elims
tff(fact_5353_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = foldr(A,A,times_times(A),Xs,one_one(A)) ) ).
% prod_list.eq_foldr
tff(fact_5354_partition__rev_Opelims,axiom,
! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa),Xb)))
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ~ aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ( ( Y = partition_rev(A,X,
$ite(aa(A,$o,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),
Xs3) )
=> ~ aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)))) ) ) ) ) ) ) ).
% partition_rev.pelims
tff(fact_5355_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa),Xb) = Y )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ( Y != aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wk(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X3)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) ) ) ) ) ).
% quicksort_by_rel.elims
tff(fact_5356_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wk(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R3),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R3),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ).
% quicksort_by_rel.simps(2)
tff(fact_5357_set__quicksort__by__rel,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).
% set_quicksort_by_rel
tff(fact_5358_quicksort__by__rel__permutes,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A),Xs: list(A)] : mset(A,aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),Xs)) = mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).
% quicksort_by_rel_permutes
tff(fact_5359_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),nil(A)) = Sl2 ).
% quicksort_by_rel.simps(1)
tff(fact_5360_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list(A),R3: fun(A,fun(A,$o)),Xs: list(A)] :
( ( Sl2 != nil(A) )
=> ( aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R3,nil(A)),Xs)),Sl2) ) ) ).
% quicksort_by_rel_remove_acc_guared
tff(fact_5361_quicksort__by__rel__remove__acc,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A),Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R3,nil(A)),Xs)),Sl2) ).
% quicksort_by_rel_remove_acc
tff(fact_5362_sorted__wrt__quicksort__by__rel,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
| aa(A,$o,aa(A,fun(A,$o),R3,Y2),X3) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R3,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),R3,X3),Z4) ) )
=> sorted_wrt(A,R3,aa(list(A),list(A),quicksort_by_rel(A,R3,nil(A)),Xs)) ) ) ).
% sorted_wrt_quicksort_by_rel
tff(fact_5363_sorted__quicksort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),quicksort_by_rel(A,ord_less_eq(A),nil(A)),Xs)) ) ).
% sorted_quicksort_by_rel
tff(fact_5364_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
=> ( aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wk(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R3),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R3),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ) ) ).
% quicksort_by_rel.psimps(2)
tff(fact_5365_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa),Xb) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
=> ( ( Y = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wk(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X3)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)))) ) ) ) ) ) ).
% quicksort_by_rel.pelims
tff(fact_5366_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)))] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22)))
=> ( ! [R8: fun(A,fun(A,$o)),Sl: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))))
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R8),Sl),nil(A)) )
=> ( ! [R8: fun(A,fun(A,$o)),Sl: list(A),X3: A,Xs3: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R8),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3))))
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R8),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R8),Sl),Y5) ) )
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R8),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R8),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),quicksort_by_rel(A,R8,Sl),Y5))),Xb2) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R8),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3)) ) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,A0),A1),A22) ) ) ) ).
% quicksort_by_rel.pinduct
tff(fact_5367_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),Sl2: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),nil(A))))
=> ( aa(list(A),list(A),quicksort_by_rel(A,R3,Sl2),nil(A)) = Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
tff(fact_5368_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs)))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
| ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,Fs)) ) ) ) ).
% in_measures(2)
tff(fact_5369_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
=> ( hd(A,slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).
% slice_head
tff(fact_5370_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),insert(A,X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ).
% List.set_insert
tff(fact_5371_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,nil(fun(A,nat)))) ).
% in_measures(1)
tff(fact_5372_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( hd(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),hd(A,Xs)),hd(B,Ys)) ) ) ) ).
% hd_zip
tff(fact_5373_measures__less,axiom,
! [A: $tType,F2: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs))) ) ).
% measures_less
tff(fact_5374_measures__lesseq,axiom,
! [A: $tType,F2: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,Fs))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs))) ) ) ).
% measures_lesseq
tff(fact_5375_sorted__hd__min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> ! [X4: A] :
( member(A,X4,aa(list(A),set(A),set2(A),Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),hd(A,Xs)),X4) ) ) ) ) ).
% sorted_hd_min
tff(fact_5376_sorted__hd__last,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( ( L != nil(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),hd(A,L)),last(A,L)) ) ) ) ).
% sorted_hd_last
tff(fact_5377_hd__last__singletonI,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( hd(A,Xs) = last(A,Xs) )
=> ( distinct(A,Xs)
=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,Xs)),nil(A)) ) ) ) ) ).
% hd_last_singletonI
tff(fact_5378_hd__butlast,axiom,
! [A: $tType,Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))
=> ( hd(A,butlast(A,Xs)) = hd(A,Xs) ) ) ).
% hd_butlast
tff(fact_5379_rev__split__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,tl(A,L))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,L)),nil(A))) = rev(A,L) ) ) ).
% rev_split_conv
tff(fact_5380_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_vk(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),map_of(A,C,Xs)) ).
% map_of_map
tff(fact_5381_prod__list__def,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).
% prod_list_def
tff(fact_5382_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list(A),X: A] :
( ( L != nil(A) )
=> ( ( ( X = hd(A,L) )
| member(A,X,aa(list(A),set(A),set2(A),tl(A,L))) )
<=> member(A,X,aa(list(A),set(A),set2(A),L)) ) ) ).
% in_hd_or_tl_conv
tff(fact_5383_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( member(A,X,aa(list(A),set(A),set2(A),tl(A,Xs)))
=> member(A,X,aa(list(A),set(A),set2(A),Xs)) ) ).
% in_set_tlD
tff(fact_5384_tl__obtain__elem,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( tl(A,Xs) = nil(A) )
=> ~ ! [E2: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) ) ) ).
% tl_obtain_elem
tff(fact_5385_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( X != hd(A,Xs) )
=> ( member(A,X,aa(list(A),set(A),set2(A),Xs))
=> member(A,X,aa(list(A),set(A),set2(A),tl(A,Xs))) ) ) ).
% not_hd_in_tl
tff(fact_5386_tl__last,axiom,
! [A: $tType,Xs: list(A)] :
( ( tl(A,Xs) != nil(A) )
=> ( last(A,Xs) = last(A,tl(A,Xs)) ) ) ).
% tl_last
tff(fact_5387_rev__butlast__is__tl__rev,axiom,
! [A: $tType,L: list(A)] : rev(A,butlast(A,L)) = tl(A,rev(A,L)) ).
% rev_butlast_is_tl_rev
tff(fact_5388_tl__subset,axiom,
! [A: $tType,Xs: list(A),A3: set(A)] :
( ( Xs != nil(A) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),tl(A,Xs))),A3) ) ) ).
% tl_subset
tff(fact_5389_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list(A),N2: nat] :
( ( Xs != nil(A) )
=> ( aa(nat,A,nth(A,tl(A,Xs)),N2) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N2)) ) ) ).
% Misc.nth_tl
tff(fact_5390_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),P: fun(B,fun(A,$o))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ! [N3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs))
=> aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,Ys),N3)),aa(nat,A,nth(A,Xs),N3)) )
=> ! [N9: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N9),aa(list(A),nat,size_size(list(A)),tl(A,Xs)))
=> aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,tl(B,Ys)),N9)),aa(nat,A,nth(A,tl(A,Xs)),N9)) ) ) ) ).
% list_take_induct_tl2
tff(fact_5391_distinct__hd__tl,axiom,
! [A: $tType,Xs: list(A),X: A] :
( distinct(A,Xs)
=> ( ( X = hd(A,Xs) )
=> ~ member(A,X,aa(list(A),set(A),set2(A),tl(A,Xs))) ) ) ).
% distinct_hd_tl
tff(fact_5392_butlast__rev__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( butlast(A,rev(A,Xs)) = rev(A,tl(A,Xs)) ) ) ).
% butlast_rev_tl
tff(fact_5393_remove1__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( remove1(A,hd(A,Xs),Xs) = tl(A,Xs) ) ) ).
% remove1_tl
tff(fact_5394_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N: nat,L: list(A),M2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L))
=> ( aa(list(nat),list(A),map(nat,A,nth(A,L)),upt(M2,N)) = drop(A,M2,take(A,N,L)) ) ) ).
% map_nth_upt_drop_take_conv
tff(fact_5395_sum__list__triv,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [R2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_bd(A,fun(B,A),R2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).
% sum_list_triv
tff(fact_5396_set__insort__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_ul(A,A),X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).
% set_insort_insert
tff(fact_5397_upt__0__eq__Nil__conv,axiom,
! [J2: nat] :
( ( upt(zero_zero(nat),J2) = nil(nat) )
<=> ( J2 = zero_zero(nat) ) ) ).
% upt_0_eq_Nil_conv
tff(fact_5398_upt__merge,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J2),K) )
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J2)),upt(J2,K)) = upt(I2,K) ) ) ).
% upt_merge
tff(fact_5399_map__add__upt_H,axiom,
! [Ofs: nat,A4: nat,B3: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_wm(nat,fun(nat,nat),Ofs)),upt(A4,B3)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),Ofs)) ).
% map_add_upt'
tff(fact_5400_upt__eq__append__conv,axiom,
! [I2: nat,J2: nat,Xs: list(nat),Ys: list(nat)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
=> ( ( upt(I2,J2) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys) )
<=> ? [K5: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),K5)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K5),J2)
& ( upt(I2,K5) = Xs )
& ( upt(K5,J2) = Ys ) ) ) ) ).
% upt_eq_append_conv
tff(fact_5401_upt__append,axiom,
! [I2: nat,J2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I2)),upt(I2,J2)) = upt(zero_zero(nat),J2) ) ) ).
% upt_append
tff(fact_5402_butlast__upt,axiom,
! [M: nat,N2: nat] : butlast(nat,upt(M,N2)) = upt(M,minus_minus(nat,N2,one_one(nat))) ).
% butlast_upt
tff(fact_5403_filter__upt__take__conv,axiom,
! [A: $tType,P: fun(A,$o),M: nat,L: list(A),N2: nat] : filter2(nat,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_wn(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),P),M),L),upt(N2,M)) = filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_wo(fun(A,$o),fun(list(A),fun(nat,$o)),P),L),upt(N2,M)) ).
% filter_upt_take_conv
tff(fact_5404_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),C2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F2),C2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs))),C2) ) ).
% sum_list_mult_const
tff(fact_5405_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F2: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cr(A,fun(fun(B,A),fun(B,A)),C2),F2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs))) ) ).
% sum_list_const_mult
tff(fact_5406_upt__eq__lel__conv,axiom,
! [L: nat,Ha: nat,Is1: list(nat),I2: nat,Is2: list(nat)] :
( ( upt(L,Ha) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),Is2)) )
<=> ( ( Is1 = upt(L,I2) )
& ( Is2 = upt(aa(nat,nat,suc,I2),Ha) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),I2)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Ha) ) ) ).
% upt_eq_lel_conv
tff(fact_5407_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: fun(nat,$o)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U4)
=> ( ! [I3: nat] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I3)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),U4) )
=> ~ aa(nat,$o,P,I3) )
=> ( filter2(nat,P,upt(zero_zero(nat),U)) = filter2(nat,P,upt(zero_zero(nat),U4)) ) ) ) ).
% upt_filter_extend
tff(fact_5408_sum__list__replicate,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N2: nat,C2: A] : groups8242544230860333062m_list(A,replicate(A,N2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),C2) ) ).
% sum_list_replicate
tff(fact_5409_filter__upt__last,axiom,
! [A: $tType,P: fun(A,$o),L: list(A),Js: list(nat),J2: nat,I2: nat] :
( ( filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_wo(fun(A,$o),fun(list(A),fun(nat,$o)),P),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),I2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ~ aa(A,$o,P,aa(nat,A,nth(A,L),I2)) ) ) ) ).
% filter_upt_last
tff(fact_5410_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list(A),X5: set(A),F2: fun(A,nat)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
=> ( aa(set(A),$o,finite_finite2(A),X5)
=> ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_wp(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X5) ) ) ) ).
% sum_list_map_eq_sum_count2
tff(fact_5411_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F2: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_wq(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ).
% sum_list_map_eq_sum_count
tff(fact_5412_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X,Xa) = Y )
=> ( aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X),Xa))
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),nil(A))) ) ) )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs22) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) ) )
=> ~ ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X12: A,X23: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)) )
=> ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs3) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs3)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
tff(fact_5413_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,$o))] : rel_option(A,B,R3) = relcompp(option(A),option(product_prod(A,B)),option(B),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_wr(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R3)),map_option(product_prod(A,B),A,product_fst(A,B)))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_wr(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R3)),map_option(product_prod(A,B),B,product_snd(A,B)))) ).
% option.rel_compp_Grp
tff(fact_5414_option_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,$o)),A4: option(A),B3: option(B)] :
( aa(option(B),$o,aa(option(A),fun(option(B),$o),rel_option(A,B,R3),A4),B3)
<=> ? [Z3: option(product_prod(A,B))] :
( member(option(product_prod(A,B)),Z3,aa(fun(option(product_prod(A,B)),$o),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_wr(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),R3)))
& ( aa(option(product_prod(A,B)),option(A),map_option(product_prod(A,B),A,product_fst(A,B)),Z3) = A4 )
& ( aa(option(product_prod(A,B)),option(B),map_option(product_prod(A,B),B,product_snd(A,B)),Z3) = B3 ) ) ) ).
% option.in_rel
tff(fact_5415_ordering__top__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
<=> ( ordering(A,Less_eq,Less)
& ordering_top_axioms(A,Less_eq,Top) ) ) ).
% ordering_top_def
tff(fact_5416_ordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
=> ordering(A,Less_eq,Less) ) ).
% ordering_dualI
tff(fact_5417_ordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
| ( A5 = B2 ) ) )
=> ( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B2),A5) )
=> ( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A5),A5)
=> ( ! [A5: A,B2: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less,A5),C3) ) )
=> ordering(A,Less_eq,Less) ) ) ) ) ).
% ordering_strictI
tff(fact_5418_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( ( A4 != B3 )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),B3) ) ) ) ).
% ordering.not_eq_order_implies_strict
tff(fact_5419_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ( A4 != B3 ) ) ) ).
% ordering.strict_implies_not_eq
tff(fact_5420_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
& ( A4 != B3 ) ) ) ) ).
% ordering.strict_iff_order
tff(fact_5421_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
| ( A4 = B3 ) ) ) ) ).
% ordering.order_iff_strict
tff(fact_5422_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A4)
=> ( A4 = B3 ) ) ) ) ).
% ordering.antisym
tff(fact_5423_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( ordering(A,Less_eq,Less)
=> ( ( A4 = B3 )
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
& aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A4) ) ) ) ).
% ordering.eq_iff
tff(fact_5424_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering(A,Less_eq,Less) ) ).
% ordering_top.axioms(1)
tff(fact_5425_order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.ordering_axioms
tff(fact_5426_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,aTP_Lamp_ws(A,fun(A,$o)),aTP_Lamp_wt(A,fun(A,$o))) ) ).
% dual_order.ordering_axioms
tff(fact_5427_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering(A,Less_eq,Less)
=> ( ordering_top_axioms(A,Less_eq,Top)
=> ordering_top(A,Less_eq,Less,Top) ) ) ).
% ordering_top.intro
tff(fact_5428_less__length__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: fun(A,$o),L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
& ! [J3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J3),I2)
=> aa(A,$o,P,aa(nat,A,nth(A,L),J3)) ) ) ) ).
% less_length_takeWhile_conv
tff(fact_5429_eq__len__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: fun(A,$o),L: list(A)] :
( ( I2 = aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)) )
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),L))
& ! [J3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I2)
=> aa(A,$o,P,aa(nat,A,nth(A,L),J3)) )
& ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
=> ~ aa(A,$o,P,aa(nat,A,nth(A,L),I2)) ) ) ) ).
% eq_len_takeWhile_conv
tff(fact_5430_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,minus_minus(set(B),A3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_5431_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list(A),P: fun(B,$o),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),$o),comp(B,$o,product_prod(A,B),P),product_snd(A,B)),zip(A,B,Xs,Ys)) ).
% zip_takeWhile_snd
tff(fact_5432_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,bot_bot(set(B))) = nil(B) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_5433_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A3) = nil(B) )
<=> ( A3 = bot_bot(set(B)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_5434_drop__takeWhile,axiom,
! [A: $tType,I2: nat,P: fun(A,$o),L: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)))
=> ( drop(A,I2,takeWhile(A,P,L)) = takeWhile(A,P,drop(A,I2,L)) ) ) ).
% drop_takeWhile
tff(fact_5435_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = insort_key(A,B,Less_eq,F2,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,minus_minus(set(B),A3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_5436_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),X: B,A3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( ~ member(B,X,A3)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = insort_key(A,B,Less_eq,F2,X,sorted8670434370408473282of_set(A,B,Less_eq,F2,A3)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_5437_length__dropWhile__takeWhile,axiom,
! [A: $tType,X: nat,P: fun(A,$o),Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).
% length_dropWhile_takeWhile
tff(fact_5438_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ul(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).
% sort_quicksort_by_rel
tff(fact_5439_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).
% prod.comm_monoid_list_set_axioms
tff(fact_5440_group_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( semigroup(A,F2)
=> ( group_axioms(A,F2,Z2,Inverse)
=> group(A,F2,Z2,Inverse) ) ) ).
% group.intro
tff(fact_5441_group_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> semigroup(A,F2) ) ).
% group.axioms(1)
tff(fact_5442_semigroup_Oassoc,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A4: A,B3: A,C2: A] :
( semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A4),B3)),C2) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,aa(A,fun(A,A),F2,B3),C2)) ) ) ).
% semigroup.assoc
tff(fact_5443_semigroup_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A5: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B2)),C3) = aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B2),C3))
=> semigroup(A,F2) ) ).
% semigroup.intro
tff(fact_5444_semigroup__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semigroup(A,F2)
<=> ! [A7: A,B7: A,C5: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A7),B7)),C5) = aa(A,A,aa(A,fun(A,A),F2,A7),aa(A,A,aa(A,fun(A,A),F2,B7),C5)) ) ).
% semigroup_def
tff(fact_5445_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_5446_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_5447_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
=> semigroup(A,F2) ) ).
% monoid.axioms(1)
tff(fact_5448_sort__mergesort,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ul(A,A)) = mergesort(A) ) ) ).
% sort_mergesort
tff(fact_5449_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ul(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% sort_mergesort_by_rel
tff(fact_5450_group__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
<=> ( semigroup(A,F2)
& group_axioms(A,F2,Z2,Inverse) ) ) ).
% group_def
tff(fact_5451_monoid__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
<=> ( semigroup(A,F2)
& monoid_axioms(A,F2,Z2) ) ) ).
% monoid_def
tff(fact_5452_monoid_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semigroup(A,F2)
=> ( monoid_axioms(A,F2,Z2)
=> monoid(A,F2,Z2) ) ) ).
% monoid.intro
tff(fact_5453_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),coset(A,Xs)) = coset(A,removeAll(A,X,Xs)) ).
% insert_code(2)
tff(fact_5454_monoid__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F2,Z2)
<=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A7) = A7
& ! [A7: A] : aa(A,A,aa(A,fun(A,A),F2,A7),Z2) = A7 ) ) ).
% monoid_axioms_def
tff(fact_5455_monoid__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),Z2) = A5
=> monoid_axioms(A,F2,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_5456_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
=> monoid_axioms(A,F2,Z2) ) ).
% monoid.axioms(2)
tff(fact_5457_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),remdups(A)),linorder_sort_key(A,A,aTP_Lamp_ul(A,A))) = mergesort_remdups(A) ) ) ).
% remdup_sort_mergesort_remdups
tff(fact_5458_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).
% prod_list.comm_monoid_list_axioms
tff(fact_5459_lazI,axiom,
! [A: $tType,B: $tType,A4: list(A),B3: list(B),P: fun(A,fun(B,$o))] :
( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
=> ( ! [I3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),B3))
=> aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,A4),I3)),aa(nat,B,nth(B,B3),I3)) )
=> list_all_zip(A,B,P,A4,B3) ) ) ).
% lazI
tff(fact_5460_laz__conj,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),A4: list(A),B3: list(B)] :
( list_all_zip(A,B,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_wu(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q),A4,B3)
<=> ( list_all_zip(A,B,P,A4,B3)
& list_all_zip(A,B,Q,A4,B3) ) ) ).
% laz_conj
tff(fact_5461_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),A3: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_wv(fun(A,$o),fun(A,fun(B,$o)),P),A3,B5)
<=> ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B5) )
& ! [X2: A] :
( member(A,X2,aa(list(A),set(A),set2(A),A3))
=> aa(A,$o,P,X2) ) ) ) ).
% laz_weak_Pa
tff(fact_5462_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: fun(B,$o),A3: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_ow(fun(B,$o),fun(A,fun(B,$o)),P),A3,B5)
<=> ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B5) )
& ! [X2: B] :
( member(B,X2,aa(list(B),set(B),set2(B),B5))
=> aa(B,$o,P,X2) ) ) ) ).
% laz_weak_Pb
tff(fact_5463_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: fun(A,fun(B,fun(C,$o))),A3: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_ww(fun(A,fun(B,fun(C,$o))),fun(A,fun(B,$o)),P),A3,B5)
=> ~ ! [C7: list(C)] :
( list_all_zip(A,C,aTP_Lamp_wx(fun(A,fun(B,fun(C,$o))),fun(A,fun(C,$o)),P),A3,C7)
=> ~ list_all_zip(B,C,aTP_Lamp_wy(fun(A,fun(B,fun(C,$o))),fun(B,fun(C,$o)),P),B5,C7) ) ) ).
% laz_swap_ex
tff(fact_5464_laz__eq,axiom,
! [A: $tType,A4: list(A),B3: list(A)] :
( list_all_zip(A,A,fequal(A),A4,B3)
<=> ( A4 = B3 ) ) ).
% laz_eq
tff(fact_5465_laz__len,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),A4: list(A),B3: list(B)] :
( list_all_zip(A,B,P,A4,B3)
=> ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) ) ) ).
% laz_len
tff(fact_5466_list__all__zip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : list_all_zip(A,B,P,nil(A),nil(B)) ).
% list_all_zip.simps(1)
tff(fact_5467_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),A4: A,As: list(A),B3: B,Bs: list(B)] :
( list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs))
<=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
& list_all_zip(A,B,P,As,Bs) ) ) ).
% list_all_zip.simps(2)
tff(fact_5468_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),F2: fun(C,A),As: list(C),Bs: list(B)] :
( list_all_zip(A,B,P,aa(list(C),list(A),map(C,A,F2),As),Bs)
<=> list_all_zip(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_wz(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),P),F2),As,Bs) ) ).
% list_all_zip_map1
tff(fact_5469_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,$o)),As: list(A),F2: fun(C,B),Bs: list(C)] :
( list_all_zip(A,B,P,As,aa(list(C),list(B),map(C,B,F2),Bs))
<=> list_all_zip(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_xa(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),P),F2),As,Bs) ) ).
% list_all_zip_map2
tff(fact_5470_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),V: A,Va2: list(A)] : ~ list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2),nil(B)) ).
% list_all_zip.simps(3)
tff(fact_5471_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),V: B,Va2: list(B)] : ~ list_all_zip(A,B,P,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va2)) ).
% list_all_zip.simps(4)
tff(fact_5472_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B),Y: $o] :
( ( list_all_zip(A,B,X,Xa,Xb)
<=> (Y) )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ (Y) ) )
=> ( ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ( (Y)
<=> ~ ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) ) ) )
=> ( ( ? [V3: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)
=> ( ( Xb = nil(B) )
=> (Y) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V3: B,Va: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)
=> (Y) ) ) ) ) ) ) ).
% list_all_zip.elims(1)
tff(fact_5473_list__all__zip_Oelims_I2_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X,Xa,Xb)
=> ( ( ( Xa = nil(A) )
=> ( Xb != nil(B) ) )
=> ~ ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ~ ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) ) ) ) ) ).
% list_all_zip.elims(2)
tff(fact_5474_list__all__zip_Oelims_I3_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X,Xa,Xb)
=> ( ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) ) )
=> ( ( ? [V3: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)
=> ( Xb != nil(B) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V3: B,Va: list(B)] : Xb != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) ) ) ) ) ).
% list_all_zip.elims(3)
tff(fact_5475_length__remdups__card,axiom,
! [A: $tType,L: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),L)) = finite_card(A,aa(list(A),set(A),set2(A),L)) ).
% length_remdups_card
tff(fact_5476_list__all__zip__alt,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),As: list(A),Bs: list(B)] :
( list_all_zip(A,B,P,As,Bs)
<=> ( ( aa(list(A),nat,size_size(list(A)),As) = aa(list(B),nat,size_size(list(B)),Bs) )
& ! [I4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),As))
=> aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,As),I4)),aa(nat,B,nth(B,Bs),I4)) ) ) ) ).
% list_all_zip_alt
tff(fact_5477_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X,Xa,Xb)
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2))))
=> ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) ) ) )
=> ( ! [V3: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( ( Xb = nil(B) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V3: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)))) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
tff(fact_5478_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X,Xa,Xb)
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) )
=> ~ ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2))))
=> ~ ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
tff(fact_5479_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B),Y: $o] :
( ( list_all_zip(A,B,X,Xa,Xb)
<=> (Y) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
=> ( ! [A5: A,As2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
=> ! [B2: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2) )
=> ( ( (Y)
<=> ( aa(B,$o,aa(A,fun(B,$o),X,A5),B2)
& list_all_zip(A,B,X,As2,Bs2) ) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2)))) ) ) )
=> ( ! [V3: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
=> ( ( Xb = nil(B) )
=> ( ~ (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V3: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
=> ( ~ (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)))) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
tff(fact_5480_nths__empty,axiom,
! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).
% nths_empty
tff(fact_5481_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),L: list(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( aa(list(B),list(A),map(B,A,hilbert_inv_into(A,B,top_top(set(A)),F2)),aa(list(A),list(B),map(A,B,F2),L)) = L ) ) ).
% inj_map_inv_f
tff(fact_5482_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
=> ordering_axioms(A,Less_eq,Less) ) ).
% ordering.axioms(2)
tff(fact_5483_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
& ( A5 != B2 ) ) )
=> ( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A5)
=> ( A5 = B2 ) ) )
=> ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_axioms.intro
tff(fact_5484_ordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering_axioms(A,Less_eq,Less)
<=> ( ! [A7: A,B7: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A7),B7)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),B7)
& ( A7 != B7 ) ) )
& ! [A7: A,B7: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),B7)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B7),A7)
=> ( A7 = B7 ) ) ) ) ) ).
% ordering_axioms_def
tff(fact_5485_ordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_def
tff(fact_5486_ordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
=> ( ordering_axioms(A,Less_eq,Less)
=> ordering(A,Less_eq,Less) ) ) ).
% ordering.intro
tff(fact_5487_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
! [A: $tType,Ha: multiset(A)] : bot_bot(set(multiset(A))) != set_atMost(multiset(A),subseteq_mset(A),Ha) ).
% subset_mset.not_empty_eq_Iic_eq_empty
tff(fact_5488_partial__preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
<=> ( ! [A7: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),A7)
& ! [A7: A,B7: A,C5: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),B7)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B7),C5)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),C5) ) ) ) ) ).
% partial_preordering_def
tff(fact_5489_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),A4: A,B3: A,C2: A] :
( partial_preordering(A,Less_eq)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),C2) ) ) ) ).
% partial_preordering.trans
tff(fact_5490_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),A5)
=> ( ! [A5: A,B2: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),C3) ) )
=> partial_preordering(A,Less_eq) ) ) ).
% partial_preordering.intro
tff(fact_5491_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),A4: A] :
( partial_preordering(A,Less_eq)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),A4) ) ).
% partial_preordering.refl
tff(fact_5492_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,ord_less_eq(A)) ) ).
% order.partial_preordering_axioms
tff(fact_5493_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% ordering.axioms(1)
tff(fact_5494_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,aTP_Lamp_xb(A,fun(A,$o))) ) ).
% dual_order.partial_preordering_axioms
tff(fact_5495_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> groups_monoid_list(A,times_times(A),one_one(A)) ) ).
% prod_list.monoid_list_axioms
tff(fact_5496_sym__INTER,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [X3: A] :
( member(A,X3,S)
=> sym(B,aa(A,set(product_prod(B,B)),R2,X3)) )
=> sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).
% sym_INTER
tff(fact_5497_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
=> comm_monoid_axioms(A,F2,Z2) ) ).
% comm_monoid.axioms(2)
tff(fact_5498_sym__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( sym(A,converse(A,A,R2))
<=> sym(A,R2) ) ).
% sym_converse
tff(fact_5499_symD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B3: A,A4: A] :
( sym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ).
% symD
tff(fact_5500_symE,axiom,
! [A: $tType,R2: set(product_prod(A,A)),B3: A,A4: A] :
( sym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2) ) ) ).
% symE
tff(fact_5501_symI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A5),R2) )
=> sym(A,R2) ) ).
% symI
tff(fact_5502_sym__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( sym(A,R2)
<=> ! [X2: A,Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R2) ) ) ).
% sym_def
tff(fact_5503_sym__Int,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( sym(A,R2)
=> ( sym(A,S2)
=> sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S2)) ) ) ).
% sym_Int
tff(fact_5504_sym__inv__image,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),F2: fun(B,A)] :
( sym(A,R2)
=> sym(B,inv_image(A,B,R2,F2)) ) ).
% sym_inv_image
tff(fact_5505_sym__Id__on,axiom,
! [A: $tType,A3: set(A)] : sym(A,id_on(A,A3)) ).
% sym_Id_on
tff(fact_5506_sym__Id,axiom,
! [A: $tType] : sym(A,id2(A)) ).
% sym_Id
tff(fact_5507_sym__conv__converse__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( sym(A,R2)
<=> ( converse(A,A,R2) = R2 ) ) ).
% sym_conv_converse_eq
tff(fact_5508_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),Z2) = A5
=> comm_monoid_axioms(A,F2,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_5509_comm__monoid__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F2,Z2)
<=> ! [A7: A] : aa(A,A,aa(A,fun(A,A),F2,A7),Z2) = A7 ) ).
% comm_monoid_axioms_def
tff(fact_5510_sym__Un,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( sym(A,R2)
=> ( sym(A,S2)
=> sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ).
% sym_Un
tff(fact_5511_sym__Int__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),converse(A,A,R2))) ).
% sym_Int_converse
tff(fact_5512_sym__Un__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),converse(A,A,R2))) ).
% sym_Un_converse
tff(fact_5513_sym__UNION,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [X3: A] :
( member(A,X3,S)
=> sym(B,aa(A,set(product_prod(B,B)),R2,X3)) )
=> sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).
% sym_UNION
tff(fact_5514_comm__monoid__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
<=> ( abel_semigroup(A,F2)
& comm_monoid_axioms(A,F2,Z2) ) ) ).
% comm_monoid_def
tff(fact_5515_comm__monoid_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( abel_semigroup(A,F2)
=> ( comm_monoid_axioms(A,F2,Z2)
=> comm_monoid(A,F2,Z2) ) ) ).
% comm_monoid.intro
tff(fact_5516_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,aTP_Lamp_xb(A,fun(A,$o)),aTP_Lamp_tc(A,fun(A,$o))) ) ).
% dual_order.preordering_axioms
tff(fact_5517_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
=> semigroup(A,F2) ) ).
% abel_semigroup.axioms(1)
tff(fact_5518_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
=> abel_semigroup(A,F2) ) ).
% comm_monoid.axioms(1)
tff(fact_5519_preordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
=> preordering(A,Less_eq,Less) ) ).
% preordering_dualI
tff(fact_5520_preordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
| ( A5 = B2 ) ) )
=> ( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B2),A5) )
=> ( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A5),A5)
=> ( ! [A5: A,B2: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less,A5),C3) ) )
=> preordering(A,Less_eq,Less) ) ) ) ) ).
% preordering_strictI
tff(fact_5521_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3) ) ) ).
% preordering.strict_implies_order
tff(fact_5522_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A4) ) ) ) ).
% preordering.strict_iff_not
tff(fact_5523_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),C2) ) ) ) ).
% preordering.strict_trans2
tff(fact_5524_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),C2) ) ) ) ).
% preordering.strict_trans1
tff(fact_5525_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),C2) ) ) ) ).
% preordering.strict_trans
tff(fact_5526_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A] :
( preordering(A,Less_eq,Less)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4) ) ).
% preordering.irrefl
tff(fact_5527_preordering_Oasym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: A,B3: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B3),A4) ) ) ).
% preordering.asym
tff(fact_5528_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),B3: A,A4: A,C2: A] :
( abel_semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,B3),aa(A,A,aa(A,fun(A,A),F2,A4),C2)) = aa(A,A,aa(A,fun(A,A),F2,A4),aa(A,A,aa(A,fun(A,A),F2,B3),C2)) ) ) ).
% abel_semigroup.left_commute
tff(fact_5529_abel__semigroup_Ocommute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A4: A,B3: A] :
( abel_semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A4),B3) = aa(A,A,aa(A,fun(A,A),F2,B3),A4) ) ) ).
% abel_semigroup.commute
tff(fact_5530_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_5531_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_5532_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% preordering.axioms(1)
tff(fact_5533_order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.preordering_axioms
tff(fact_5534_preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
=> ( preordering_axioms(A,Less_eq,Less)
=> preordering(A,Less_eq,Less) ) ) ).
% preordering.intro
tff(fact_5535_preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& preordering_axioms(A,Less_eq,Less) ) ) ).
% preordering_def
tff(fact_5536_abel__semigroup_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semigroup(A,F2)
=> ( abel_s757365448890700780axioms(A,F2)
=> abel_semigroup(A,F2) ) ) ).
% abel_semigroup.intro
tff(fact_5537_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A5: A,B2: A] : aa(A,A,aa(A,fun(A,A),F2,A5),B2) = aa(A,A,aa(A,fun(A,A),F2,B2),A5)
=> abel_s757365448890700780axioms(A,F2) ) ).
% abel_semigroup_axioms.intro
tff(fact_5538_abel__semigroup__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F2)
<=> ! [A7: A,B7: A] : aa(A,A,aa(A,fun(A,A),F2,A7),B7) = aa(A,A,aa(A,fun(A,A),F2,B7),A7) ) ).
% abel_semigroup_axioms_def
tff(fact_5539_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B2)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A5) ) )
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering_axioms.intro
tff(fact_5540_preordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering_axioms(A,Less_eq,Less)
<=> ! [A7: A,B7: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A7),B7)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A7),B7)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B7),A7) ) ) ) ).
% preordering_axioms_def
tff(fact_5541_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
=> abel_s757365448890700780axioms(A,F2) ) ).
% abel_semigroup.axioms(2)
tff(fact_5542_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering.axioms(2)
tff(fact_5543_abel__semigroup__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
<=> ( semigroup(A,F2)
& abel_s757365448890700780axioms(A,F2) ) ) ).
% abel_semigroup_def
tff(fact_5544_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X) != zero_zero(nat) ).
% prod.size_neq
tff(fact_5545_curr__def,axiom,
! [A: $tType,C: $tType,B: $tType,A3: set(A),F2: fun(product_prod(A,B),C),X4: A] :
bNF_Wellorder_curr(A,B,C,A3,F2,X4) = $ite(member(A,X4,A3),aa(A,fun(B,C),aTP_Lamp_ij(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2),X4),undefined(fun(B,C))) ).
% curr_def
tff(fact_5546_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
! [A: $tType,L: multiset(A)] : bot_bot(set(multiset(A))) != set_atLeast(multiset(A),subseteq_mset(A),L) ).
% subset_mset.not_empty_eq_Ici_eq_empty
tff(fact_5547_comm__monoid__set_Oinsert_H,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,I: set(B),P2: fun(B,A),I2: B] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aa(set(B),fun(fun(B,A),fun(B,$o)),aTP_Lamp_xc(A,fun(set(B),fun(fun(B,A),fun(B,$o))),Z2),I),P2)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I2),I)) = $ite(member(B,I2,I),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),I),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,P2,I2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),I))) ) ) ) ).
% comm_monoid_set.insert'
tff(fact_5548_find__SomeD_I2_J,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),X: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
=> member(A,X,aa(list(A),set(A),set2(A),Xs)) ) ).
% find_SomeD(2)
tff(fact_5549_in__range_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ! [X3: nat] :
( member(nat,X3,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),lim(product_unit,H)) ) ) ) ) ) ).
% in_range.pelims(3)
tff(fact_5550_find__SomeD_I1_J,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),X: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
=> aa(A,$o,P,X) ) ).
% find_SomeD(1)
tff(fact_5551_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).
% prod.comm_monoid_set_axioms
tff(fact_5552_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F2: fun(A,fun(A,A)),Z2: A,P2: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F2,Z2),P2),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty'
tff(fact_5553_in__range_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
<=> (Y) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ! [X2: nat] :
( member(nat,X2,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,H)) ) )
=> ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ) ) ).
% in_range.pelims(1)
tff(fact_5554_in__range_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ~ ! [X4: nat] :
( member(nat,X4,As2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H)) ) ) ) ) ) ).
% in_range.pelims(2)
tff(fact_5555_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ( As2 = bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
tff(fact_5556_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ( As2 != bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
tff(fact_5557_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
<=> (Y) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H: heap_ext(product_unit),As2: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2) )
=> ( ( (Y)
<=> ( As2 = bot_bot(set(nat)) ) )
=> ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ) ) ).
% one_assn_raw.pelims(1)
tff(fact_5558_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N2: nat] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_xd(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(nat,set(nat),set_ord_atMost(nat),N2)) ) ) ).
% comm_monoid_set.in_pairs_0
tff(fact_5559_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N2: nat] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_xd(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F2),G)),set_or1337092689740270186AtMost(nat,M,N2)) ) ) ).
% comm_monoid_set.in_pairs
tff(fact_5560_comm__monoid__set_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,I: set(B),A3: fun(B,set(C)),G: fun(C,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),I)
=> ( ! [X3: B] :
( member(B,X3,I)
=> aa(set(C),$o,finite_finite2(C),aa(B,set(C),A3,X3)) )
=> ( ! [X3: B] :
( member(B,X3,I)
=> ! [Xa4: B] :
( member(B,Xa4,I)
=> ( ( X3 != Xa4 )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A3,X3)),aa(B,set(C),A3,Xa4)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F2,Z2),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A3),I))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_xe(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F2),Z2),A3),G)),I) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
tff(fact_5561_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty
tff(fact_5562_comm__monoid__mult__class_Oprod__def,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ( groups7121269368397514597t_prod(A,B) = groups_comm_monoid_F(B,A,times_times(B),one_one(B)) ) ) ).
% comm_monoid_mult_class.prod_def
tff(fact_5563_comm__monoid__set_Oinsert,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(B),X: B,G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( ~ member(B,X,A3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A3)) ) ) ) ) ).
% comm_monoid_set.insert
tff(fact_5564_comm__monoid__set_Oinsert__if,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(B),G: fun(B,A),X: B] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = $ite(member(B,X,A3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A3),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A3))) ) ) ) ).
% comm_monoid_set.insert_if
tff(fact_5565_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,C6: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( ! [X3: set(B)] :
( member(set(B),X3,C6)
=> aa(set(B),$o,finite_finite2(B),X3) )
=> ( ! [X3: set(B)] :
( member(set(B),X3,C6)
=> ! [Xa4: set(B)] :
( member(set(B),Xa4,C6)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X3),Xa4) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C6)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups_comm_monoid_F(A,set(B),F2,Z2)),groups_comm_monoid_F(A,B,F2,Z2)),G),C6) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
tff(fact_5566_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(B),G: fun(B,A),X: B] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),minus_minus(set(B),A3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ).
% comm_monoid_set.insert_remove
tff(fact_5567_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(B),X: B,G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( member(B,X,A3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A3) = aa(A,A,aa(A,fun(A,A),F2,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),minus_minus(set(B),A3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% comm_monoid_set.remove
tff(fact_5568_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(B),B5: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),A3)
=> ( aa(set(B),$o,finite_finite2(B),B5)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B5) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B5)) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B5)) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
tff(fact_5569_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,S: set(B),A4: B,B3: fun(B,A),C2: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(B),$o,finite_finite2(B),S)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_xf(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A4),B3),C2)),S) = $ite(member(B,A4,S),aa(A,A,aa(A,fun(A,A),F2,aa(B,A,B3,A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),C2),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B)))))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),C2),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B)))))) ) ) ) ).
% comm_monoid_set.delta_remove
tff(fact_5570_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),K: nat,N2: nat] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_xg(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F2),Z2),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),K))) ) ) ).
% comm_monoid_set.nat_group
tff(fact_5571_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_la(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int_def
tff(fact_5572_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,R2),X),Y)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% Enum.rtranclp_rtrancl_eq
tff(fact_5573_rtrancl__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : transitive_rtrancl(A,X4) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_rtranclp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),X4)))) ).
% rtrancl_def
tff(fact_5574_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
=> aa(B,$o,aa(A,fun(B,$o),P,A5),B2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).
% converse_rtranclp_induct2
tff(fact_5575_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Xa: A,Xb: B,Za: A,Zb: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
=> ~ ! [A5: A,B2: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2))
=> ~ aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)) ) ) ) ).
% converse_rtranclpE2
tff(fact_5576_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
=> ( ! [A5: A,B2: B,Aa2: A,Ba: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))
=> ( aa(B,$o,aa(A,fun(B,$o),P,A5),B2)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% rtranclp_induct2
tff(fact_5577_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X4: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3),transitive_rtrancl(A,R2)) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_5578_next_Osimps,axiom,
! [V: 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),V),W)) = $let(
v: code_natural,
v:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one2)))))))))))))))),
$let(
w2: code_natural,
w2:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(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),bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(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_5579_dir__image__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F2: fun(B,A)] : bNF_We2720479622203943262_image(B,A,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_pb(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R2),F2)) ).
% dir_image_def
tff(fact_5580_accp__acc__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X4: A] :
( aa(A,$o,accp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4)
<=> member(A,X4,acc(A,R2)) ) ).
% accp_acc_eq
tff(fact_5581_acc__induct__rule,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(A,A4,acc(A,R2))
=> ( ! [X3: A] :
( member(A,X3,acc(A,R2))
=> ( ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2)
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,A4) ) ) ).
% acc_induct_rule
tff(fact_5582_not__acc__down,axiom,
! [A: $tType,X: A,R3: set(product_prod(A,A))] :
( ~ member(A,X,acc(A,R3))
=> ~ ! [Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X),R3)
=> member(A,Z4,acc(A,R3)) ) ) ).
% not_acc_down
tff(fact_5583_acc__downward,axiom,
! [A: $tType,B3: A,R2: set(product_prod(A,A)),A4: A] :
( member(A,B3,acc(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> member(A,A4,acc(A,R2)) ) ) ).
% acc_downward
tff(fact_5584_acc__induct,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(A,A4,acc(A,R2))
=> ( ! [X3: A] :
( member(A,X3,acc(A,R2))
=> ( ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2)
=> aa(A,$o,P,Y5) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,A4) ) ) ).
% acc_induct
tff(fact_5585_acc_Ointros,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),R2)
=> member(A,Y2,acc(A,R2)) )
=> member(A,X,acc(A,R2)) ) ).
% acc.intros
tff(fact_5586_acc_Osimps,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A))] :
( member(A,A4,acc(A,R2))
<=> ? [X2: A] :
( ( A4 = X2 )
& ! [Xa2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),R2)
=> member(A,Xa2,acc(A,R2)) ) ) ) ).
% acc.simps
tff(fact_5587_acc_Ocases,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A))] :
( member(A,A4,acc(A,R2))
=> ! [Y5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),A4),R2)
=> member(A,Y5,acc(A,R2)) ) ) ).
% acc.cases
tff(fact_5588_acc__subset__induct,axiom,
! [A: $tType,D4: set(A),R3: set(product_prod(A,A)),X: A,P: fun(A,$o)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),acc(A,R3))
=> ( ! [X3: A,Z4: A] :
( member(A,X3,D4)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X3),R3)
=> member(A,Z4,D4) ) )
=> ( member(A,X,D4)
=> ( ! [X3: A] :
( member(A,X3,D4)
=> ( ! [Z6: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X3),R3)
=> aa(A,$o,P,Z6) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,X) ) ) ) ) ).
% acc_subset_induct
tff(fact_5589_acc__downwards,axiom,
! [A: $tType,A4: A,R2: set(product_prod(A,A)),B3: A] :
( member(A,A4,acc(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),transitive_rtrancl(A,R2))
=> member(A,B3,acc(A,R2)) ) ) ).
% acc_downwards
tff(fact_5590_acc__downwards__aux,axiom,
! [A: $tType,B3: A,A4: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),transitive_rtrancl(A,R2))
=> ( member(A,A4,acc(A,R2))
=> member(A,B3,acc(A,R2)) ) ) ).
% acc_downwards_aux
tff(fact_5591_acc__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : acc(A,X4) = aa(fun(A,$o),set(A),collect(A),accp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),X4))) ).
% acc_def
tff(fact_5592_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),bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),K),aTP_Lamp_xi(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_xj(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_5593_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,minus_minus(code_natural,X,one_one(code_natural)),Xa))) ) ) ).
% iterate.elims
tff(fact_5594_iterate_Osimps,axiom,
! [A: $tType,B: $tType,K: code_natural,F2: fun(B,fun(A,product_prod(B,A))),X: B] :
aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),X) = $ite(K = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F2,X),iterate(B,A,minus_minus(code_natural,K,one_one(code_natural)),F2))) ).
% iterate.simps
tff(fact_5595_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb)))
=> ~ ( ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,minus_minus(code_natural,X,one_one(code_natural)),Xa))) )
=> ~ aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb))) ) ) ) ).
% iterate.pelims
tff(fact_5596_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o),A3: set(A)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
=> ( comple1602240252501008431_chain(A,Ord,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,A3)
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,aa(set(A),A,Lub,A3)) ) ) ) ) ).
% ccpo.admissibleD
tff(fact_5597_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: fun(A,fun(A,$o)),P: fun(A,$o),Lub: fun(set(A),A)] :
( ! [A10: set(A)] :
( comple1602240252501008431_chain(A,Ord,A10)
=> ( ( A10 != bot_bot(set(A)) )
=> ( ! [X4: A] :
( member(A,X4,A10)
=> aa(A,$o,P,X4) )
=> aa(A,$o,P,aa(set(A),A,Lub,A10)) ) ) )
=> comple1908693960933563346ssible(A,Lub,Ord,P) ) ).
% ccpo.admissibleI
tff(fact_5598_ccpo_Oadmissible__def,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
<=> ! [A11: set(A)] :
( comple1602240252501008431_chain(A,Ord,A11)
=> ( ( A11 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( member(A,X2,A11)
=> aa(A,$o,P,X2) )
=> aa(A,$o,P,aa(set(A),A,Lub,A11)) ) ) ) ) ).
% ccpo.admissible_def
tff(fact_5599_fixp__induct,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,$o),F2: fun(A,A)] :
( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
=> ( aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))))
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,P,aa(A,A,F2,X3)) )
=> aa(A,$o,P,comple115746919287870866o_fixp(A,F2)) ) ) ) ) ) ).
% fixp_induct
tff(fact_5600_prod__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> comm_monoid_mset(A,times_times(A),one_one(A)) ) ).
% prod_mset.comm_monoid_mset_axioms
tff(fact_5601_old_Ounit_Orec,axiom,
! [A: $tType,F1: A] : product_rec_unit(A,F1,product_Unity) = F1 ).
% old.unit.rec
tff(fact_5602_unit__abs__eta__conv,axiom,
! [A: $tType,F2: fun(product_unit,A)] : aTP_Lamp_xk(fun(product_unit,A),fun(product_unit,A),F2) = F2 ).
% unit_abs_eta_conv
tff(fact_5603_bot__unit__def,axiom,
bot_bot(product_unit) = product_Unity ).
% bot_unit_def
tff(fact_5604_sup__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).
% sup_unit_def
tff(fact_5605_fixp__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F2: fun(A,A),G: fun(A,A)] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F2)
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),G)
=> ( ! [Z9: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z9)),aa(A,A,G,Z9))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),comple115746919287870866o_fixp(A,F2)),comple115746919287870866o_fixp(A,G)) ) ) ) ) ).
% fixp_mono
tff(fact_5606_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
tff(fact_5607_Inf__unit__def,axiom,
! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Inf_Inf(product_unit),Uu) = product_Unity ).
% Inf_unit_def
tff(fact_5608_uminus__unit__def,axiom,
! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).
% uminus_unit_def
tff(fact_5609_top__unit__def,axiom,
top_top(product_unit) = product_Unity ).
% top_unit_def
tff(fact_5610_Unity__def,axiom,
product_Unity = aa($o,product_unit,product_Abs_unit,$true) ).
% Unity_def
tff(fact_5611_Sup__unit__def,axiom,
! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Sup_Sup(product_unit),Uu) = product_Unity ).
% Sup_unit_def
tff(fact_5612_inf__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),inf_inf(product_unit),Uu),Uv) = product_Unity ).
% inf_unit_def
tff(fact_5613_UNIV__unit,axiom,
top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert2(product_unit),product_Unity),bot_bot(set(product_unit))) ).
% UNIV_unit
tff(fact_5614_default__unit__def,axiom,
default_default(product_unit) = product_Unity ).
% default_unit_def
tff(fact_5615_abstract__filter__def,axiom,
! [A: $tType,F2: fun(product_unit,filter(A))] : abstract_filter(A,F2) = aa(product_unit,filter(A),F2,product_Unity) ).
% abstract_filter_def
tff(fact_5616_CODE__ABORT__def,axiom,
! [A: $tType,F2: fun(product_unit,A)] : cODE_ABORT(A,F2) = aa(product_unit,A,F2,product_Unity) ).
% CODE_ABORT_def
tff(fact_5617_cone__def,axiom,
bNF_Cardinal_cone = bNF_Ca6860139660246222851ard_of(product_unit,aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert2(product_unit),product_Unity),bot_bot(set(product_unit)))) ).
% cone_def
tff(fact_5618_old_Ounit_Ocase,axiom,
! [A: $tType,F2: A] : product_case_unit(A,F2,product_Unity) = F2 ).
% old.unit.case
tff(fact_5619_scomp__fcomp,axiom,
! [A: $tType,D: $tType,E: $tType,B: $tType,C: $tType,F2: fun(A,product_prod(D,E)),G: fun(D,fun(E,C)),Ha: fun(C,B)] : fcomp(A,C,B,product_scomp(A,D,E,C,F2,G),Ha) = product_scomp(A,D,E,B,F2,aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_xl(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),G),Ha)) ).
% scomp_fcomp
tff(fact_5620_unit_Ocase__distrib,axiom,
! [B: $tType,A: $tType,Ha: fun(B,A),F2: B,Unit: product_unit] : aa(B,A,Ha,product_case_unit(B,F2,Unit)) = product_case_unit(A,aa(B,A,Ha,F2),Unit) ).
% unit.case_distrib
tff(fact_5621_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F2: fun(A,E),G: fun(E,product_prod(C,D)),Ha: fun(C,fun(D,B))] : product_scomp(A,C,D,B,fcomp(A,E,product_prod(C,D),F2,G),Ha) = fcomp(A,E,B,F2,product_scomp(E,C,D,B,G,Ha)) ).
% fcomp_scomp
tff(fact_5622_symp__INF,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,fun(B,fun(B,$o)))] :
( ! [X3: A] :
( member(A,X3,S)
=> symp(B,aa(A,fun(B,fun(B,$o)),R2,X3)) )
=> symp(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Inf_Inf(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),S))) ) ).
% symp_INF
tff(fact_5623_times__integer__code_I3_J,axiom,
! [M: num,N2: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Pos(M)),code_Pos(N2)) = code_Pos(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_integer_code(3)
tff(fact_5624_ord__to__filter__compat,axiom,
! [A: $tType,R0: set(product_prod(A,A))] : bNF_Wellorder_compat(set(product_prod(A,A)),set(A),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),inf_inf(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),bNF_We4044943003108391690rdLess(A,A)),product_Sigma(set(product_prod(A,A)),set(product_prod(A,A)),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert2(set(product_prod(A,A))),R0),bot_bot(set(set(product_prod(A,A)))))),aTP_Lamp_xm(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),R0))),bNF_We413866401316099525erIncl(A,R0),bNF_We8469521843155493636filter(A,R0)) ).
% ord_to_filter_compat
tff(fact_5625_sympD,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),B3: A,A4: A] :
( symp(A,R2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),R2,A4),B3) ) ) ).
% sympD
tff(fact_5626_sympE,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),B3: A,A4: A] :
( symp(A,R2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),R2,A4),B3) ) ) ).
% sympE
tff(fact_5627_sympI,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,A5),B2)
=> aa(A,$o,aa(A,fun(A,$o),R2,B2),A5) )
=> symp(A,R2) ) ).
% sympI
tff(fact_5628_symp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( symp(A,R2)
<=> ! [X2: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,X2),Y3)
=> aa(A,$o,aa(A,fun(A,$o),R2,Y3),X2) ) ) ).
% symp_def
tff(fact_5629_symp__sup,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( symp(A,R2)
=> ( symp(A,S2)
=> symp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),R2),S2)) ) ) ).
% symp_sup
tff(fact_5630_symp__inf,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( symp(A,R2)
=> ( symp(A,S2)
=> symp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S2)) ) ) ).
% symp_inf
tff(fact_5631_symp__conversep,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( symp(A,R3)
<=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),conversep(A,A,R3)),R3) ) ).
% symp_conversep
tff(fact_5632_symp__sym__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( symp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> sym(A,R2) ) ).
% symp_sym_eq
tff(fact_5633_symp__SUP,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,fun(B,fun(B,$o)))] :
( ! [X3: A] :
( member(A,X3,S)
=> symp(B,aa(A,fun(B,fun(B,$o)),R2,X3)) )
=> symp(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Sup_Sup(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),S))) ) ).
% symp_SUP
tff(fact_5634_compat__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
( bNF_Wellorder_compat(A,B,R2,R5,F2)
<=> ! [A7: A,B7: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),R2)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,A7)),aa(A,B,F2,B7)),R5) ) ) ).
% compat_def
tff(fact_5635_times__integer__code_I6_J,axiom,
! [M: num,N2: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(M)),code_Neg(N2)) = code_Pos(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_integer_code(6)
tff(fact_5636_times__integer__code_I5_J,axiom,
! [M: num,N2: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(M)),code_Pos(N2)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_integer_code(5)
tff(fact_5637_times__integer__code_I4_J,axiom,
! [M: num,N2: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Pos(M)),code_Neg(N2)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_integer_code(4)
tff(fact_5638_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R5: set(product_prod(A,A)),R2: set(product_prod(B,B)),F2: fun(B,A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R5)
=> ( bNF_Wellorder_iso(B,A,R2,R5,F2)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,hilbert_inv_into(B,A,aa(set(product_prod(B,B)),set(B),field2(B),R2),F2),X)),aa(A,B,hilbert_inv_into(B,A,aa(set(product_prod(B,B)),set(B),field2(B),R2),F2),Y)),R2) ) ) ).
% iso_backward
tff(fact_5639_iso__iff2,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
( bNF_Wellorder_iso(A,B,R2,R5,F2)
<=> ( bij_betw(A,B,F2,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(B,B)),set(B),field2(B),R5))
& ! [X2: A] :
( member(A,X2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ! [Xa2: A] :
( member(A,Xa2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R2)
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,X2)),aa(A,B,F2,Xa2)),R5) ) ) ) ) ) ).
% iso_iff2
tff(fact_5640_bot__in__iterates,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F2: fun(A,A)] : member(A,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))),comple6359979572994053840erates(A,F2)) ) ).
% bot_in_iterates
tff(fact_5641_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( bNF_Wellorder_iso(A,B,R2,R5,F2)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F2,X)),aa(A,B,F2,Y)),R5) ) ) ).
% iso_forward
tff(fact_5642_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa: 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),Xa)) = 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,Xa)) ).
% times_natural.rep_eq
tff(fact_5643_sndsp_Ocases,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B),A4: B] :
( basic_sndsp(A,B,P2,A4)
=> ( A4 = aa(product_prod(A,B),B,product_snd(A,B),P2) ) ) ).
% sndsp.cases
tff(fact_5644_sndsp_Osimps,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B),A4: B] :
( basic_sndsp(A,B,P2,A4)
<=> ( A4 = aa(product_prod(A,B),B,product_snd(A,B),P2) ) ) ).
% sndsp.simps
tff(fact_5645_sndsp_Ointros,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B)] : basic_sndsp(A,B,P2,aa(product_prod(A,B),B,product_snd(A,B),P2)) ).
% sndsp.intros
tff(fact_5646_transp__INF,axiom,
! [B: $tType,A: $tType,S: set(A),R2: fun(A,fun(B,fun(B,$o)))] :
( ! [X3: A] :
( member(A,X3,S)
=> transp(B,aa(A,fun(B,fun(B,$o)),R2,X3)) )
=> transp(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Inf_Inf(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),S))) ) ).
% transp_INF
tff(fact_5647_pred__prod__beta,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),Xy: product_prod(A,B)] :
( basic_pred_prod(A,B,P,Q,Xy)
<=> ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),Xy))
& aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),Xy)) ) ) ).
% pred_prod_beta
tff(fact_5648_times__int__code_I3_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),times_times(int),pos(M)),pos(N2)) = pos(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_int_code(3)
tff(fact_5649_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P1: fun(A,$o),P24: fun(B,$o),A4: A,B3: B] :
( basic_pred_prod(A,B,P1,P24,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
<=> ( aa(A,$o,P1,A4)
& aa(B,$o,P24,B3) ) ) ).
% pred_prod_inject
tff(fact_5650_transpD,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A,Y: A,Z2: A] :
( transp(A,R2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),R2,X),Z2) ) ) ) ).
% transpD
tff(fact_5651_transpE,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A,Y: A,Z2: A] :
( transp(A,R2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),R2,X),Z2) ) ) ) ).
% transpE
tff(fact_5652_transpI,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( ! [X3: A,Y2: A,Z4: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y2),Z4)
=> aa(A,$o,aa(A,fun(A,$o),R2,X3),Z4) ) )
=> transp(A,R2) ) ).
% transpI
tff(fact_5653_transp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( transp(A,R2)
<=> ! [X2: A,Y3: A,Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),R2,X2),Y3)
=> ( aa(A,$o,aa(A,fun(A,$o),R2,Y3),Z3)
=> aa(A,$o,aa(A,fun(A,$o),R2,X2),Z3) ) ) ) ).
% transp_def
tff(fact_5654_transp__equality,axiom,
! [A: $tType] : transp(A,fequal(A)) ).
% transp_equality
tff(fact_5655_transp__less,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,ord_less(A)) ) ).
% transp_less
tff(fact_5656_transp__singleton,axiom,
! [A: $tType,A4: A] : transp(A,aTP_Lamp_xn(A,fun(A,fun(A,$o)),A4)) ).
% transp_singleton
tff(fact_5657_transp__empty,axiom,
! [A: $tType] : transp(A,aTP_Lamp_xo(A,fun(A,$o))) ).
% transp_empty
tff(fact_5658_transp__gr,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,aTP_Lamp_tc(A,fun(A,$o))) ) ).
% transp_gr
tff(fact_5659_transp__le,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,ord_less_eq(A)) ) ).
% transp_le
tff(fact_5660_transp__ge,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,aTP_Lamp_xb(A,fun(A,$o))) ) ).
% transp_ge
tff(fact_5661_pred__prod__split,axiom,
! [A: $tType,B: $tType,P: fun($o,$o),Q: fun(A,$o),R3: fun(B,$o),Xy: product_prod(A,B)] :
( aa($o,$o,P,basic_pred_prod(A,B,Q,R3,Xy))
<=> ! [X2: A,Y3: B] :
( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3) )
=> aa($o,$o,P,
( aa(A,$o,Q,X2)
& aa(B,$o,R3,Y3) )) ) ) ).
% pred_prod_split
tff(fact_5662_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P1: fun(A,$o),A4: A,P24: fun(B,$o),B3: B] :
( aa(A,$o,P1,A4)
=> ( aa(B,$o,P24,B3)
=> basic_pred_prod(A,B,P1,P24,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)) ) ) ).
% pred_prod.intros
tff(fact_5663_pred__prod_Osimps,axiom,
! [A: $tType,B: $tType,P1: fun(A,$o),P24: fun(B,$o),A4: product_prod(A,B)] :
( basic_pred_prod(A,B,P1,P24,A4)
<=> ? [A7: A,B7: B] :
( ( A4 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
& aa(A,$o,P1,A7)
& aa(B,$o,P24,B7) ) ) ).
% pred_prod.simps
tff(fact_5664_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P1: fun(A,$o),P24: fun(B,$o),A4: product_prod(A,B)] :
( basic_pred_prod(A,B,P1,P24,A4)
=> ~ ! [A5: A,B2: B] :
( ( A4 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) )
=> ( aa(A,$o,P1,A5)
=> ~ aa(B,$o,P24,B2) ) ) ) ).
% pred_prod.cases
tff(fact_5665_transp__inf,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),S2: fun(A,fun(A,$o))] :
( transp(A,R2)
=> ( transp(A,S2)
=> transp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S2)) ) ) ).
% transp_inf
tff(fact_5666_transp__relcompp,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( transp(A,R2)
<=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),relcompp(A,A,A,R2,R2)),R2) ) ).
% transp_relcompp
tff(fact_5667_transp__relcompp__less__eq,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( transp(A,R2)
=> aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),relcompp(A,A,A,R2,R2)),R2) ) ).
% transp_relcompp_less_eq
tff(fact_5668_transp__trans__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( transp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> trans(A,R2) ) ).
% transp_trans_eq
tff(fact_5669_transp__trans,axiom,
! [A: $tType,R2: fun(A,fun(A,$o))] :
( transp(A,R2)
<=> trans(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).
% transp_trans
tff(fact_5670_times__int__code_I4_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),times_times(int),pos(M)),neg(N2)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_int_code(4)
tff(fact_5671_times__int__code_I5_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),pos(N2)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_int_code(5)
tff(fact_5672_times__int__code_I6_J,axiom,
! [M: num,N2: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),neg(N2)) = pos(aa(num,num,aa(num,fun(num,num),times_times(num),M),N2)) ).
% times_int_code(6)
tff(fact_5673_repeat__mset__def,axiom,
! [A: $tType] : repeat_mset(A) = aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(nat,fun(multiset(A),multiset(A))),map_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(nat),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat)))) ).
% repeat_mset_def
tff(fact_5674_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X) != zero_zero(nat) ).
% sum.size_neq
tff(fact_5675_Nitpick_OEx1__unfold,axiom,
! [A: $tType,P: fun(A,$o)] :
( ? [X2: A] :
( aa(A,$o,P,X2)
& ! [Y3: A] :
( aa(A,$o,P,Y3)
=> ( Y3 = X2 ) ) )
<=> ? [X2: A] : aa(fun(A,$o),set(A),collect(A),P) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).
% Nitpick.Ex1_unfold
tff(fact_5676_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(3)
tff(fact_5677_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(4)
tff(fact_5678_asymp__greater,axiom,
! [A: $tType] :
( preorder(A)
=> asymp(A,aTP_Lamp_tc(A,fun(A,$o))) ) ).
% asymp_greater
tff(fact_5679_Union__plus,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(sum_sum(B,C),set(A)),A3: set(B),B5: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(sum_sum(B,C)),set(set(A)),image2(sum_sum(B,C),set(A),F2),sum_Plus(B,C,A3,B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xp(fun(sum_sum(B,C),set(A)),fun(B,set(A)),F2)),A3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aTP_Lamp_xq(fun(sum_sum(B,C),set(A)),fun(C,set(A)),F2)),B5))) ).
% Union_plus
tff(fact_5680_Union__sum,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(sum_sum(B,C),set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(sum_sum(B,C)),set(set(A)),image2(sum_sum(B,C),set(A),F2),top_top(set(sum_sum(B,C))))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xp(fun(sum_sum(B,C),set(A)),fun(B,set(A)),F2)),top_top(set(B))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aTP_Lamp_xq(fun(sum_sum(B,C),set(A)),fun(C,set(A)),F2)),top_top(set(C))))) ).
% Union_sum
tff(fact_5681_prod_OPlus,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [A3: set(A),B5: set(B),G: fun(sum_sum(A,B),C)] :
( aa(set(A),$o,finite_finite2(A),A3)
=> ( aa(set(B),$o,finite_finite2(B),B5)
=> ( aa(set(sum_sum(A,B)),C,aa(fun(sum_sum(A,B),C),fun(set(sum_sum(A,B)),C),groups7121269368397514597t_prod(sum_sum(A,B),C),G),sum_Plus(A,B,A3,B5)) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,sum_sum(A,B)),fun(A,C),comp(sum_sum(A,B),C,A,G),sum_Inl(A,B))),A3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(B,sum_sum(A,B)),fun(B,C),comp(sum_sum(A,B),C,B,G),sum_Inr(B,A))),B5)) ) ) ) ) ).
% prod.Plus
tff(fact_5682_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(C,B)] : aa(fun(A,sum_sum(A,C)),fun(A,B),comp(sum_sum(A,C),B,A,sum_case_sum(A,B,C,F2,G)),sum_Inl(A,C)) = F2 ).
% case_sum_o_inj(1)
tff(fact_5683_case__sum__o__inj_I2_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(C,B),G: fun(A,B)] : aa(fun(A,sum_sum(C,A)),fun(A,B),comp(sum_sum(C,A),B,A,sum_case_sum(C,B,A,F2,G)),sum_Inr(A,C)) = G ).
% case_sum_o_inj(2)
tff(fact_5684_asympD,axiom,
! [A: $tType,R3: fun(A,fun(A,$o)),X: A,Y: A] :
( asymp(A,R3)
=> ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),R3,Y),X) ) ) ).
% asympD
tff(fact_5685_asymp_Ointros,axiom,
! [A: $tType,R3: fun(A,fun(A,$o))] :
( ! [A5: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),R3,A5),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),R3,B2),A5) )
=> asymp(A,R3) ) ).
% asymp.intros
tff(fact_5686_asymp_Osimps,axiom,
! [A: $tType,A4: fun(A,fun(A,$o))] :
( asymp(A,A4)
<=> ? [R7: fun(A,fun(A,$o))] :
( ( A4 = R7 )
& ! [X2: A,Xa2: A] :
( aa(A,$o,aa(A,fun(A,$o),R7,X2),Xa2)
=> ~ aa(A,$o,aa(A,fun(A,$o),R7,Xa2),X2) ) ) ) ).
% asymp.simps
tff(fact_5687_asymp_Ocases,axiom,
! [A: $tType,A4: fun(A,fun(A,$o))] :
( asymp(A,A4)
=> ! [A12: A,B13: A] :
( aa(A,$o,aa(A,fun(A,$o),A4,A12),B13)
=> ~ aa(A,$o,aa(A,fun(A,$o),A4,B13),A12) ) ) ).
% asymp.cases
tff(fact_5688_Basic__BNF__LFPs_OInl__def__alt,axiom,
! [B: $tType,A: $tType,X4: A] : aa(A,sum_sum(A,B),sum_Inl(A,B),X4) = aa(sum_sum(A,B),sum_sum(A,B),basic_BNF_xtor(sum_sum(A,B)),aa(sum_sum(A,B),sum_sum(A,B),bNF_id_bnf(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X4))) ).
% Basic_BNF_LFPs.Inl_def_alt
tff(fact_5689_Basic__BNF__LFPs_OInr__def__alt,axiom,
! [B: $tType,A: $tType,X4: A] : aa(A,sum_sum(B,A),sum_Inr(A,B),X4) = aa(sum_sum(B,A),sum_sum(B,A),basic_BNF_xtor(sum_sum(B,A)),aa(sum_sum(B,A),sum_sum(B,A),bNF_id_bnf(sum_sum(B,A)),aa(A,sum_sum(B,A),sum_Inr(A,B),X4))) ).
% Basic_BNF_LFPs.Inr_def_alt
tff(fact_5690_asymp__less,axiom,
! [A: $tType] :
( preorder(A)
=> asymp(A,ord_less(A)) ) ).
% asymp_less
tff(fact_5691_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa: fun(A,nat),X: fun(B,nat),X1: A] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa,X),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X1)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(1)
tff(fact_5692_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa: fun(A,nat),X: fun(B,nat),X22: B] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa,X),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(2)
tff(fact_5693_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A13: set(B)] :
( bNF_Ca8970107618336181345der_on(A,top_top(set(A)),R2)
=> ( bNF_Cardinal_cexp(sum_sum(B,product_unit),A,bNF_Cardinal_csum(B,product_unit,bNF_Ca6860139660246222851ard_of(B,A13),bNF_Cardinal_cone),R2) = bNF_Ca6860139660246222851ard_of(fun(A,sum_sum(B,product_unit)),bNF_Wellorder_Func(A,sum_sum(B,product_unit),top_top(set(A)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(set(sum_sum(B,product_unit)),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),sup_sup(set(sum_sum(B,product_unit))),aa(set(B),set(sum_sum(B,product_unit)),image2(B,sum_sum(B,product_unit),sum_Inl(B,product_unit)),A13)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(sum_sum(B,product_unit),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),insert2(sum_sum(B,product_unit)),aa(product_unit,sum_sum(B,product_unit),sum_Inr(product_unit,B),product_Unity)),bot_bot(set(sum_sum(B,product_unit))))))) ) ) ).
% card_order_csum_cone_cexp_def
tff(fact_5694_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(A,sum_sum(A,B),sum_Inl(A,B),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).
% sum_set_simps(1)
tff(fact_5695_sum__set__simps_I4_J,axiom,
! [B: $tType,A: $tType,X: A] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(A,sum_sum(B,A),sum_Inr(A,B),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).
% sum_set_simps(4)
tff(fact_5696_asymp__asym__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( asymp(A,aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),R3))
<=> asym(A,R3) ) ).
% asymp_asym_eq
tff(fact_5697_sum__set__simps_I2_J,axiom,
! [B: $tType,A: $tType,X: B] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(B,sum_sum(A,B),sum_Inr(B,A),X)) = bot_bot(set(A)) ).
% sum_set_simps(2)
tff(fact_5698_sum__set__simps_I3_J,axiom,
! [B: $tType,A: $tType,X: B] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(B,sum_sum(B,A),sum_Inl(B,A),X)) = bot_bot(set(A)) ).
% sum_set_simps(3)
tff(fact_5699_asym__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( asym(A,R3)
<=> ! [X2: A,Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R3) ) ) ).
% asym_iff
tff(fact_5700_asymD,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A] :
( asym(A,R3)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),R3) ) ) ).
% asymD
tff(fact_5701_asym_Ointros,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [A5: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B2),R3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A5),R3) )
=> asym(A,R3) ) ).
% asym.intros
tff(fact_5702_asym_Osimps,axiom,
! [A: $tType,A4: set(product_prod(A,A))] :
( asym(A,A4)
<=> ? [R7: set(product_prod(A,A))] :
( ( A4 = R7 )
& ! [X2: A,Xa2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),R7)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),R7) ) ) ) ).
% asym.simps
tff(fact_5703_asym_Ocases,axiom,
! [A: $tType,A4: set(product_prod(A,A))] :
( asym(A,A4)
=> ! [A12: A,B13: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),B13),A4)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B13),A12),A4) ) ) ).
% asym.cases
tff(fact_5704_asym__inv__image,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),F2: fun(B,A)] :
( asym(A,R3)
=> asym(B,inv_image(A,B,R3,F2)) ) ).
% asym_inv_image
tff(fact_5705_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] : basic_setl(A,B) = sum_case_sum(A,set(A),B,aTP_Lamp_bc(A,set(A)),aTP_Lamp_di(B,set(A))) ).
% sum_set_defs(1)
tff(fact_5706_sum__set__defs_I2_J,axiom,
! [A: $tType,B: $tType] : basic_setr(A,B) = sum_case_sum(A,set(B),B,aTP_Lamp_ax(A,set(B)),aTP_Lamp_xr(B,set(B))) ).
% sum_set_defs(2)
tff(fact_5707_sum_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),nat),comp(sum_sum(C,D),nat,sum_sum(A,B),basic_BNF_size_sum(C,D,F2,Fa)),sum_map_sum(A,C,B,D,G,Ga)) = basic_BNF_size_sum(A,B,aa(fun(A,C),fun(A,nat),comp(C,nat,A,F2),G),aa(fun(B,D),fun(B,nat),comp(D,nat,B,Fa),Ga)) ).
% sum.size_gen_o_map
tff(fact_5708_times__natural_Otransfer,axiom,
aa(fun(code_natural,fun(code_natural,code_natural)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),$o),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),times_times(nat)),times_times(code_natural)) ).
% times_natural.transfer
tff(fact_5709_adm__wf__def,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B))] :
( adm_wf(A,B,R3,F4)
<=> ! [F6: fun(A,B),G4: fun(A,B),X2: A] :
( ! [Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X2),R3)
=> ( aa(A,B,F6,Z3) = aa(A,B,G4,Z3) ) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),F4,F6),X2) = aa(A,B,aa(fun(A,B),fun(A,B),F4,G4),X2) ) ) ) ).
% adm_wf_def
tff(fact_5710_map__sum__Inl__conv,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,Fl: fun(C,A),Fr: fun(D,B),S2: sum_sum(C,D),Y: A] :
( ( aa(sum_sum(C,D),sum_sum(A,B),sum_map_sum(C,A,D,B,Fl,Fr),S2) = aa(A,sum_sum(A,B),sum_Inl(A,B),Y) )
<=> ? [X2: C] :
( ( S2 = aa(C,sum_sum(C,D),sum_Inl(C,D),X2) )
& ( Y = aa(C,A,Fl,X2) ) ) ) ).
% map_sum_Inl_conv
tff(fact_5711_map__sum__Inr__conv,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Fl: fun(C,A),Fr: fun(D,B),S2: sum_sum(C,D),Y: B] :
( ( aa(sum_sum(C,D),sum_sum(A,B),sum_map_sum(C,A,D,B,Fl,Fr),S2) = aa(B,sum_sum(A,B),sum_Inr(B,A),Y) )
<=> ? [X2: D] :
( ( S2 = aa(D,sum_sum(C,D),sum_Inr(D,C),X2) )
& ( Y = aa(D,B,Fr,X2) ) ) ) ).
% map_sum_Inr_conv
tff(fact_5712_map__sum__o__inj_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(D,C)] : aa(fun(A,sum_sum(A,D)),fun(A,sum_sum(B,C)),comp(sum_sum(A,D),sum_sum(B,C),A,sum_map_sum(A,B,D,C,F2,G)),sum_Inl(A,D)) = aa(fun(A,B),fun(A,sum_sum(B,C)),comp(B,sum_sum(B,C),A,sum_Inl(B,C)),F2) ).
% map_sum_o_inj(1)
tff(fact_5713_map__sum__o__inj_I2_J,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F2: fun(D,B),G: fun(A,C)] : aa(fun(A,sum_sum(D,A)),fun(A,sum_sum(B,C)),comp(sum_sum(D,A),sum_sum(B,C),A,sum_map_sum(D,B,A,C,F2,G)),sum_Inr(A,D)) = aa(fun(A,C),fun(A,sum_sum(B,C)),comp(C,sum_sum(B,C),A,sum_Inr(C,B)),G) ).
% map_sum_o_inj(2)
tff(fact_5714_sum_Orel__compp__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(C,$o)),R23: fun(B,fun(D,$o))] : bNF_rel_sum(A,C,B,D,R12,R23) = relcompp(sum_sum(A,B),sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),conversep(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_xs(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),R12),R23)),sum_map_sum(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)))),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_xs(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),R12),R23)),sum_map_sum(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).
% sum.rel_compp_Grp
tff(fact_5715_sum_Oin__rel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A4: sum_sum(A,C),B3: sum_sum(B,D)] :
( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,R12,R23),A4),B3)
<=> ? [Z3: sum_sum(product_prod(A,B),product_prod(C,D))] :
( member(sum_sum(product_prod(A,B),product_prod(C,D)),Z3,aa(fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),set(sum_sum(product_prod(A,B),product_prod(C,D))),collect(sum_sum(product_prod(A,B),product_prod(C,D))),aa(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_xt(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o)),R12),R23)))
& ( aa(sum_sum(product_prod(A,B),product_prod(C,D)),sum_sum(A,C),sum_map_sum(product_prod(A,B),A,product_prod(C,D),C,product_fst(A,B),product_fst(C,D)),Z3) = A4 )
& ( aa(sum_sum(product_prod(A,B),product_prod(C,D)),sum_sum(B,D),sum_map_sum(product_prod(A,B),B,product_prod(C,D),D,product_snd(A,B),product_snd(C,D)),Z3) = B3 ) ) ) ).
% sum.in_rel
tff(fact_5716_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: set(B),T3: fun(B,fun(A,$o))] :
( type_definition(A,B,Rep,Abs,S)
=> ( ! [X3: B,Xa4: A] :
( aa(A,$o,aa(B,fun(A,$o),T3,X3),Xa4)
<=> ( X3 = aa(A,B,Rep,Xa4) ) )
=> quotient(B,A,bNF_eq_onp(B,aTP_Lamp_nz(set(B),fun(B,$o),S)),Abs,Rep,T3) ) ) ).
% typedef_to_Quotient
tff(fact_5717_rel__sum_Ointros_I1_J,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(B,$o)),A4: A,C2: B,R23: fun(C,fun(D,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R12,A4),C2)
=> aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,R12,R23),aa(A,sum_sum(A,C),sum_Inl(A,C),A4)),aa(B,sum_sum(B,D),sum_Inl(B,D),C2)) ) ).
% rel_sum.intros(1)
tff(fact_5718_rel__sum_Ointros_I2_J,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R23: fun(A,fun(B,$o)),B3: A,D3: B,R12: fun(C,fun(D,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R23,B3),D3)
=> aa(sum_sum(D,B),$o,aa(sum_sum(C,A),fun(sum_sum(D,B),$o),bNF_rel_sum(C,D,A,B,R12,R23),aa(A,sum_sum(C,A),sum_Inr(A,C),B3)),aa(B,sum_sum(D,B),sum_Inr(B,D),D3)) ) ).
% rel_sum.intros(2)
tff(fact_5719_Quotient__filter,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(A,$o)),Abs: fun(A,B),Rep: fun(B,A),T3: fun(A,fun(B,$o))] :
( quotient(A,B,R3,Abs,Rep,T3)
=> quotient(filter(A),filter(B),rel_filter(A,A,R3),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),Abs),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Rep),rel_filter(A,B,T3)) ) ).
% Quotient_filter
tff(fact_5720_rel__sum_Osimps,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: sum_sum(A,C),A22: sum_sum(B,D)] :
( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,R12,R23),A1),A22)
<=> ( ? [A7: A,C5: B] :
( ( A1 = aa(A,sum_sum(A,C),sum_Inl(A,C),A7) )
& ( A22 = aa(B,sum_sum(B,D),sum_Inl(B,D),C5) )
& aa(B,$o,aa(A,fun(B,$o),R12,A7),C5) )
| ? [B7: C,D5: D] :
( ( A1 = aa(C,sum_sum(A,C),sum_Inr(C,A),B7) )
& ( A22 = aa(D,sum_sum(B,D),sum_Inr(D,B),D5) )
& aa(D,$o,aa(C,fun(D,$o),R23,B7),D5) ) ) ) ).
% rel_sum.simps
tff(fact_5721_rel__sum_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: sum_sum(A,C),A22: sum_sum(B,D)] :
( aa(sum_sum(B,D),$o,aa(sum_sum(A,C),fun(sum_sum(B,D),$o),bNF_rel_sum(A,B,C,D,R12,R23),A1),A22)
=> ( ! [A5: A] :
( ( A1 = aa(A,sum_sum(A,C),sum_Inl(A,C),A5) )
=> ! [C3: B] :
( ( A22 = aa(B,sum_sum(B,D),sum_Inl(B,D),C3) )
=> ~ aa(B,$o,aa(A,fun(B,$o),R12,A5),C3) ) )
=> ~ ! [B2: C] :
( ( A1 = aa(C,sum_sum(A,C),sum_Inr(C,A),B2) )
=> ! [D2: D] :
( ( A22 = aa(D,sum_sum(B,D),sum_Inr(D,B),D2) )
=> ~ aa(D,$o,aa(C,fun(D,$o),R23,B2),D2) ) ) ) ) ).
% rel_sum.cases
tff(fact_5722_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),T3: fun(B,fun(A,$o))] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ! [X3: B,Xa4: A] :
( aa(A,$o,aa(B,fun(A,$o),T3,X3),Xa4)
<=> ( X3 = aa(A,B,Rep,Xa4) ) )
=> quotient(B,A,fequal(B),Abs,Rep,T3) ) ) ).
% UNIV_typedef_to_Quotient
tff(fact_5723_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,$o),T3: fun(B,fun(A,$o))] :
( type_definition(A,B,Rep,Abs,aa(fun(B,$o),set(B),collect(B),P))
=> ( ! [X3: B,Xa4: A] :
( aa(A,$o,aa(B,fun(A,$o),T3,X3),Xa4)
<=> ( X3 = aa(A,B,Rep,Xa4) ) )
=> quotient(B,A,bNF_eq_onp(B,P),Abs,Rep,T3) ) ) ).
% open_typedef_to_Quotient
tff(fact_5724_Quotient__eq__onp__typedef,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,$o))] :
( quotient(A,B,bNF_eq_onp(A,P),Abs,Rep,Cr)
=> type_definition(B,A,Rep,Abs,aa(fun(A,$o),set(A),collect(A),P)) ) ).
% Quotient_eq_onp_typedef
tff(fact_5725_Quotient__eq__onp__type__copy,axiom,
! [B: $tType,A: $tType,Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,$o))] :
( quotient(A,B,fequal(A),Abs,Rep,Cr)
=> type_definition(B,A,Rep,Abs,top_top(set(A))) ) ).
% Quotient_eq_onp_type_copy
tff(fact_5726_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F2,Z2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic5214292709420241887eutr_F(A,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic5214292709420241887eutr_F(A,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% semilattice_neutr_set.insert_remove
tff(fact_5727_semilattice__neutr__set_Oinsert,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F2,Z2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic5214292709420241887eutr_F(A,F2,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic5214292709420241887eutr_F(A,F2,Z2,A3)) ) ) ) ).
% semilattice_neutr_set.insert
tff(fact_5728_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(A)] :
( lattic5652469242046573047tr_set(A,F2,Z2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),F2,X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic5214292709420241887eutr_F(A,F2,Z2,A3),A3) ) ) ) ) ).
% semilattice_neutr_set.closed
tff(fact_5729_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F2,Z2)
=> ( lattic5214292709420241887eutr_F(A,F2,Z2,bot_bot(set(A))) = Z2 ) ) ).
% semilattice_neutr_set.empty
tff(fact_5730_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F2,Z2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic5214292709420241887eutr_F(A,F2,Z2,A3) = aa(A,A,aa(A,fun(A,A),F2,X),lattic5214292709420241887eutr_F(A,F2,Z2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% semilattice_neutr_set.remove
tff(fact_5731_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P2: fun(B,$o),F2: fun(C,B)] :
( nO_MATCH(fun(A,A),fun(B,$o),id(A),P2)
=> ( basic_pred_fun(C,B,aTP_Lamp_xu(C,$o),P2,F2)
<=> basic_pred_fun(C,$o,aTP_Lamp_xu(C,$o),id($o),aa(fun(C,B),fun(C,$o),comp(B,$o,C,P2),F2)) ) ) ).
% pred_fun_True_id
tff(fact_5732_Fpow__not__empty,axiom,
! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) != bot_bot(set(set(A))) ).
% Fpow_not_empty
tff(fact_5733_times__natural_Oabs__eq,axiom,
! [Xa: 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,Xa)),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),Xa),X)) ).
% times_natural.abs_eq
tff(fact_5734_empty__in__Fpow,axiom,
! [A: $tType,A3: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,A3)) ).
% empty_in_Fpow
tff(fact_5735_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_5736_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_5737_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N2: multiset(A),M: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),N2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2))
<=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2) = N2 ) ) ).
% subset_mset.min_arg_le(1)
tff(fact_5738_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset(A),N2: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2))
<=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N2) = M ) ) ).
% subset_mset.min_arg_le(2)
tff(fact_5739_ord_Omin__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),A4: A,B3: A] :
aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A4),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3),A4,B3) ).
% ord.min_def
tff(fact_5740_ord_Omin_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] : min(A,Less_eq) = min(A,Less_eq) ).
% ord.min.cong
tff(fact_5741_prop__match,axiom,
! [A: $tType,P: fun(A,$o),Al: list(A),E4: A,E5: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( list_all(A,P,Al)
=> ( ~ aa(A,$o,P,E4)
=> ( ~ aa(A,$o,P,E5)
=> ( list_all(A,P,Bl)
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),Bl2)) )
<=> ( ( Al = Al2 )
& ( E4 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
tff(fact_5742_prop__matchD,axiom,
! [A: $tType,Al: list(A),E4: A,Bl: list(A),Al2: list(A),E5: A,Bl2: list(A),P: fun(A,$o)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E4),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),Bl2)) )
=> ( list_all(A,P,Al)
=> ( ~ aa(A,$o,P,E4)
=> ( ~ aa(A,$o,P,E5)
=> ( list_all(A,P,Bl)
=> ( ( Al = Al2 )
& ( E4 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
tff(fact_5743_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_5744_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,F2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ).
% semilattice_set.insert_remove
tff(fact_5745_semilattice__set_Oremove,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( member(A,X,A3)
=> ( lattic1715443433743089157tice_F(A,F2,A3) = $ite(minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,F2,minus_minus(set(A),A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ) ) ).
% semilattice_set.remove
tff(fact_5746_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_5747_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,F2,X,A3) ) ) ) ).
% semilattice_set.eq_fold
tff(fact_5748_semilattice__set_Osingleton,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ) ).
% semilattice_set.singleton
tff(fact_5749_semilattice__set_Oclosed,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A)] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),F2,X3),Y2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))
=> member(A,lattic1715443433743089157tice_F(A,F2,A3),A3) ) ) ) ) ).
% semilattice_set.closed
tff(fact_5750_semilattice__set_Oinsert,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,F2,A3)) ) ) ) ) ).
% semilattice_set.insert
tff(fact_5751_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),X: A] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ~ member(A,X,A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(A,A,aa(A,fun(A,A),F2,X),lattic1715443433743089157tice_F(A,F2,A3)) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
tff(fact_5752_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Ha: fun(A,A),N: set(A)] :
( lattic149705377957585745ce_set(A,F2)
=> ( ! [X3: A,Y2: A] : aa(A,A,Ha,aa(A,A,aa(A,fun(A,A),F2,X3),Y2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Ha,X3)),aa(A,A,Ha,Y2))
=> ( aa(set(A),$o,finite_finite2(A),N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,Ha,lattic1715443433743089157tice_F(A,F2,N)) = lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),image2(A,A,Ha),N)) ) ) ) ) ) ).
% semilattice_set.hom_commute
tff(fact_5753_semilattice__set_Osubset,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),B5: set(A)] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
=> ( aa(A,A,aa(A,fun(A,A),F2,lattic1715443433743089157tice_F(A,F2,B5)),lattic1715443433743089157tice_F(A,F2,A3)) = lattic1715443433743089157tice_F(A,F2,A3) ) ) ) ) ) ).
% semilattice_set.subset
tff(fact_5754_semilattice__set_Ounion,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: set(A),B5: set(A)] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> ( ( B5 != bot_bot(set(A)) )
=> ( lattic1715443433743089157tice_F(A,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B5)) = aa(A,A,aa(A,fun(A,A),F2,lattic1715443433743089157tice_F(A,F2,A3)),lattic1715443433743089157tice_F(A,F2,B5)) ) ) ) ) ) ) ).
% semilattice_set.union
tff(fact_5755_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: set(A),B5: set(A)] :
( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite2(A),B5)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,lattic1715443433743089157tice_F(A,F2,B5)),lattic1715443433743089157tice_F(A,F2,A3)) ) ) ) ) ).
% semilattice_order_set.subset_imp
tff(fact_5756_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: set(A),X: A] :
( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),lattic1715443433743089157tice_F(A,F2,A3))
=> ! [A12: A] :
( member(A,A12,A3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A12) ) ) ) ) ) ).
% semilattice_order_set.boundedE
tff(fact_5757_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: set(A),X: A] :
( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( member(A,A5,A3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A5) )
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),lattic1715443433743089157tice_F(A,F2,A3)) ) ) ) ) ).
% semilattice_order_set.boundedI
tff(fact_5758_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: set(A),X: A] :
( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite2(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),lattic1715443433743089157tice_F(A,F2,A3))
<=> ! [X2: A] :
( member(A,X2,A3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),X2) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
tff(fact_5759_ATP_Olambda__1,axiom,
! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_aa(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(set(A),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).
% ATP.lambda_1
tff(fact_5760_ATP_Olambda__2,axiom,
! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_ga(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).
% ATP.lambda_2
tff(fact_5761_ATP_Olambda__3,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aTP_Lamp_dq(product_prod(A,A),$o),Uu)
<=> ( aa(product_prod(A,A),A,product_fst(A,A),Uu) = aa(product_prod(A,A),A,product_snd(A,A),Uu) ) ) ).
% ATP.lambda_3
tff(fact_5762_ATP_Olambda__4,axiom,
! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_ey(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_4
tff(fact_5763_ATP_Olambda__5,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_ex(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).
% ATP.lambda_5
tff(fact_5764_ATP_Olambda__6,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_id(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_6
tff(fact_5765_ATP_Olambda__7,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( aa(A,$o,aTP_Lamp_tj(A,$o),Uu)
<=> dvd_dvd(A,Uu,one_one(A)) ) ) ).
% ATP.lambda_7
tff(fact_5766_ATP_Olambda__8,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,list(A),aTP_Lamp_vr(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).
% ATP.lambda_8
tff(fact_5767_ATP_Olambda__9,axiom,
! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_uk(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_9
tff(fact_5768_ATP_Olambda__10,axiom,
! [B: $tType,Uu: B] : aa(B,set(B),aTP_Lamp_xr(B,set(B)),Uu) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),bot_bot(set(B))) ).
% ATP.lambda_10
tff(fact_5769_ATP_Olambda__11,axiom,
! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_bc(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).
% ATP.lambda_11
tff(fact_5770_ATP_Olambda__12,axiom,
! [Uu: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aTP_Lamp_ky(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_12
tff(fact_5771_ATP_Olambda__13,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_xi(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_xh(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_13
tff(fact_5772_ATP_Olambda__14,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_tv(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)) ).
% ATP.lambda_14
tff(fact_5773_ATP_Olambda__15,axiom,
! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_fl(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).
% ATP.lambda_15
tff(fact_5774_ATP_Olambda__16,axiom,
! [A: $tType,Uu: list(A)] :
( aa(list(A),$o,aTP_Lamp_wa(list(A),$o),Uu)
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_16
tff(fact_5775_ATP_Olambda__17,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_vp(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_vo(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_17
tff(fact_5776_ATP_Olambda__18,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_oq(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_op(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).
% ATP.lambda_18
tff(fact_5777_ATP_Olambda__19,axiom,
! [A: $tType,Uu: multiset(A)] : aa(multiset(A),set(A),aTP_Lamp_rr(multiset(A),set(A)),Uu) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_rq(multiset(A),fun(A,$o),Uu)) ).
% ATP.lambda_19
tff(fact_5778_ATP_Olambda__20,axiom,
! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_gh(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_gg(nat,fun(nat,$o),Uu)) ).
% ATP.lambda_20
tff(fact_5779_ATP_Olambda__21,axiom,
! [A: $tType,Uu: fun(A,nat)] :
( aa(fun(A,nat),$o,aTP_Lamp_sc(fun(A,nat),$o),Uu)
<=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_sb(fun(A,nat),fun(A,$o),Uu))) ) ).
% ATP.lambda_21
tff(fact_5780_ATP_Olambda__22,axiom,
! [A: $tType,Uu: fun(A,$o)] :
( aa(fun(A,$o),$o,aTP_Lamp_th(fun(A,$o),$o),Uu)
<=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bx(fun(A,$o),fun(A,$o),Uu))) ) ).
% ATP.lambda_22
tff(fact_5781_ATP_Olambda__23,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_oc(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),set_ord_atLeast(A,Uu)) ) ).
% ATP.lambda_23
tff(fact_5782_ATP_Olambda__24,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ob(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),set_ord_atLeast(A,Uu)) ) ).
% ATP.lambda_24
tff(fact_5783_ATP_Olambda__25,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_lq(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ).
% ATP.lambda_25
tff(fact_5784_ATP_Olambda__26,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_lp(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ).
% ATP.lambda_26
tff(fact_5785_ATP_Olambda__27,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aTP_Lamp_ml(product_prod(A,A),$o),Uu)
<=> ? [X2: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2) ) ).
% ATP.lambda_27
tff(fact_5786_ATP_Olambda__28,axiom,
! [A: $tType,B: $tType,Uu: product_prod(product_prod($o,A),product_prod($o,B))] :
( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_mv(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu)
<=> ? [X2: A,Y3: B] : Uu = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$true),X2)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) ) ).
% ATP.lambda_28
tff(fact_5787_ATP_Olambda__29,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] :
aa(nat,A,aTP_Lamp_gx(nat,fun(nat,A),Uu),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_29
tff(fact_5788_ATP_Olambda__30,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] :
aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_vv(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).
% ATP.lambda_30
tff(fact_5789_ATP_Olambda__31,axiom,
! [A: $tType,Uu: list(A),Uua: nat] :
aa(nat,option(A),aTP_Lamp_um(list(A),fun(nat,option(A)),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).
% ATP.lambda_31
tff(fact_5790_ATP_Olambda__32,axiom,
! [Uu: int,Uua: int] :
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rm(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)),abs_abs(int,Uu))) ).
% ATP.lambda_32
tff(fact_5791_ATP_Olambda__33,axiom,
! [A: $tType,Uu: set(fun(A,nat)),Uua: A] :
aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_sd(set(fun(A,nat)),fun(A,nat)),Uu),Uua) = $ite(Uu = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_ru(A,fun(fun(A,nat),nat),Uua)),Uu))) ).
% ATP.lambda_33
tff(fact_5792_ATP_Olambda__34,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] :
aa(nat,A,aTP_Lamp_gy(nat,fun(nat,A),Uu),Uua) = $ite(~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_34
tff(fact_5793_ATP_Olambda__35,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_sz(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),bot_bot(A)) ) ).
% ATP.lambda_35
tff(fact_5794_ATP_Olambda__36,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_le(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
<=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).
% ATP.lambda_36
tff(fact_5795_ATP_Olambda__37,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_kv(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_37
tff(fact_5796_ATP_Olambda__38,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hw(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_38
tff(fact_5797_ATP_Olambda__39,axiom,
! [Uu: code_integer,Uua: code_integer] :
aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_kx(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
l2: int,
l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),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_39
tff(fact_5798_ATP_Olambda__40,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_kw(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_40
tff(fact_5799_ATP_Olambda__41,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( aa(list(A),$o,aTP_Lamp_vf(set(A),fun(list(A),$o),Uu),Uua)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)
& distinct(A,Uua) ) ) ).
% ATP.lambda_41
tff(fact_5800_ATP_Olambda__42,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( aa(list(A),$o,aTP_Lamp_ve(set(A),fun(list(A),$o),Uu),Uua)
<=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
& distinct(A,Uua) ) ) ).
% ATP.lambda_42
tff(fact_5801_ATP_Olambda__43,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ia(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_43
tff(fact_5802_ATP_Olambda__44,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_fz(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),image(A,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).
% ATP.lambda_44
tff(fact_5803_ATP_Olambda__45,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hb(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).
% ATP.lambda_45
tff(fact_5804_ATP_Olambda__46,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ic(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),minus_minus(A,divide_divide(A,Uu,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_46
tff(fact_5805_ATP_Olambda__47,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hm(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).
% ATP.lambda_47
tff(fact_5806_ATP_Olambda__48,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
( aa(set(set(A)),$o,aTP_Lamp_ab(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
<=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
& ( Uua != bot_bot(set(set(A))) ) ) ) ).
% ATP.lambda_48
tff(fact_5807_ATP_Olambda__49,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% ATP.lambda_49
tff(fact_5808_ATP_Olambda__50,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_qv(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua)
<=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Uu),Uua)
& ! [A7: A,B7: A,C5: A] :
( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),Uua)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5),Uu) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),Uu) ) ) ) ).
% ATP.lambda_50
tff(fact_5809_ATP_Olambda__51,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_gk(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).
% ATP.lambda_51
tff(fact_5810_ATP_Olambda__52,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_fx(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).
% ATP.lambda_52
tff(fact_5811_ATP_Olambda__53,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_ti(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),Uua),bot_bot(set(product_prod(A,B))))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).
% ATP.lambda_53
tff(fact_5812_ATP_Olambda__54,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(A,product_prod(B,C))] :
( aa(fun(A,product_prod(B,C)),$o,aTP_Lamp_kt(fun(B,fun(C,$o)),fun(fun(A,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)),$o),ord_less_eq(set(product_prod(B,C))),aa(set(A),set(product_prod(B,C)),image2(A,product_prod(B,C),Uua),top_top(set(A)))),aa(fun(product_prod(B,C),$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_54
tff(fact_5813_ATP_Olambda__55,axiom,
! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_ng(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
<=> ( ( Uua = bot_bot(set(A)) )
| ? [A11: set(A),A7: A] :
( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),A11) )
& aa(set(A),$o,Uu,A11) ) ) ) ).
% ATP.lambda_55
tff(fact_5814_ATP_Olambda__56,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ek(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))),Uu) ).
% ATP.lambda_56
tff(fact_5815_ATP_Olambda__57,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_gq(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(A,B)),fun(C,B),comp(product_prod(A,B),B,C,product_snd(A,B)),Uu)),top_top(set(C))) ).
% ATP.lambda_57
tff(fact_5816_ATP_Olambda__58,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).
% ATP.lambda_58
tff(fact_5817_ATP_Olambda__59,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).
% ATP.lambda_59
tff(fact_5818_ATP_Olambda__60,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).
% ATP.lambda_60
tff(fact_5819_ATP_Olambda__61,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_qm(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ( aa(A,$o,Uu,Uua)
& ! [Y3: A] :
( aa(A,$o,Uu,Y3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3) ) ) ) ) ).
% ATP.lambda_61
tff(fact_5820_ATP_Olambda__62,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_qs(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ( aa(A,$o,Uu,Uua)
& ! [Y3: A] :
( aa(A,$o,Uu,Y3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).
% ATP.lambda_62
tff(fact_5821_ATP_Olambda__63,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_dc(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).
% ATP.lambda_63
tff(fact_5822_ATP_Olambda__64,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_ez(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).
% ATP.lambda_64
tff(fact_5823_ATP_Olambda__65,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: A] :
( aa(A,$o,aTP_Lamp_ei(fun(A,B),fun(A,$o),Uu),Uua)
<=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).
% ATP.lambda_65
tff(fact_5824_ATP_Olambda__66,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_lt(set(A),fun(set(A),$o),Uu),Uua)
<=> ( aa(set(A),$o,finite_finite2(A),Uua)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).
% ATP.lambda_66
tff(fact_5825_ATP_Olambda__67,axiom,
! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_fo(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
& ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uua) ) ) ).
% ATP.lambda_67
tff(fact_5826_ATP_Olambda__68,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_am(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
& ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uua) ) ) ).
% ATP.lambda_68
tff(fact_5827_ATP_Olambda__69,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: option(product_prod(A,B))] :
( aa(option(product_prod(A,B)),$o,aTP_Lamp_wr(fun(A,fun(B,$o)),fun(option(product_prod(A,B)),$o),Uu),Uua)
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),set_option(product_prod(A,B),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).
% ATP.lambda_69
tff(fact_5828_ATP_Olambda__70,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: multiset(product_prod(A,B))] :
( aa(multiset(product_prod(A,B)),$o,aTP_Lamp_si(fun(A,fun(B,$o)),fun(multiset(product_prod(A,B)),$o),Uu),Uua)
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).
% ATP.lambda_70
tff(fact_5829_ATP_Olambda__71,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
( aa(list(product_prod(A,B)),$o,aTP_Lamp_wj(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),Uu),Uua)
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).
% ATP.lambda_71
tff(fact_5830_ATP_Olambda__72,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_vi(product_prod(A,B),fun(product_prod(A,B),$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu)),aa(product_prod(A,B),A,product_fst(A,B),Uua)) ) ) ).
% ATP.lambda_72
tff(fact_5831_ATP_Olambda__73,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_vu(product_prod(A,B),fun(product_prod(A,B),$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu)) ) ) ).
% ATP.lambda_73
tff(fact_5832_ATP_Olambda__74,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hu(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_74
tff(fact_5833_ATP_Olambda__75,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_vx(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).
% ATP.lambda_75
tff(fact_5834_ATP_Olambda__76,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( aa(list(A),$o,aTP_Lamp_ut(nat,fun(list(A),$o),Uu),Uua)
<=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).
% ATP.lambda_76
tff(fact_5835_ATP_Olambda__77,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(set(product_prod(A,A))),aTP_Lamp_xm(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),Uu),Uua) = image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert2(set(product_prod(A,A))),Uu),bot_bot(set(set(product_prod(A,A)))))) ).
% ATP.lambda_77
tff(fact_5836_ATP_Olambda__78,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ie(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).
% ATP.lambda_78
tff(fact_5837_ATP_Olambda__79,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ig(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).
% ATP.lambda_79
tff(fact_5838_ATP_Olambda__80,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gu(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% ATP.lambda_80
tff(fact_5839_ATP_Olambda__81,axiom,
! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_gr(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu)),Uua)) ).
% ATP.lambda_81
tff(fact_5840_ATP_Olambda__82,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hy(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_82
tff(fact_5841_ATP_Olambda__83,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_fe(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = image(B,A,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).
% ATP.lambda_83
tff(fact_5842_ATP_Olambda__84,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_fg(fun(A,B),fun(B,set(A)),Uu),Uua) = vimage(A,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).
% ATP.lambda_84
tff(fact_5843_ATP_Olambda__85,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_lf(set(A),fun(A,$o),Uu),Uua)
<=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).
% ATP.lambda_85
tff(fact_5844_ATP_Olambda__86,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_db(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).
% ATP.lambda_86
tff(fact_5845_ATP_Olambda__87,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ug(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).
% ATP.lambda_87
tff(fact_5846_ATP_Olambda__88,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kg(A,fun(nat,A),Uu),Uua) = minus_minus(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_88
tff(fact_5847_ATP_Olambda__89,axiom,
! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_wb(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua) = map_add(A,B,Uua,map_of(A,B,Uu)) ).
% ATP.lambda_89
tff(fact_5848_ATP_Olambda__90,axiom,
! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_vg(list(set(A)),fun(set(A),$o),Uu),Uua)
<=> member(set(A),Uua,aa(list(set(A)),set(set(A)),set2(set(A)),Uu)) ) ).
% ATP.lambda_90
tff(fact_5849_ATP_Olambda__91,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( aa(A,$o,aTP_Lamp_wf(list(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).
% ATP.lambda_91
tff(fact_5850_ATP_Olambda__92,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( aa(list(A),$o,aTP_Lamp_us(nat,fun(list(A),$o),Uu),Uua)
<=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_92
tff(fact_5851_ATP_Olambda__93,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_bu(set(A),fun(set(A),$o),Uu),Uua)
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).
% ATP.lambda_93
tff(fact_5852_ATP_Olambda__94,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kr(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_94
tff(fact_5853_ATP_Olambda__95,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_eo(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_95
tff(fact_5854_ATP_Olambda__96,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xb(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_96
tff(fact_5855_ATP_Olambda__97,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_nr(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_97
tff(fact_5856_ATP_Olambda__98,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ws(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_98
tff(fact_5857_ATP_Olambda__99,axiom,
! [Uu: nat,Uua: nat] :
( aa(nat,$o,aTP_Lamp_dm(nat,fun(nat,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).
% ATP.lambda_99
tff(fact_5858_ATP_Olambda__100,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ks(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_100
tff(fact_5859_ATP_Olambda__101,axiom,
! [A: $tType] :
( unboun7993243217541854897norder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_ns(A,fun(A,$o),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_101
tff(fact_5860_ATP_Olambda__102,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ep(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_102
tff(fact_5861_ATP_Olambda__103,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tc(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_103
tff(fact_5862_ATP_Olambda__104,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wt(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_104
tff(fact_5863_ATP_Olambda__105,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_hf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_105
tff(fact_5864_ATP_Olambda__106,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kd(nat,fun(nat,nat),Uu),Uua) = minus_minus(nat,Uua,Uu) ).
% ATP.lambda_106
tff(fact_5865_ATP_Olambda__107,axiom,
! [A: $tType,Uu: A,Uua: multiset(A)] : aa(multiset(A),nat,aTP_Lamp_rs(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ).
% ATP.lambda_107
tff(fact_5866_ATP_Olambda__108,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_ew(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).
% ATP.lambda_108
tff(fact_5867_ATP_Olambda__109,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_km(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).
% ATP.lambda_109
tff(fact_5868_ATP_Olambda__110,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wm(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).
% ATP.lambda_110
tff(fact_5869_ATP_Olambda__111,axiom,
! [Uu: nat,Uua: nat] :
( aa(nat,$o,aTP_Lamp_gg(nat,fun(nat,$o),Uu),Uua)
<=> dvd_dvd(nat,Uua,Uu) ) ).
% ATP.lambda_111
tff(fact_5870_ATP_Olambda__112,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_dt(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).
% ATP.lambda_112
tff(fact_5871_ATP_Olambda__113,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_it(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).
% ATP.lambda_113
tff(fact_5872_ATP_Olambda__114,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_is(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).
% ATP.lambda_114
tff(fact_5873_ATP_Olambda__115,axiom,
! [B: $tType,Uu: set(B),Uua: B] :
( aa(B,$o,aTP_Lamp_nz(set(B),fun(B,$o),Uu),Uua)
<=> member(B,Uua,Uu) ) ).
% ATP.lambda_115
tff(fact_5874_ATP_Olambda__116,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_kq(set(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,Uu) ) ) ).
% ATP.lambda_116
tff(fact_5875_ATP_Olambda__117,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_pe(set(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,Uu) ) ) ).
% ATP.lambda_117
tff(fact_5876_ATP_Olambda__118,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Uu),Uua)
<=> member(A,Uua,Uu) ) ).
% ATP.lambda_118
tff(fact_5877_ATP_Olambda__119,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(C,set(B))] : aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_tm(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,Uua),Uu) ).
% ATP.lambda_119
tff(fact_5878_ATP_Olambda__120,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_ai(A,fun(A,$o),Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_120
tff(fact_5879_ATP_Olambda__121,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_gj(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).
% ATP.lambda_121
tff(fact_5880_ATP_Olambda__122,axiom,
! [A: $tType,Uu: multiset(A),Uua: A] :
( aa(A,$o,aTP_Lamp_rq(multiset(A),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua)) ) ).
% ATP.lambda_122
tff(fact_5881_ATP_Olambda__123,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] :
( aa(A,$o,aTP_Lamp_sb(fun(A,nat),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).
% ATP.lambda_123
tff(fact_5882_ATP_Olambda__124,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: A] :
( aa(A,$o,aTP_Lamp_rt(set(multiset(A)),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_rs(A,fun(multiset(A),nat),Uua)),Uu))) ) ).
% ATP.lambda_124
tff(fact_5883_ATP_Olambda__125,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_fk(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).
% ATP.lambda_125
tff(fact_5884_ATP_Olambda__126,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_im(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).
% ATP.lambda_126
tff(fact_5885_ATP_Olambda__127,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ik(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).
% ATP.lambda_127
tff(fact_5886_ATP_Olambda__128,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: C] : aa(C,set(A),aTP_Lamp_xq(fun(sum_sum(B,C),set(A)),fun(C,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(C,sum_sum(B,C),sum_Inr(C,B),Uua)) ).
% ATP.lambda_128
tff(fact_5887_ATP_Olambda__129,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_xp(fun(sum_sum(B,C),set(A)),fun(B,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(B,sum_sum(B,C),sum_Inl(B,C),Uua)) ).
% ATP.lambda_129
tff(fact_5888_ATP_Olambda__130,axiom,
! [Uu: fun(nat,$o),Uua: nat] :
( aa(nat,$o,aTP_Lamp_oe(fun(nat,$o),fun(nat,$o),Uu),Uua)
<=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_130
tff(fact_5889_ATP_Olambda__131,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_131
tff(fact_5890_ATP_Olambda__132,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_og(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).
% ATP.lambda_132
tff(fact_5891_ATP_Olambda__133,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_tn(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).
% ATP.lambda_133
tff(fact_5892_ATP_Olambda__134,axiom,
! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_ru(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).
% ATP.lambda_134
tff(fact_5893_ATP_Olambda__135,axiom,
! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_xk(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).
% ATP.lambda_135
tff(fact_5894_ATP_Olambda__136,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_vd(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_vc(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).
% ATP.lambda_136
tff(fact_5895_ATP_Olambda__137,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_la(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_kz(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_137
tff(fact_5896_ATP_Olambda__138,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_fy(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_fx(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).
% ATP.lambda_138
tff(fact_5897_ATP_Olambda__139,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: A] : aa(A,nat,aTP_Lamp_rw(set(multiset(A)),fun(A,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_rs(A,fun(multiset(A),nat),Uua)),Uu)) ).
% ATP.lambda_139
tff(fact_5898_ATP_Olambda__140,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_vw(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_vv(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).
% ATP.lambda_140
tff(fact_5899_ATP_Olambda__141,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_xh(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),bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_141
tff(fact_5900_ATP_Olambda__142,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gt(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),minus_minus(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_142
tff(fact_5901_ATP_Olambda__143,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gs(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_143
tff(fact_5902_ATP_Olambda__144,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( aa(A,$o,aTP_Lamp_we(list(A),fun(A,$o),Uu),Uua)
<=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).
% ATP.lambda_144
tff(fact_5903_ATP_Olambda__145,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_xj(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),modulo_modulo(code_natural,Uua,Uu)) ).
% ATP.lambda_145
tff(fact_5904_ATP_Olambda__146,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_op(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_146
tff(fact_5905_ATP_Olambda__147,axiom,
! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : aa(A,B,aTP_Lamp_vt(list(product_prod(A,B)),fun(A,B),Uu),Uua) = aa(option(B),B,the2(B),aa(A,option(B),map_of(A,B,Uu),Uua)) ).
% ATP.lambda_147
tff(fact_5906_ATP_Olambda__148,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_gl(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).
% ATP.lambda_148
tff(fact_5907_ATP_Olambda__149,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_fi(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_149
tff(fact_5908_ATP_Olambda__150,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_by(set(A),fun(A,$o),Uu),Uua)
<=> ~ member(A,Uua,Uu) ) ).
% ATP.lambda_150
tff(fact_5909_ATP_Olambda__151,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pk(A,fun(A,$o)),Uu),Uua)
<=> ( Uu != Uua ) ) ).
% ATP.lambda_151
tff(fact_5910_ATP_Olambda__152,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_oh(A,fun(A,$o),Uu),Uua)
<=> ( Uua != Uu ) ) ) ).
% ATP.lambda_152
tff(fact_5911_ATP_Olambda__153,axiom,
! [A: $tType] :
( ( linorder(A)
& no_bot(A) )
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_nq(A,fun(A,$o),Uu),Uua)
<=> ( Uua != Uu ) ) ) ).
% ATP.lambda_153
tff(fact_5912_ATP_Olambda__154,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ce(A,fun(A,$o)),Uu),Uua)
<=> ( Uua != Uu ) ) ).
% ATP.lambda_154
tff(fact_5913_ATP_Olambda__155,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_gm(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A)))) ).
% ATP.lambda_155
tff(fact_5914_ATP_Olambda__156,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_gv(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),Uu),Uua) = transitive_trancl(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).
% ATP.lambda_156
tff(fact_5915_ATP_Olambda__157,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ha(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).
% ATP.lambda_157
tff(fact_5916_ATP_Olambda__158,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_ty(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).
% ATP.lambda_158
tff(fact_5917_ATP_Olambda__159,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_uc(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).
% ATP.lambda_159
tff(fact_5918_ATP_Olambda__160,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_dy(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).
% ATP.lambda_160
tff(fact_5919_ATP_Olambda__161,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_vj(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).
% ATP.lambda_161
tff(fact_5920_ATP_Olambda__162,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_vs(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).
% ATP.lambda_162
tff(fact_5921_ATP_Olambda__163,axiom,
! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_lv(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = aa(set(C),filter(C),principal(C),aa(D,set(C),Uu,Uua)) ).
% ATP.lambda_163
tff(fact_5922_ATP_Olambda__164,axiom,
! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_lz(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = aa(set(D),filter(D),principal(D),aa(C,set(D),Uu,Uua)) ).
% ATP.lambda_164
tff(fact_5923_ATP_Olambda__165,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_lk(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),aa(B,set(A),Uu,Uua)) ).
% ATP.lambda_165
tff(fact_5924_ATP_Olambda__166,axiom,
! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_ly(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = aa(set(E),filter(E),principal(E),aa(A,set(E),Uu,Uua)) ).
% ATP.lambda_166
tff(fact_5925_ATP_Olambda__167,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_lj(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = aa(set(B),filter(B),principal(B),aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_167
tff(fact_5926_ATP_Olambda__168,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_gi(fun(A,set(B)),fun(A,nat),Uu),Uua) = finite_card(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_168
tff(fact_5927_ATP_Olambda__169,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: A] : aa(A,set(B),aTP_Lamp_jp(fun(A,fun(B,$o)),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),Uu,Uua)) ).
% ATP.lambda_169
tff(fact_5928_ATP_Olambda__170,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_fh(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,Uu,Uua)) ).
% ATP.lambda_170
tff(fact_5929_ATP_Olambda__171,axiom,
! [B: $tType,Uu: fun(B,$o),Uua: B] :
( aa(B,$o,aTP_Lamp_at(fun(B,$o),fun(B,$o),Uu),Uua)
<=> ~ aa(B,$o,Uu,Uua) ) ).
% ATP.lambda_171
tff(fact_5930_ATP_Olambda__172,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_bx(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ~ aa(A,$o,Uu,Uua) ) ).
% ATP.lambda_172
tff(fact_5931_ATP_Olambda__173,axiom,
! [B: $tType,A: $tType] :
( finite_finite(A)
=> ! [Uu: fun(B,fun(A,$o)),Uua: B] :
( aa(B,$o,aTP_Lamp_qf(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
<=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_12) ) ) ).
% ATP.lambda_173
tff(fact_5932_ATP_Olambda__174,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
( aa(A,$o,aTP_Lamp_qb(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
<=> ! [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).
% ATP.lambda_174
tff(fact_5933_ATP_Olambda__175,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
( aa(A,$o,aTP_Lamp_mf(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
<=> ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).
% ATP.lambda_175
tff(fact_5934_ATP_Olambda__176,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_ls(set(A),fun(set(A),filter(set(A))),Uu),Uua) = aa(set(set(A)),filter(set(A)),principal(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_lr(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ).
% ATP.lambda_176
tff(fact_5935_ATP_Olambda__177,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o)] : aa(fun(B,$o),filter(product_prod(A,B)),aa(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_os(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),Uu),Uua) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),principal(product_prod(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_je(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua)))) ).
% ATP.lambda_177
tff(fact_5936_ATP_Olambda__178,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A] :
( aa(A,$o,aTP_Lamp_mg(fun(B,fun(A,$o)),fun(A,$o),Uu),Uua)
<=> ? [X2: B] : aa(A,$o,aa(B,fun(A,$o),Uu,X2),Uua) ) ).
% ATP.lambda_178
tff(fact_5937_ATP_Olambda__179,axiom,
! [A: $tType,Uu: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_me(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ? [X2: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu,X2),Uua) ) ).
% ATP.lambda_179
tff(fact_5938_ATP_Olambda__180,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] :
( aa(B,$o,aTP_Lamp_tu(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
<=> ? [A7: A] : aa(B,$o,aa(A,fun(B,$o),Uu,A7),Uua) ) ).
% ATP.lambda_180
tff(fact_5939_ATP_Olambda__181,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_mn(set(A),fun(set(A),$o),Uu),Uua)
<=> ? [B10: set(A)] :
( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B10) )
& member(set(A),Uu,pow2(A,B10)) ) ) ).
% ATP.lambda_181
tff(fact_5940_ATP_Olambda__182,axiom,
! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
( aa(filter(A),$o,aTP_Lamp_pp(set(filter(A)),fun(filter(A),$o),Uu),Uua)
<=> ! [X2: filter(A)] :
( member(filter(A),X2,Uu)
=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X2) ) ) ).
% ATP.lambda_182
tff(fact_5941_ATP_Olambda__183,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_pg(set(A),fun(A,$o),Uu),Uua)
<=> ! [X2: A] :
( member(A,X2,Uu)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X2) ) ) ) ).
% ATP.lambda_183
tff(fact_5942_ATP_Olambda__184,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_ph(set(A),fun(A,$o),Uu),Uua)
<=> ! [X2: A] :
( member(A,X2,Uu)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Uua) ) ) ) ).
% ATP.lambda_184
tff(fact_5943_ATP_Olambda__185,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
( aa(multiset(A),$o,aTP_Lamp_ro(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,Uu)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Uua),X2) ) ) ).
% ATP.lambda_185
tff(fact_5944_ATP_Olambda__186,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
( aa(multiset(A),$o,aTP_Lamp_rn(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
<=> ! [X2: multiset(A)] :
( member(multiset(A),X2,Uu)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),Uua) ) ) ).
% ATP.lambda_186
tff(fact_5945_ATP_Olambda__187,axiom,
! [A: $tType,Uu: set(filter(A)),Uua: fun(A,$o)] :
( aa(fun(A,$o),$o,aTP_Lamp_pz(set(filter(A)),fun(fun(A,$o),$o),Uu),Uua)
<=> ! [X2: filter(A)] :
( member(filter(A),X2,Uu)
=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uua),X2) ) ) ).
% ATP.lambda_187
tff(fact_5946_ATP_Olambda__188,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_ql(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ! [Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3)
=> aa(A,$o,Uu,Y3) ) ) ) ).
% ATP.lambda_188
tff(fact_5947_ATP_Olambda__189,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_pl(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
<=> ! [X2: A] :
( member(A,X2,Uua)
=> ! [Xa2: A] :
( member(A,Xa2,Uua)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),Uu)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),Uu) ) ) ) ) ).
% ATP.lambda_189
tff(fact_5948_ATP_Olambda__190,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
( aa(A,$o,aTP_Lamp_mm(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
<=> ? [Y3: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y3),Uu) ) ).
% ATP.lambda_190
tff(fact_5949_ATP_Olambda__191,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_mz(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ? [X2: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Uu,X2)),Uua) ) ).
% ATP.lambda_191
tff(fact_5950_ATP_Olambda__192,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_tw(A,fun(product_prod(A,B),$o),Uu),Uua)
<=> ? [V5: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V5) ) ).
% ATP.lambda_192
tff(fact_5951_ATP_Olambda__193,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
( aa(A,$o,aTP_Lamp_mk(fun(B,A),fun(A,$o),Uu),Uua)
<=> ? [X2: B] : Uua = aa(B,A,Uu,X2) ) ).
% ATP.lambda_193
tff(fact_5952_ATP_Olambda__194,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod($o,A),product_prod($o,B))] :
( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_mu(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu),Uua)
<=> ? [X2: A,Y3: B] :
( ( Uua = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$false),X2)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) )
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y3),Uu) ) ) ).
% ATP.lambda_194
tff(fact_5953_ATP_Olambda__195,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_uf(fun(A,option(B)),fun(product_prod(A,B),$o),Uu),Uua)
<=> ? [A7: A,B7: B] :
( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
& ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B7) ) ) ) ).
% ATP.lambda_195
tff(fact_5954_ATP_Olambda__196,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_pt(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
<=> ? [X9: set(A),Y8: set(A)] :
( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X9),Y8) )
& ( X9 != bot_bot(set(A)) )
& ! [X2: A] :
( member(A,X2,Y8)
=> ? [Xa2: A] :
( member(A,Xa2,X9)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X2),Uu) ) ) ) ) ).
% ATP.lambda_196
tff(fact_5955_ATP_Olambda__197,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ay(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).
% ATP.lambda_197
tff(fact_5956_ATP_Olambda__198,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_et(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).
% ATP.lambda_198
tff(fact_5957_ATP_Olambda__199,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_tp(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ).
% ATP.lambda_199
tff(fact_5958_ATP_Olambda__200,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_tq(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).
% ATP.lambda_200
tff(fact_5959_ATP_Olambda__201,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_dk(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).
% ATP.lambda_201
tff(fact_5960_ATP_Olambda__202,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_uo(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).
% ATP.lambda_202
tff(fact_5961_ATP_Olambda__203,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,set(B),aTP_Lamp_jf(fun(B,$o),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),Uu) ).
% ATP.lambda_203
tff(fact_5962_ATP_Olambda__204,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_vh(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).
% ATP.lambda_204
tff(fact_5963_ATP_Olambda__205,axiom,
! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_uz(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).
% ATP.lambda_205
tff(fact_5964_ATP_Olambda__206,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_kj(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),minus_minus(code_integer,Uub,aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_206
tff(fact_5965_ATP_Olambda__207,axiom,
! [Uu: num,Uua: nat,Uub: nat] :
aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ix(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),minus_minus(nat,Uub,aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_207
tff(fact_5966_ATP_Olambda__208,axiom,
! [Uu: num,Uua: int,Uub: int] :
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_iy(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),minus_minus(int,Uub,aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_208
tff(fact_5967_ATP_Olambda__209,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: num,Uua: A,Uub: A] :
aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ja(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),minus_minus(A,Uub,aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).
% ATP.lambda_209
tff(fact_5968_ATP_Olambda__210,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_eg(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).
% ATP.lambda_210
tff(fact_5969_ATP_Olambda__211,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ar(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).
% ATP.lambda_211
tff(fact_5970_ATP_Olambda__212,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aq(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).
% ATP.lambda_212
tff(fact_5971_ATP_Olambda__213,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_sk(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,zero_zero(A)) ) ).
% ATP.lambda_213
tff(fact_5972_ATP_Olambda__214,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_rz(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,one_one(A)) ) ).
% ATP.lambda_214
tff(fact_5973_ATP_Olambda__215,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_sj(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,zero_zero(A)) ) ).
% ATP.lambda_215
tff(fact_5974_ATP_Olambda__216,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_ry(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,one_one(A)) ) ).
% ATP.lambda_216
tff(fact_5975_ATP_Olambda__217,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_ku(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),minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua),one_one(code_integer))),minus_minus(code_integer,abs_abs(code_integer,Uu),Uub))) ).
% ATP.lambda_217
tff(fact_5976_ATP_Olambda__218,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: list(A)] :
aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_wi(fun(A,$o),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).
% ATP.lambda_218
tff(fact_5977_ATP_Olambda__219,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_jb(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).
% ATP.lambda_219
tff(fact_5978_ATP_Olambda__220,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_dg(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).
% ATP.lambda_220
tff(fact_5979_ATP_Olambda__221,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_ko(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_gl(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).
% ATP.lambda_221
tff(fact_5980_ATP_Olambda__222,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_fj(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_fi(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).
% ATP.lambda_222
tff(fact_5981_ATP_Olambda__223,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_uj(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).
% ATP.lambda_223
tff(fact_5982_ATP_Olambda__224,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xd(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),Uu,aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub))),aa(nat,A,Uua,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)))) ).
% ATP.lambda_224
tff(fact_5983_ATP_Olambda__225,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_iq(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).
% ATP.lambda_225
tff(fact_5984_ATP_Olambda__226,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_226
tff(fact_5985_ATP_Olambda__227,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_if(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).
% ATP.lambda_227
tff(fact_5986_ATP_Olambda__228,axiom,
! [B: $tType,A: $tType] :
( finite_finite(A)
=> ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_qe(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).
% ATP.lambda_228
tff(fact_5987_ATP_Olambda__229,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_jg(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).
% ATP.lambda_229
tff(fact_5988_ATP_Olambda__230,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,B)),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_va(fun(A,fun(B,B)),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(A,fun(B,B),Uu,Uub),Uua) ).
% ATP.lambda_230
tff(fact_5989_ATP_Olambda__231,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wl(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uua) ) ).
% ATP.lambda_231
tff(fact_5990_ATP_Olambda__232,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rl(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_rk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_232
tff(fact_5991_ATP_Olambda__233,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_rj(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_ri(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).
% ATP.lambda_233
tff(fact_5992_ATP_Olambda__234,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_rh(rat,fun(int,fun(int,$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,$o)),aTP_Lamp_rg(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).
% ATP.lambda_234
tff(fact_5993_ATP_Olambda__235,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rf(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_re(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_235
tff(fact_5994_ATP_Olambda__236,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rd(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_rc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_236
tff(fact_5995_ATP_Olambda__237,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rb(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_ra(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_237
tff(fact_5996_ATP_Olambda__238,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: filter(B),Uub: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_qp(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aTP_Lamp_qo(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uub)),Uua) ) ).
% ATP.lambda_238
tff(fact_5997_ATP_Olambda__239,axiom,
! [A: $tType] :
( field(A)
=> ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).
% ATP.lambda_239
tff(fact_5998_ATP_Olambda__240,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: sum_sum(product_prod(A,C),product_prod(B,D))] :
( aa(sum_sum(product_prod(A,C),product_prod(B,D)),$o,aa(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_xs(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),$o)),Uu),Uua),Uub)
<=> ( aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(A,C)),basic_setl(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),Uu)))
& aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(B,D)),basic_setr(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(B,D),$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),Uua))) ) ) ).
% ATP.lambda_240
tff(fact_5999_ATP_Olambda__241,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(D,$o)),Uub: sum_sum(product_prod(A,B),product_prod(C,D))] :
( aa(sum_sum(product_prod(A,B),product_prod(C,D)),$o,aa(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_xt(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(sum_sum(product_prod(A,B),product_prod(C,D)),$o)),Uu),Uua),Uub)
<=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(sum_sum(product_prod(A,B),product_prod(C,D)),set(product_prod(A,B)),basic_setl(product_prod(A,B),product_prod(C,D)),Uub)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu)))
& aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),aa(sum_sum(product_prod(A,B),product_prod(C,D)),set(product_prod(C,D)),basic_setr(product_prod(A,B),product_prod(C,D)),Uub)),aa(fun(product_prod(C,D),$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),Uua))) ) ) ).
% ATP.lambda_241
tff(fact_6000_ATP_Olambda__242,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(B,fun(D,$o)),Uub: product_prod(product_prod(A,C),product_prod(B,D))] :
( aa(product_prod(product_prod(A,C),product_prod(B,D)),$o,aa(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o),aTP_Lamp_ta(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),fun(product_prod(product_prod(A,C),product_prod(B,D)),$o)),Uu),Uua),Uub)
<=> ( aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),basic_fsts(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),Uu)))
& aa(set(product_prod(B,D)),$o,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),$o),ord_less_eq(set(product_prod(B,D))),basic_snds(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),$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),Uua))) ) ) ).
% ATP.lambda_242
tff(fact_6001_ATP_Olambda__243,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,fun(D,$o)),Uub: product_prod(product_prod(A,B),product_prod(C,D))] :
( aa(product_prod(product_prod(A,B),product_prod(C,D)),$o,aa(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o),aTP_Lamp_tb(fun(A,fun(B,$o)),fun(fun(C,fun(D,$o)),fun(product_prod(product_prod(A,B),product_prod(C,D)),$o)),Uu),Uua),Uub)
<=> ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),basic_fsts(product_prod(A,B),product_prod(C,D),Uub)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu)))
& aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),basic_snds(product_prod(A,B),product_prod(C,D),Uub)),aa(fun(product_prod(C,D),$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),Uua))) ) ) ).
% ATP.lambda_243
tff(fact_6002_ATP_Olambda__244,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_uw(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& ? [Xys2: list(A),X2: A,Y3: A,Xs7: list(A),Ys5: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs7)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),Uu) ) ) ) ).
% ATP.lambda_244
tff(fact_6003_ATP_Olambda__245,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_pr(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
& ! [X2: A] :
( member(A,X2,Uua)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X2),Uu) ) ) ) ).
% ATP.lambda_245
tff(fact_6004_ATP_Olambda__246,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ps(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
& ! [X2: A] :
( member(A,X2,Uua)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Uub),Uu) ) ) ) ).
% ATP.lambda_246
tff(fact_6005_ATP_Olambda__247,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_pq(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
& ! [X2: A] :
( member(A,X2,Uua)
=> ( ( Uub != X2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X2),Uu) ) ) ) ) ).
% ATP.lambda_247
tff(fact_6006_ATP_Olambda__248,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_qa(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,aa(set(product_prod(A,A)),set(A),field2(A),Uu))
& ! [X2: A] :
( member(A,X2,Uua)
=> ( ( Uub != X2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Uub),Uu) ) ) ) ) ).
% ATP.lambda_248
tff(fact_6007_ATP_Olambda__249,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ee(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_249
tff(fact_6008_ATP_Olambda__250,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ch(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
| member(A,Uub,Uua) ) ) ).
% ATP.lambda_250
tff(fact_6009_ATP_Olambda__251,axiom,
! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cx(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub = Uu )
| member(A,Uub,Uua) ) ) ).
% ATP.lambda_251
tff(fact_6010_ATP_Olambda__252,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_jw(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uub),Uua)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).
% ATP.lambda_252
tff(fact_6011_ATP_Olambda__253,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ef(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_253
tff(fact_6012_ATP_Olambda__254,axiom,
! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_qn(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uua) ) ) ).
% ATP.lambda_254
tff(fact_6013_ATP_Olambda__255,axiom,
! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_po(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uua,Uu)
& member(A,Uub,Uu) ) ) ).
% ATP.lambda_255
tff(fact_6014_ATP_Olambda__256,axiom,
! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
( aa(set(A),$o,aa(A,fun(set(A),$o),aTP_Lamp_oo(set(set(A)),fun(A,fun(set(A),$o)),Uu),Uua),Uub)
<=> ( member(set(A),Uub,Uu)
& member(A,Uua,Uub) ) ) ).
% ATP.lambda_256
tff(fact_6015_ATP_Olambda__257,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cd(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& member(A,Uub,Uua) ) ) ).
% ATP.lambda_257
tff(fact_6016_ATP_Olambda__258,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xn(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uua = Uu )
& ( Uub = Uu ) ) ) ).
% ATP.lambda_258
tff(fact_6017_ATP_Olambda__259,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_dv(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
<=> ( minus_minus(set(A),Uub,Uu) = minus_minus(set(A),Uua,Uu) ) ) ).
% ATP.lambda_259
tff(fact_6018_ATP_Olambda__260,axiom,
! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_wp(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uu,Uub)),aa(A,nat,Uua,Uub)) ).
% ATP.lambda_260
tff(fact_6019_ATP_Olambda__261,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_wq(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uua,Uub)),aa(A,nat,Uu,Uub)) ).
% ATP.lambda_261
tff(fact_6020_ATP_Olambda__262,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_nu(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uua)
=> aa(A,$o,Uu,Uub) ) ) ).
% ATP.lambda_262
tff(fact_6021_ATP_Olambda__263,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bv(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_263
tff(fact_6022_ATP_Olambda__264,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jn(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uua)
& aa(A,$o,Uu,Uub) ) ) ).
% ATP.lambda_264
tff(fact_6023_ATP_Olambda__265,axiom,
! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cz(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_265
tff(fact_6024_ATP_Olambda__266,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jq(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uua = Uub )
& aa(A,$o,Uu,Uua) ) ) ).
% ATP.lambda_266
tff(fact_6025_ATP_Olambda__267,axiom,
! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_da(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_267
tff(fact_6026_ATP_Olambda__268,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ej(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).
% ATP.lambda_268
tff(fact_6027_ATP_Olambda__269,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ji(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).
% ATP.lambda_269
tff(fact_6028_ATP_Olambda__270,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_jd(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).
% ATP.lambda_270
tff(fact_6029_ATP_Olambda__271,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_dd(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).
% ATP.lambda_271
tff(fact_6030_ATP_Olambda__272,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_lg(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
<=> ( member(B,Uub,Uua)
& ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_272
tff(fact_6031_ATP_Olambda__273,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bz(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ~ member(A,Uub,Uua) ) ) ).
% ATP.lambda_273
tff(fact_6032_ATP_Olambda__274,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: multiset(A),Uub: A] : aa(A,nat,aa(multiset(A),fun(A,nat),aTP_Lamp_rx(fun(A,nat),fun(multiset(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(nat,nat,suc,aa(A,nat,Uu,Uub))) ).
% ATP.lambda_274
tff(fact_6033_ATP_Olambda__275,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fn(nat,fun(nat,fun(nat,$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_275
tff(fact_6034_ATP_Olambda__276,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hj(A,fun(A,fun(A,A)),Uu),Uua),Uub) = minus_minus(A,divide_divide(A,Uub,Uu),Uua) ) ).
% ATP.lambda_276
tff(fact_6035_ATP_Olambda__277,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub),Uua) ) ).
% ATP.lambda_277
tff(fact_6036_ATP_Olambda__278,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).
% ATP.lambda_278
tff(fact_6037_ATP_Olambda__279,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_279
tff(fact_6038_ATP_Olambda__280,axiom,
! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C,Uub: B] :
( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_sv(set(product_prod(C,B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
<=> 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_280
tff(fact_6039_ATP_Olambda__281,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_sy(set(product_prod(B,C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
<=> 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_281
tff(fact_6040_ATP_Olambda__282,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_eh(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).
% ATP.lambda_282
tff(fact_6041_ATP_Olambda__283,axiom,
! [C: $tType,A: $tType,Uu: set(product_prod(A,C)),Uua: A,Uub: C] :
( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_su(set(product_prod(A,C)),fun(A,fun(C,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uua),Uub),Uu) ) ).
% ATP.lambda_283
tff(fact_6042_ATP_Olambda__284,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).
% ATP.lambda_284
tff(fact_6043_ATP_Olambda__285,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_er(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ).
% ATP.lambda_285
tff(fact_6044_ATP_Olambda__286,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_jr(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua),Uu) ) ).
% ATP.lambda_286
tff(fact_6045_ATP_Olambda__287,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cv(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ).
% ATP.lambda_287
tff(fact_6046_ATP_Olambda__288,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_om(fun(A,B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).
% ATP.lambda_288
tff(fact_6047_ATP_Olambda__289,axiom,
! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_ou(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub))
& aa(B,$o,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub)) ) ) ).
% ATP.lambda_289
tff(fact_6048_ATP_Olambda__290,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_oi(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).
% ATP.lambda_290
tff(fact_6049_ATP_Olambda__291,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cn(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_291
tff(fact_6050_ATP_Olambda__292,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_292
tff(fact_6051_ATP_Olambda__293,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_293
tff(fact_6052_ATP_Olambda__294,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_du(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = minus_minus(set(B),aa(A,set(B),Uu,Uub),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_294
tff(fact_6053_ATP_Olambda__295,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_nf(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),aa(A,assn,Uua,Uub)) ).
% ATP.lambda_295
tff(fact_6054_ATP_Olambda__296,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_dp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_296
tff(fact_6055_ATP_Olambda__297,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_297
tff(fact_6056_ATP_Olambda__298,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_do(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_298
tff(fact_6057_ATP_Olambda__299,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_299
tff(fact_6058_ATP_Olambda__300,axiom,
! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_sa(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).
% ATP.lambda_300
tff(fact_6059_ATP_Olambda__301,axiom,
! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_rv(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).
% ATP.lambda_301
tff(fact_6060_ATP_Olambda__302,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gn(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).
% ATP.lambda_302
tff(fact_6061_ATP_Olambda__303,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_or(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).
% ATP.lambda_303
tff(fact_6062_ATP_Olambda__304,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uu,Uub)
=> aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_304
tff(fact_6063_ATP_Olambda__305,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ag(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).
% ATP.lambda_305
tff(fact_6064_ATP_Olambda__306,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uu,Uub)
| aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_306
tff(fact_6065_ATP_Olambda__307,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_af(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).
% ATP.lambda_307
tff(fact_6066_ATP_Olambda__308,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uu,Uub)
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_308
tff(fact_6067_ATP_Olambda__309,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wd(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).
% ATP.lambda_309
tff(fact_6068_ATP_Olambda__310,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_no(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
<=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).
% ATP.lambda_310
tff(fact_6069_ATP_Olambda__311,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uu,Uub)
<=> aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_311
tff(fact_6070_ATP_Olambda__312,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nt(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).
% ATP.lambda_312
tff(fact_6071_ATP_Olambda__313,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_bm(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).
% ATP.lambda_313
tff(fact_6072_ATP_Olambda__314,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_qt(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uua,Uub)
& ! [Y3: A] :
( aa(A,$o,Uua,Y3)
=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y3) ) ) ) ).
% ATP.lambda_314
tff(fact_6073_ATP_Olambda__315,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ne(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),ex_assn(A,Uua)) ).
% ATP.lambda_315
tff(fact_6074_ATP_Olambda__316,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tx(fun(A,option(B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).
% ATP.lambda_316
tff(fact_6075_ATP_Olambda__317,axiom,
! [A: $tType,B: $tType] :
( unboun7993243217541854897norder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ny(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).
% ATP.lambda_317
tff(fact_6076_ATP_Olambda__318,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).
% ATP.lambda_318
tff(fact_6077_ATP_Olambda__319,axiom,
! [A: $tType,B: $tType] :
( ( dense_linorder(B)
& no_bot(B) )
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).
% ATP.lambda_319
tff(fact_6078_ATP_Olambda__320,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_320
tff(fact_6079_ATP_Olambda__321,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_nb(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uub)),Uua) ).
% ATP.lambda_321
tff(fact_6080_ATP_Olambda__322,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bp(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = minus_minus(set(A),aa(B,set(A),Uu,Uub),Uua) ).
% ATP.lambda_322
tff(fact_6081_ATP_Olambda__323,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_be(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).
% ATP.lambda_323
tff(fact_6082_ATP_Olambda__324,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_nd(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),Uua) ).
% ATP.lambda_324
tff(fact_6083_ATP_Olambda__325,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_bn(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).
% ATP.lambda_325
tff(fact_6084_ATP_Olambda__326,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_nc(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(A,assn,Uu,Uub)),Uua) ).
% ATP.lambda_326
tff(fact_6085_ATP_Olambda__327,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ed(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).
% ATP.lambda_327
tff(fact_6086_ATP_Olambda__328,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_fb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).
% ATP.lambda_328
tff(fact_6087_ATP_Olambda__329,axiom,
! [E: $tType,F3: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F3,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F3,B),aa(fun(C,fun(D,B)),fun(E,fun(F3,B)),aTP_Lamp_sp(fun(E,fun(F3,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F3,B))),Uu),Uua),Uub) = product_scomp(F3,C,D,B,aa(E,fun(F3,product_prod(C,D)),Uu,Uub),Uua) ).
% ATP.lambda_329
tff(fact_6088_ATP_Olambda__330,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_oz(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),aa(A,filter(B),Uu,Uub)),Uua) ).
% ATP.lambda_330
tff(fact_6089_ATP_Olambda__331,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,fun(A,fun(C,$o))),Uua: fun(C,fun(B,$o)),Uub: D] : aa(D,fun(A,fun(B,$o)),aa(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o))),aTP_Lamp_sw(fun(D,fun(A,fun(C,$o))),fun(fun(C,fun(B,$o)),fun(D,fun(A,fun(B,$o)))),Uu),Uua),Uub) = relcompp(A,C,B,aa(D,fun(A,fun(C,$o)),Uu,Uub),Uua) ).
% ATP.lambda_331
tff(fact_6090_ATP_Olambda__332,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ea(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).
% ATP.lambda_332
tff(fact_6091_ATP_Olambda__333,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_fm(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub),Uua) ).
% ATP.lambda_333
tff(fact_6092_ATP_Olambda__334,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_qg(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
<=> ( aa(A,$o,Uu,Uub)
=> (Uua) ) ) ).
% ATP.lambda_334
tff(fact_6093_ATP_Olambda__335,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_cb(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> member(B,aa(A,B,Uu,Uub),Uua) ) ).
% ATP.lambda_335
tff(fact_6094_ATP_Olambda__336,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_dh(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
<=> member(A,aa(B,A,Uua,Uub),Uu) ) ).
% ATP.lambda_336
tff(fact_6095_ATP_Olambda__337,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,Uu: fun(D,fun(E,C)),Uua: fun(C,B),Uub: D] : aa(D,fun(E,B),aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_xl(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),Uu),Uua),Uub) = fcomp(E,C,B,aa(D,fun(E,C),Uu,Uub),Uua) ).
% ATP.lambda_337
tff(fact_6096_ATP_Olambda__338,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(B)),Uua: fun(B,set(A)),Uub: C] : aa(C,set(A),aa(fun(B,set(A)),fun(C,set(A)),aTP_Lamp_sq(fun(C,set(B)),fun(fun(B,set(A)),fun(C,set(A))),Uu),Uua),Uub) = bind(B,A,aa(C,set(B),Uu,Uub),Uua) ).
% ATP.lambda_338
tff(fact_6097_ATP_Olambda__339,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_nl(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
<=> ( aa(A,$o,Uu,Uub)
| (Uua) ) ) ).
% ATP.lambda_339
tff(fact_6098_ATP_Olambda__340,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_qc(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
<=> ( aa(A,$o,Uu,Uub)
& (Uua) ) ) ).
% ATP.lambda_340
tff(fact_6099_ATP_Olambda__341,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_lc(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ).
% ATP.lambda_341
tff(fact_6100_ATP_Olambda__342,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_sn(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub != Uua )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ) ).
% ATP.lambda_342
tff(fact_6101_ATP_Olambda__343,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cu(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub != Uua )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ) ).
% ATP.lambda_343
tff(fact_6102_ATP_Olambda__344,axiom,
! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cw(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub != Uu )
=> aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_344
tff(fact_6103_ATP_Olambda__345,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pj(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uua != Uu )
& ( Uub != Uu ) ) ) ).
% ATP.lambda_345
tff(fact_6104_ATP_Olambda__346,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bl(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_346
tff(fact_6105_ATP_Olambda__347,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_qk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ~ aa(A,$o,Uu,Uub)
| aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_347
tff(fact_6106_ATP_Olambda__348,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_lr(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
<=> ( aa(set(A),$o,finite_finite2(A),Uub)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uub)
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).
% ATP.lambda_348
tff(fact_6107_ATP_Olambda__349,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_py(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
<=> ( aa(set(A),$o,finite_finite2(A),Uua)
& aa(set(A),$o,finite_finite2(A),Uub)
& ( Uub != bot_bot(set(A)) )
& ! [X2: A] :
( member(A,X2,Uua)
=> ? [Xa2: A] :
( member(A,Xa2,Uub)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa2),Uu) ) ) ) ) ).
% ATP.lambda_349
tff(fact_6108_ATP_Olambda__350,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_gp(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = image(A,A,converse(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))) ).
% ATP.lambda_350
tff(fact_6109_ATP_Olambda__351,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lm(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).
% ATP.lambda_351
tff(fact_6110_ATP_Olambda__352,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ll(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).
% ATP.lambda_352
tff(fact_6111_ATP_Olambda__353,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: list(A),Uub: B] : aa(B,list(A),aa(list(A),fun(B,list(A)),aTP_Lamp_vq(fun(B,A),fun(list(A),fun(B,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uub)),nil(A))) ).
% ATP.lambda_353
tff(fact_6112_ATP_Olambda__354,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).
% ATP.lambda_354
tff(fact_6113_ATP_Olambda__355,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hx(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).
% ATP.lambda_355
tff(fact_6114_ATP_Olambda__356,axiom,
! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_mb(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> member(A,Uub,minus_minus(set(A),Uua,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).
% ATP.lambda_356
tff(fact_6115_ATP_Olambda__357,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_li(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),minus_minus(nat,Uua,aa(nat,nat,suc,Uub))) ) ).
% ATP.lambda_357
tff(fact_6116_ATP_Olambda__358,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).
% ATP.lambda_358
tff(fact_6117_ATP_Olambda__359,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hz(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_359
tff(fact_6118_ATP_Olambda__360,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_vn(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).
% ATP.lambda_360
tff(fact_6119_ATP_Olambda__361,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_vo(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).
% ATP.lambda_361
tff(fact_6120_ATP_Olambda__362,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_pd(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
<=> member(B,Uub,aa(set(A),set(B),image2(A,B,Uu),Uua)) ) ) ).
% ATP.lambda_362
tff(fact_6121_ATP_Olambda__363,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fs(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_363
tff(fact_6122_ATP_Olambda__364,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_add(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fr(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_364
tff(fact_6123_ATP_Olambda__365,axiom,
! [B: $tType,A: $tType] :
( unboun7993243217541854897norder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_od(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).
% ATP.lambda_365
tff(fact_6124_ATP_Olambda__366,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_oj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).
% ATP.lambda_366
tff(fact_6125_ATP_Olambda__367,axiom,
! [B: $tType,A: $tType] :
( unboun7993243217541854897norder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ok(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).
% ATP.lambda_367
tff(fact_6126_ATP_Olambda__368,axiom,
! [A: $tType,Uu: nat,Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(nat,fun(fun(A,nat),fun(A,nat)),aTP_Lamp_tf(nat,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(A,nat,Uua,Uub)) ).
% ATP.lambda_368
tff(fact_6127_ATP_Olambda__369,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cr(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_369
tff(fact_6128_ATP_Olambda__370,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: nat,Uub: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_tg(fun(A,nat),fun(nat,fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(A,nat,Uu,Uub)) ).
% ATP.lambda_370
tff(fact_6129_ATP_Olambda__371,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bs(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = minus_minus(set(A),Uu,aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_371
tff(fact_6130_ATP_Olambda__372,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bf(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_372
tff(fact_6131_ATP_Olambda__373,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bo(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_373
tff(fact_6132_ATP_Olambda__374,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ec(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_374
tff(fact_6133_ATP_Olambda__375,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_pa(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = aa(filter(C),filter(product_prod(B,C)),aa(filter(B),fun(filter(C),filter(product_prod(B,C))),prod_filter(B,C),Uu),aa(A,filter(C),Uua,Uub)) ).
% ATP.lambda_375
tff(fact_6134_ATP_Olambda__376,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_qy(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).
% ATP.lambda_376
tff(fact_6135_ATP_Olambda__377,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_qx(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).
% ATP.lambda_377
tff(fact_6136_ATP_Olambda__378,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,$o)),Uua: fun(D,fun(C,fun(B,$o))),Uub: D] : aa(D,fun(A,fun(B,$o)),aa(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o))),aTP_Lamp_sx(fun(A,fun(C,$o)),fun(fun(D,fun(C,fun(B,$o))),fun(D,fun(A,fun(B,$o)))),Uu),Uua),Uub) = relcompp(A,C,B,Uu,aa(D,fun(C,fun(B,$o)),Uua,Uub)) ).
% ATP.lambda_378
tff(fact_6137_ATP_Olambda__379,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_vk(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).
% ATP.lambda_379
tff(fact_6138_ATP_Olambda__380,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_dz(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).
% ATP.lambda_380
tff(fact_6139_ATP_Olambda__381,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_on(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Uu),aa(C,filter(B),Uua,Uub)) ).
% ATP.lambda_381
tff(fact_6140_ATP_Olambda__382,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dl(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = image(B,A,Uu,aa(C,set(B),Uua,Uub)) ).
% ATP.lambda_382
tff(fact_6141_ATP_Olambda__383,axiom,
! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
<=> ( (Uu)
=> aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_383
tff(fact_6142_ATP_Olambda__384,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_dx(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> member(A,Uub,aa(B,set(A),Uu,Uua)) ) ).
% ATP.lambda_384
tff(fact_6143_ATP_Olambda__385,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_dj(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_385
tff(fact_6144_ATP_Olambda__386,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_bb(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_386
tff(fact_6145_ATP_Olambda__387,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ft(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_387
tff(fact_6146_ATP_Olambda__388,axiom,
! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nk($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
<=> ( (Uu)
| aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_388
tff(fact_6147_ATP_Olambda__389,axiom,
! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_qd($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
<=> ( (Uu)
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_389
tff(fact_6148_ATP_Olambda__390,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_eq(set(A),fun(set(A),fun(A,set(A))),Uu),Uua),Uub) = minus_minus(set(A),Uu,Uua) ).
% ATP.lambda_390
tff(fact_6149_ATP_Olambda__391,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_dn(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).
% ATP.lambda_391
tff(fact_6150_ATP_Olambda__392,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_kk(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).
% ATP.lambda_392
tff(fact_6151_ATP_Olambda__393,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_bq(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),Uua) ).
% ATP.lambda_393
tff(fact_6152_ATP_Olambda__394,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_iw(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).
% ATP.lambda_394
tff(fact_6153_ATP_Olambda__395,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_bj(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),Uua) ).
% ATP.lambda_395
tff(fact_6154_ATP_Olambda__396,axiom,
! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_of(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_396
tff(fact_6155_ATP_Olambda__397,axiom,
! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tt(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).
% ATP.lambda_397
tff(fact_6156_ATP_Olambda__398,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ij(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).
% ATP.lambda_398
tff(fact_6157_ATP_Olambda__399,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_wo(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
<=> aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ).
% ATP.lambda_399
tff(fact_6158_ATP_Olambda__400,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(multiset(A),B),Uua: fun(C,multiset(A)),Uub: C] : aa(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_rp(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(C,multiset(A),Uua,Uub)) ) ).
% ATP.lambda_400
tff(fact_6159_ATP_Olambda__401,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_ir(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D,fun(B,C),Uu,aa(A,D,Uua,Uub)) ).
% ATP.lambda_401
tff(fact_6160_ATP_Olambda__402,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ca(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).
% ATP.lambda_402
tff(fact_6161_ATP_Olambda__403,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cl(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).
% ATP.lambda_403
tff(fact_6162_ATP_Olambda__404,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_404
tff(fact_6163_ATP_Olambda__405,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_np(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
<=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_405
tff(fact_6164_ATP_Olambda__406,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,$o),aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_wz(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Uu),Uua),Uub) = aa(A,fun(B,$o),Uu,aa(C,A,Uua,Uub)) ).
% ATP.lambda_406
tff(fact_6165_ATP_Olambda__407,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_lw(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).
% ATP.lambda_407
tff(fact_6166_ATP_Olambda__408,axiom,
! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,$o),Uub: C] :
( aa(C,$o,aa(fun(D,$o),fun(C,$o),aTP_Lamp_st(fun(C,D),fun(fun(D,$o),fun(C,$o)),Uu),Uua),Uub)
<=> aa(D,$o,Uua,aa(C,D,Uu,Uub)) ) ).
% ATP.lambda_408
tff(fact_6167_ATP_Olambda__409,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,$o),Uub: B] :
( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_qo(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
<=> aa(A,$o,Uua,aa(B,A,Uu,Uub)) ) ).
% ATP.lambda_409
tff(fact_6168_ATP_Olambda__410,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ad(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_410
tff(fact_6169_ATP_Olambda__411,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_tr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_411
tff(fact_6170_ATP_Olambda__412,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_1(C)
=> ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_gc(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_412
tff(fact_6171_ATP_Olambda__413,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_qw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).
% ATP.lambda_413
tff(fact_6172_ATP_Olambda__414,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_ka(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).
% ATP.lambda_414
tff(fact_6173_ATP_Olambda__415,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_vc(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).
% ATP.lambda_415
tff(fact_6174_ATP_Olambda__416,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ov(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,Uu,Uua) ) ).
% ATP.lambda_416
tff(fact_6175_ATP_Olambda__417,axiom,
! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_wv(fun(A,$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> aa(A,$o,Uu,Uua) ) ).
% ATP.lambda_417
tff(fact_6176_ATP_Olambda__418,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_il(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).
% ATP.lambda_418
tff(fact_6177_ATP_Olambda__419,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_jv(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ju(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).
% ATP.lambda_419
tff(fact_6178_ATP_Olambda__420,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ww(fun(A,fun(B,fun(C,$o))),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> ? [X_12: C] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,Uua),Uub),X_12) ) ).
% ATP.lambda_420
tff(fact_6179_ATP_Olambda__421,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fu(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).
% ATP.lambda_421
tff(fact_6180_ATP_Olambda__422,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fv(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).
% ATP.lambda_422
tff(fact_6181_ATP_Olambda__423,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_kc(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).
% ATP.lambda_423
tff(fact_6182_ATP_Olambda__424,axiom,
! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_en(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = finite_card(B,aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_em(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).
% ATP.lambda_424
tff(fact_6183_ATP_Olambda__425,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: A,Uub: C] :
( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_wx(fun(A,fun(B,fun(C,$o))),fun(A,fun(C,$o)),Uu),Uua),Uub)
<=> ? [B7: B] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,Uua),B7),Uub) ) ).
% ATP.lambda_425
tff(fact_6184_ATP_Olambda__426,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,$o))),Uua: B,Uub: C] :
( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_wy(fun(A,fun(B,fun(C,$o))),fun(B,fun(C,$o)),Uu),Uua),Uub)
<=> ? [A7: A] : aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),Uu,A7),Uua),Uub) ) ).
% ATP.lambda_426
tff(fact_6185_ATP_Olambda__427,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
( aa(A,$o,aa(set(C),fun(A,$o),aTP_Lamp_mj(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Uu),Uua),Uub)
<=> ? [B7: B] : member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B7),Uua) ) ).
% ATP.lambda_427
tff(fact_6186_ATP_Olambda__428,axiom,
! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_uq(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ? [I4: nat] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).
% ATP.lambda_428
tff(fact_6187_ATP_Olambda__429,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ? [A7: A] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
& member(A,A7,Uu) ) ) ).
% ATP.lambda_429
tff(fact_6188_ATP_Olambda__430,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_up(nat,fun(list(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [I4: nat] :
( ( Uub = aa(nat,A,nth(A,Uua),I4) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uu),I4)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uua)) ) ) ).
% ATP.lambda_430
tff(fact_6189_ATP_Olambda__431,axiom,
! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,set(A)),Uub: A] :
( aa(A,$o,aa(fun(B,set(A)),fun(A,$o),aTP_Lamp_sr(set(B),fun(fun(B,set(A)),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: set(A)] :
( member(set(A),X2,aa(set(B),set(set(A)),image2(B,set(A),Uua),Uu))
& member(A,Uub,X2) ) ) ).
% ATP.lambda_431
tff(fact_6190_ATP_Olambda__432,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mt(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ? [A7: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
& member(A,A7,Uu) ) ) ) ).
% ATP.lambda_432
tff(fact_6191_ATP_Olambda__433,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mr(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ? [A7: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
& member(A,A7,Uu) ) ) ) ).
% ATP.lambda_433
tff(fact_6192_ATP_Olambda__434,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_sg(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A7: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uua),A7) )
& member(multiset(A),A7,Uu) ) ) ).
% ATP.lambda_434
tff(fact_6193_ATP_Olambda__435,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_se(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A7: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uua),A7) )
& member(multiset(A),A7,Uu) ) ) ).
% ATP.lambda_435
tff(fact_6194_ATP_Olambda__436,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( aa(set(A),$o,aa(list(B),fun(set(A),$o),aTP_Lamp_ur(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
<=> ? [I4: nat] :
( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I4)) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(B),nat,size_size(list(B)),Uua)) ) ) ).
% ATP.lambda_436
tff(fact_6195_ATP_Olambda__437,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( aa(set(A),$o,aa(list(B),fun(set(A),$o),aTP_Lamp_uu(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu),Uua),Uub)
<=> ? [A7: B] :
( ( Uub = aa(B,set(A),Uu,A7) )
& member(B,A7,aa(list(B),set(B),set2(B),Uua)) ) ) ).
% ATP.lambda_437
tff(fact_6196_ATP_Olambda__438,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_md(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [L2: B] :
( ( Uub = aa(B,A,Uu,L2) )
& member(B,L2,Uua) ) ) ).
% ATP.lambda_438
tff(fact_6197_ATP_Olambda__439,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_mh(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: B] :
( ( Uub = aa(B,A,Uu,X2) )
& aa(B,$o,Uua,X2) ) ) ).
% ATP.lambda_439
tff(fact_6198_ATP_Olambda__440,axiom,
! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
( aa(A,$o,aa(list(B),fun(A,$o),aTP_Lamp_un(fun(B,option(A)),fun(list(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: B] :
( member(B,X2,aa(list(B),set(B),set2(B),Uua))
& ( aa(B,option(A),Uu,X2) = aa(A,option(A),some(A),Uub) ) ) ) ).
% ATP.lambda_440
tff(fact_6199_ATP_Olambda__441,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_pi(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
<=> ! [X2: A] :
( member(A,X2,Uu)
=> aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X2) ) ) ).
% ATP.lambda_441
tff(fact_6200_ATP_Olambda__442,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_qj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
<=> ? [X2: A] :
( member(A,X2,Uu)
& aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X2) ) ) ).
% ATP.lambda_442
tff(fact_6201_ATP_Olambda__443,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_pu(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: B] :
( member(B,X2,Uua)
& aa(A,$o,aa(B,fun(A,$o),Uu,X2),Uub) ) ) ).
% ATP.lambda_443
tff(fact_6202_ATP_Olambda__444,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pn(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ! [X2: product_prod(A,A)] :
( member(product_prod(A,A),X2,Uu)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_pm(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)),X2) ) ) ).
% ATP.lambda_444
tff(fact_6203_ATP_Olambda__445,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_mi(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ? [P6: product_prod(A,B)] :
( ( Uub = P6 )
& member(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P6)),aa(product_prod(A,B),B,product_snd(A,B),P6)),Uua) ) ) ).
% ATP.lambda_445
tff(fact_6204_ATP_Olambda__446,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_pw(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: B] :
( member(B,X2,Uua)
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Uub),Uu) ) ) ).
% ATP.lambda_446
tff(fact_6205_ATP_Olambda__447,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_px(fun(A,B),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: A] :
( member(A,X2,Uua)
& ( aa(A,B,Uu,X2) = aa(A,B,Uu,Uub) ) ) ) ).
% ATP.lambda_447
tff(fact_6206_ATP_Olambda__448,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_pv(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X2: B] :
( member(B,X2,Uua)
& ( Uub = aa(B,A,Uu,X2) ) ) ) ).
% ATP.lambda_448
tff(fact_6207_ATP_Olambda__449,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(B),Uub: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_qu(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
<=> ? [Q4: fun(B,$o)] :
( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q4),Uua)
& ! [X2: A] :
( aa(B,$o,Q4,aa(A,B,Uu,X2))
=> aa(A,$o,Uub,X2) ) ) ) ).
% ATP.lambda_449
tff(fact_6208_ATP_Olambda__450,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,$o)),Uub: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_ma(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
<=> ? [A7: B,B7: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A7)),aa(B,A,Uu,B7)) )
& aa(B,$o,aa(B,fun(B,$o),Uua,A7),B7) ) ) ).
% ATP.lambda_450
tff(fact_6209_ATP_Olambda__451,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_pb(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
<=> ? [A17: B,A25: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A17)),aa(B,A,Uua,A25)) )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A17),A25),Uu) ) ) ).
% ATP.lambda_451
tff(fact_6210_ATP_Olambda__452,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_uv(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ? [A7: A,V4: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V4)) )
| ? [U5: list(A),Aa3: A,B7: A,Va3: list(A),W3: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B7),Uu)
& ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va3)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W3)) ) ) ) ) ).
% ATP.lambda_452
tff(fact_6211_ATP_Olambda__453,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ms(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [A7: A,B7: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B7) )
& member(A,A7,Uu)
& member(A,B7,Uua) ) ) ) ).
% ATP.lambda_453
tff(fact_6212_ATP_Olambda__454,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_mq(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [A7: A,B7: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B7) )
& member(A,A7,Uu)
& member(A,B7,Uua) ) ) ) ).
% ATP.lambda_454
tff(fact_6213_ATP_Olambda__455,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_sh(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A7: multiset(A),B7: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A7),B7) )
& member(multiset(A),A7,Uu)
& member(multiset(A),B7,Uua) ) ) ).
% ATP.lambda_455
tff(fact_6214_ATP_Olambda__456,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_sf(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A7: multiset(A),B7: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A7),B7) )
& member(multiset(A),A7,Uu)
& member(multiset(A),B7,Uua) ) ) ).
% ATP.lambda_456
tff(fact_6215_ATP_Olambda__457,axiom,
! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_qh(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
<=> ? [Q4: fun(A,$o),R7: fun(A,$o)] :
( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q4),Uu)
& aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R7),Uua)
& ! [X2: A] :
( ( aa(A,$o,Q4,X2)
& aa(A,$o,R7,X2) )
=> aa(A,$o,Uub,X2) ) ) ) ).
% ATP.lambda_457
tff(fact_6216_ATP_Olambda__458,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_qz(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A7: A,M03: multiset(A),K8: multiset(A)] :
( ( Uub = add_mset(A,A7,M03) )
& ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K8) )
& ! [B7: A] :
( member(A,B7,aa(multiset(A),set(A),set_mset(A),K8))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),A7),Uu) ) ) ) ).
% ATP.lambda_458
tff(fact_6217_ATP_Olambda__459,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_uy(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ? [Us3: list(A),Z3: A,Z10: A,Vs3: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Vs3)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Z10),Uu)
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z10),Vs3)) ) ) ) ).
% ATP.lambda_459
tff(fact_6218_ATP_Olambda__460,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_ih(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).
% ATP.lambda_460
tff(fact_6219_ATP_Olambda__461,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] :
aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_jc(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).
% ATP.lambda_461
tff(fact_6220_ATP_Olambda__462,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_ni(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),aa(B,A,the_inv_into(A,B,Uua,Uu),Uuc),Uub) ).
% ATP.lambda_462
tff(fact_6221_ATP_Olambda__463,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] :
aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_nh(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uua),Uu)),aa(B,A,the_inv_into(A,B,Uu,Uua),Uuc),Uub) ).
% ATP.lambda_463
tff(fact_6222_ATP_Olambda__464,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] :
aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_jz(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).
% ATP.lambda_464
tff(fact_6223_ATP_Olambda__465,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
aa(nat,A,Uua,Uuc),
$ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,minus_minus(nat,Uuc,aa(nat,nat,suc,zero_zero(nat))))) ) ) ).
% ATP.lambda_465
tff(fact_6224_ATP_Olambda__466,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hr(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).
% ATP.lambda_466
tff(fact_6225_ATP_Olambda__467,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_fa(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_467
tff(fact_6226_ATP_Olambda__468,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_xf(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_468
tff(fact_6227_ATP_Olambda__469,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fq(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_469
tff(fact_6228_ATP_Olambda__470,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_470
tff(fact_6229_ATP_Olambda__471,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_gf(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).
% ATP.lambda_471
tff(fact_6230_ATP_Olambda__472,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(A,option(B)),Uuc: A] :
aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_ub(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).
% ATP.lambda_472
tff(fact_6231_ATP_Olambda__473,axiom,
! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: B,Uub: A,Uuc: B] :
aa(B,option(A),aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_ue(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uua,aa(A,option(A),some(A),Uub),aa(B,option(A),Uu,Uuc)) ).
% ATP.lambda_473
tff(fact_6232_ATP_Olambda__474,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_bh(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ).
% ATP.lambda_474
tff(fact_6233_ATP_Olambda__475,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_as(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_475
tff(fact_6234_ATP_Olambda__476,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ff(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_476
tff(fact_6235_ATP_Olambda__477,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_lx(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_477
tff(fact_6236_ATP_Olambda__478,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ki(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_kh(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_478
tff(fact_6237_ATP_Olambda__479,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_kf(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_ke(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_479
tff(fact_6238_ATP_Olambda__480,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_uh(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,compow(fun(A,A),Uuc,aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).
% ATP.lambda_480
tff(fact_6239_ATP_Olambda__481,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_pf(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).
% ATP.lambda_481
tff(fact_6240_ATP_Olambda__482,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,B),Uub: A,Uuc: C] :
( aa(C,$o,aa(A,fun(C,$o),aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_xa(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Uu),Uua),Uub),Uuc)
<=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(C,B,Uua,Uuc)) ) ).
% ATP.lambda_482
tff(fact_6241_ATP_Olambda__483,axiom,
! [B: $tType,A: $tType,C: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_co(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_483
tff(fact_6242_ATP_Olambda__484,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: filter(B),Uuc: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aa(set(B),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_te(fun(B,A),fun(set(B),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_td(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_484
tff(fact_6243_ATP_Olambda__485,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: filter(B),Uuc: fun(A,$o)] :
( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_qr(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
<=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_qq(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_485
tff(fact_6244_ATP_Olambda__486,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_wu(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(B,$o,aa(A,fun(B,$o),Uu,Uub),Uuc)
& aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).
% ATP.lambda_486
tff(fact_6245_ATP_Olambda__487,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: set(A),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_lu(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),product_Sigma(A,A,Uua,aTP_Lamp_al(set(A),fun(A,set(A)),Uua)))
& aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uuc) ) ) ).
% ATP.lambda_487
tff(fact_6246_ATP_Olambda__488,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_kz(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_488
tff(fact_6247_ATP_Olambda__489,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_js(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uu)
& ~ member(A,Uub,Uua)
& ~ member(A,Uuc,Uua) ) ) ).
% ATP.lambda_489
tff(fact_6248_ATP_Olambda__490,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_kp(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
| ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua) ) ) ) ).
% ATP.lambda_490
tff(fact_6249_ATP_Olambda__491,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jt(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).
% ATP.lambda_491
tff(fact_6250_ATP_Olambda__492,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_tz(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
& aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).
% ATP.lambda_492
tff(fact_6251_ATP_Olambda__493,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,minus_minus(nat,Uub,Uuc))) ) ).
% ATP.lambda_493
tff(fact_6252_ATP_Olambda__494,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_he(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),minus_minus(nat,Uub,Uuc))) ).
% ATP.lambda_494
tff(fact_6253_ATP_Olambda__495,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),minus_minus(nat,Uub,Uuc))) ) ).
% ATP.lambda_495
tff(fact_6254_ATP_Olambda__496,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ux(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
& ? [Xys2: list(A),X2: A,Y3: A,Xs7: list(A),Ys5: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs7)) )
& ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3),Uu) ) ) ) ).
% ATP.lambda_496
tff(fact_6255_ATP_Olambda__497,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ln(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),minus_minus(nat,Uua,aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).
% ATP.lambda_497
tff(fact_6256_ATP_Olambda__498,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_cj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,aa(set(B),set(A),image2(B,A,Uu),Uua))
& aa(A,$o,Uub,Uuc) ) ) ).
% ATP.lambda_498
tff(fact_6257_ATP_Olambda__499,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_em(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(B,Uuc,Uu)
& aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).
% ATP.lambda_499
tff(fact_6258_ATP_Olambda__500,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_el(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uu)
& aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).
% ATP.lambda_500
tff(fact_6259_ATP_Olambda__501,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_gd(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uu)
& ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_501
tff(fact_6260_ATP_Olambda__502,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_lb(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uua)
& ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).
% ATP.lambda_502
tff(fact_6261_ATP_Olambda__503,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_lo(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),minus_minus(nat,Uua,Uuc))) ) ).
% ATP.lambda_503
tff(fact_6262_ATP_Olambda__504,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hc(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),minus_minus(nat,Uub,Uuc))) ).
% ATP.lambda_504
tff(fact_6263_ATP_Olambda__505,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ri(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).
% ATP.lambda_505
tff(fact_6264_ATP_Olambda__506,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_rg(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_506
tff(fact_6265_ATP_Olambda__507,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_vy(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).
% ATP.lambda_507
tff(fact_6266_ATP_Olambda__508,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_vz(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).
% ATP.lambda_508
tff(fact_6267_ATP_Olambda__509,axiom,
! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,$o),Uuc: fun(B,$o)] :
( aa(fun(B,$o),$o,aa(fun(A,$o),fun(fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_ot(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
& aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Uuc),Uua) ) ) ).
% ATP.lambda_509
tff(fact_6268_ATP_Olambda__510,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( ( Uu = Uub )
& ( Uua = Uuc ) ) ) ).
% ATP.lambda_510
tff(fact_6269_ATP_Olambda__511,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ck(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(B,Uuc,Uua)
& aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).
% ATP.lambda_511
tff(fact_6270_ATP_Olambda__512,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_dw(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).
% ATP.lambda_512
tff(fact_6271_ATP_Olambda__513,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_de(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uu)
& ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).
% ATP.lambda_513
tff(fact_6272_ATP_Olambda__514,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: set(B),Uub: fun(B,A),Uuc: B] :
( aa(B,$o,aa(fun(B,A),fun(B,$o),aa(set(B),fun(fun(B,A),fun(B,$o)),aTP_Lamp_xc(A,fun(set(B),fun(fun(B,A),fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(B,Uuc,Uua)
& ( aa(B,A,Uub,Uuc) != Uu ) ) ) ).
% ATP.lambda_514
tff(fact_6273_ATP_Olambda__515,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ui(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).
% ATP.lambda_515
tff(fact_6274_ATP_Olambda__516,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,$o),Uub: fun(A,B),Uuc: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_oa(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
& member(A,Uuc,Uu) ) ) ).
% ATP.lambda_516
tff(fact_6275_ATP_Olambda__517,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_td(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(A,$o,Uub,aa(B,A,Uu,Uuc))
& member(B,Uuc,Uua) ) ) ).
% ATP.lambda_517
tff(fact_6276_ATP_Olambda__518,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: fun(A,$o),Uuc: B] :
( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_qq(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(A,$o,Uub,aa(B,A,Uua,Uuc))
& member(B,Uuc,Uu) ) ) ).
% ATP.lambda_518
tff(fact_6277_ATP_Olambda__519,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_jo(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(A,$o,Uu,Uub)
& aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).
% ATP.lambda_519
tff(fact_6278_ATP_Olambda__520,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_kb(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_ka(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).
% ATP.lambda_520
tff(fact_6279_ATP_Olambda__521,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_wg(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).
% ATP.lambda_521
tff(fact_6280_ATP_Olambda__522,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( semiring_0(A)
& comm_monoid_add(B)
& times(B) )
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_sl(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_522
tff(fact_6281_ATP_Olambda__523,axiom,
! [B: $tType,A: $tType,C: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_co(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_523
tff(fact_6282_ATP_Olambda__524,axiom,
! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_tl(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).
% ATP.lambda_524
tff(fact_6283_ATP_Olambda__525,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [Uu: fun(A,C),Uua: fun(B,C),Uub: A,Uuc: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_mx(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,Uub)),aa(B,C,Uua,Uuc)) ) ).
% ATP.lambda_525
tff(fact_6284_ATP_Olambda__526,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_ox(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(filter(D),filter(product_prod(C,D)),aa(filter(C),fun(filter(D),filter(product_prod(C,D))),prod_filter(C,D),aa(A,filter(C),Uu,Uub)),aa(B,filter(D),Uua,Uuc)) ).
% ATP.lambda_526
tff(fact_6285_ATP_Olambda__527,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_iu(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).
% ATP.lambda_527
tff(fact_6286_ATP_Olambda__528,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ip(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).
% ATP.lambda_528
tff(fact_6287_ATP_Olambda__529,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_je(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(A,$o,Uu,Uub)
& aa(B,$o,Uua,Uuc) ) ) ).
% ATP.lambda_529
tff(fact_6288_ATP_Olambda__530,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_bw(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
=> aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ) ).
% ATP.lambda_530
tff(fact_6289_ATP_Olambda__531,axiom,
! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_nv(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Uu,Uuc))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua) ) ) ).
% ATP.lambda_531
tff(fact_6290_ATP_Olambda__532,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_1(C)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ge(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_gd(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_532
tff(fact_6291_ATP_Olambda__533,axiom,
! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kl($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
<=> ( (Uu)
& aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).
% ATP.lambda_533
tff(fact_6292_ATP_Olambda__534,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_go(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_534
tff(fact_6293_ATP_Olambda__535,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_ii(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).
% ATP.lambda_535
tff(fact_6294_ATP_Olambda__536,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: nat,Uub: list(A),Uuc: nat] :
( aa(nat,$o,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_wn(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Uu),Uua),Uub),Uuc)
<=> aa(A,$o,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc)) ) ).
% ATP.lambda_536
tff(fact_6295_ATP_Olambda__537,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_vl(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).
% ATP.lambda_537
tff(fact_6296_ATP_Olambda__538,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_vm(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).
% ATP.lambda_538
tff(fact_6297_ATP_Olambda__539,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_jm(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_jl(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_539
tff(fact_6298_ATP_Olambda__540,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_jk(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_jj(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_540
tff(fact_6299_ATP_Olambda__541,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( semiring_0(A)
& comm_monoid_add(B)
& times(B) )
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: multiset(C),Uuc: B] : aa(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_sm(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_sl(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_541
tff(fact_6300_ATP_Olambda__542,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_oy(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_ox(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).
% ATP.lambda_542
tff(fact_6301_ATP_Olambda__543,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_ra(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),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_543
tff(fact_6302_ATP_Olambda__544,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_rc(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_544
tff(fact_6303_ATP_Olambda__545,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_re(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_545
tff(fact_6304_ATP_Olambda__546,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_rk(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_546
tff(fact_6305_ATP_Olambda__547,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_fw(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),image(B,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),bot_bot(set(B)))))) ).
% ATP.lambda_547
tff(fact_6306_ATP_Olambda__548,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_mp(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ? [Y3: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y3),Uu)
& member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc),Uua) ) ) ).
% ATP.lambda_548
tff(fact_6307_ATP_Olambda__549,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_mw(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu),Uua),Uub),Uuc)
<=> ? [A7: C] :
( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
& member(C,A7,Uu) ) ) ).
% ATP.lambda_549
tff(fact_6308_ATP_Olambda__550,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_mc(fun(B,A),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ? [L2: B] :
( ( Uuc = aa(B,A,Uu,L2) )
& ( ( L2 = Uua )
| member(B,L2,Uub) ) ) ) ).
% ATP.lambda_550
tff(fact_6309_ATP_Olambda__551,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
( aa(B,$o,aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_mo(A,fun(A,fun(fun(A,B),fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ? [I4: A] :
( ( Uuc = aa(A,B,Uub,I4) )
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),I4)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),I4),Uua) ) ) ) ).
% ATP.lambda_551
tff(fact_6310_ATP_Olambda__552,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_so(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Uu),Uua),Uub),Uuc)
<=> ? [A17: A,A25: A] :
( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A17),A25) )
& member(A,A17,Uu)
& member(A,A25,Uu)
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A17)),aa(A,B,Uub,A25)),Uua) ) ) ).
% ATP.lambda_552
tff(fact_6311_ATP_Olambda__553,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,A),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_xg(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),Uub),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc)),Uuc))) ).
% ATP.lambda_553
tff(fact_6312_ATP_Olambda__554,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(C)),Uuc: fun(C,A),Uud: B] : aa(B,A,aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_xe(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),Uuc),aa(B,set(C),Uub,Uud)) ).
% ATP.lambda_554
tff(fact_6313_ATP_Olambda__555,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] :
aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_kh(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_555
tff(fact_6314_ATP_Olambda__556,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] :
aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_ke(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert2(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_556
tff(fact_6315_ATP_Olambda__557,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_wk(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),quicksort_by_rel(A,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),aa(list(A),list(A),quicksort_by_rel(A,Uu,Uua),Uud))),Uuc) ).
% ATP.lambda_557
tff(fact_6316_ATP_Olambda__558,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_vb(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).
% ATP.lambda_558
tff(fact_6317_ATP_Olambda__559,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(A,$o)),Uua: fun(A,fun(C,$o)),Uub: B,Uuc: C,Uud: A] :
( aa(A,$o,aa(C,fun(A,$o),aa(B,fun(C,fun(A,$o)),aa(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o))),aTP_Lamp_ts(fun(B,fun(A,$o)),fun(fun(A,fun(C,$o)),fun(B,fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uud)
& aa(C,$o,aa(A,fun(C,$o),Uua,Uud),Uuc) ) ) ).
% ATP.lambda_559
tff(fact_6318_ATP_Olambda__560,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hv(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),minus_minus(nat,Uu,Uud))) ) ).
% ATP.lambda_560
tff(fact_6319_ATP_Olambda__561,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hs(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),minus_minus(nat,Uu,Uud))) ) ).
% ATP.lambda_561
tff(fact_6320_ATP_Olambda__562,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_trancl(A,Uub))
| ( Uuc = Uu ) )
& ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_trancl(A,Uub))
| ( Uud = Uua ) ) ) ) ).
% ATP.lambda_562
tff(fact_6321_ATP_Olambda__563,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ht(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),minus_minus(nat,Uu,Uud))) ) ).
% ATP.lambda_563
tff(fact_6322_ATP_Olambda__564,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ju(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uud),bot_bot(set(A))))))),aa(set(product_prod(A,A)),set(A),field2(A),Uu))
& ( ( ( Uua = Uuc )
& ( Uub = Uud ) )
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),minus_minus(set(product_prod(A,A)),Uu,id2(A)))
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc),minus_minus(set(product_prod(A,A)),Uu,id2(A))) )
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& ( Uua = Uuc )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),minus_minus(set(product_prod(A,A)),Uu,id2(A))) ) ) ) ) ).
% ATP.lambda_564
tff(fact_6323_ATP_Olambda__565,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_rtrancl(A,Uub))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_rtrancl(A,Uub)) ) ) ).
% ATP.lambda_565
tff(fact_6324_ATP_Olambda__566,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_pm(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( ( Uub = Uuc )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),Uu) ) ) ).
% ATP.lambda_566
tff(fact_6325_ATP_Olambda__567,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [Uu: fun(A,C),Uua: fun(B,C),Uub: set(A),Uuc: set(B),Uud: C] :
( aa(C,$o,aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_my(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ? [A7: A,B7: B] :
( ( Uud = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,A7)),aa(B,C,Uua,B7)) )
& member(A,A7,Uub)
& member(B,B7,Uuc) ) ) ) ).
% ATP.lambda_567
tff(fact_6326_ATP_Olambda__568,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_jl(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu)
| ( ( Uub = Uud )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua) ) ) ) ).
% ATP.lambda_568
tff(fact_6327_ATP_Olambda__569,axiom,
! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_jj(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
<=> ( ( Uub = Uud )
& aa(A,$o,Uu,Uud)
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).
% ATP.lambda_569
tff(fact_6328_ATP_Olambda__570,axiom,
! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_jh($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
<=> (Uu) ) ).
% ATP.lambda_570
tff(fact_6329_ATP_Olambda__571,axiom,
! [A: $tType,B: $tType,Uu: multiset(B),Uua: A] : aa(A,multiset(B),aTP_Lamp_gw(multiset(B),fun(A,multiset(B)),Uu),Uua) = Uu ).
% ATP.lambda_571
tff(fact_6330_ATP_Olambda__572,axiom,
! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_na(assn,fun(A,assn),Uu),Uua) = Uu ).
% ATP.lambda_572
tff(fact_6331_ATP_Olambda__573,axiom,
! [A: $tType,Uu: $o,Uua: A] :
( aa(A,$o,aTP_Lamp_aj($o,fun(A,$o),(Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_573
tff(fact_6332_ATP_Olambda__574,axiom,
! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_iv(set(D),fun(C,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_574
tff(fact_6333_ATP_Olambda__575,axiom,
! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_es(set(D),fun(B,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_575
tff(fact_6334_ATP_Olambda__576,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_dr(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_576
tff(fact_6335_ATP_Olambda__577,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_ds(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_577
tff(fact_6336_ATP_Olambda__578,axiom,
! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_bi(set(B),fun(B,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_578
tff(fact_6337_ATP_Olambda__579,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aw(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_579
tff(fact_6338_ATP_Olambda__580,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_au(set(A),fun(B,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_580
tff(fact_6339_ATP_Olambda__581,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_al(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_581
tff(fact_6340_ATP_Olambda__582,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_ow(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).
% ATP.lambda_582
tff(fact_6341_ATP_Olambda__583,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_in(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).
% ATP.lambda_583
tff(fact_6342_ATP_Olambda__584,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_wc(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).
% ATP.lambda_584
tff(fact_6343_ATP_Olambda__585,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_cf(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).
% ATP.lambda_585
tff(fact_6344_ATP_Olambda__586,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ao(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_586
tff(fact_6345_ATP_Olambda__587,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ap(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_587
tff(fact_6346_ATP_Olambda__588,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_bg(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_588
tff(fact_6347_ATP_Olambda__589,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_av(B,fun(A,B),Uu),Uua) = Uu ).
% ATP.lambda_589
tff(fact_6348_ATP_Olambda__590,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_eb(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_590
tff(fact_6349_ATP_Olambda__591,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_bd(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_591
tff(fact_6350_ATP_Olambda__592,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ba(A,fun(B,A)),Uu),Uua) = Uu ).
% ATP.lambda_592
tff(fact_6351_ATP_Olambda__593,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_io(B,fun(A,A)),Uu),Uua) = Uua ).
% ATP.lambda_593
tff(fact_6352_ATP_Olambda__594,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xo(A,fun(A,$o)),Uu),Uua)
<=> $false ) ).
% ATP.lambda_594
tff(fact_6353_ATP_Olambda__595,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_kn(A,fun(B,$o)),Uu),Uua)
<=> $true ) ).
% ATP.lambda_595
tff(fact_6354_ATP_Olambda__596,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_hd(nat,nat),Uu) = Uu ).
% ATP.lambda_596
tff(fact_6355_ATP_Olambda__597,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_iz(int,int),Uu) = Uu ).
% ATP.lambda_597
tff(fact_6356_ATP_Olambda__598,axiom,
! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_ae(B,B),Uu) = Uu ).
% ATP.lambda_598
tff(fact_6357_ATP_Olambda__599,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_ul(A,A),Uu) = Uu ) ).
% ATP.lambda_599
tff(fact_6358_ATP_Olambda__600,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_cs(A,A),Uu) = Uu ) ).
% ATP.lambda_600
tff(fact_6359_ATP_Olambda__601,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_tk(A,A),Uu) = Uu ) ).
% ATP.lambda_601
tff(fact_6360_ATP_Olambda__602,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ac(A,A),Uu) = Uu ).
% ATP.lambda_602
tff(fact_6361_ATP_Olambda__603,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_az(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_603
tff(fact_6362_ATP_Olambda__604,axiom,
! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_eu(C,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_604
tff(fact_6363_ATP_Olambda__605,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_di(B,set(A)),Uu) = bot_bot(set(A)) ).
% ATP.lambda_605
tff(fact_6364_ATP_Olambda__606,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_an(B,A),Uu) = bot_bot(A) ) ).
% ATP.lambda_606
tff(fact_6365_ATP_Olambda__607,axiom,
! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_ev(A,set(D)),Uu) = bot_bot(set(D)) ).
% ATP.lambda_607
tff(fact_6366_ATP_Olambda__608,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ax(A,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_608
tff(fact_6367_ATP_Olambda__609,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_ct(A,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_609
tff(fact_6368_ATP_Olambda__610,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_ak(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_610
tff(fact_6369_ATP_Olambda__611,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_ud(B,option(A)),Uu) = none(A) ).
% ATP.lambda_611
tff(fact_6370_ATP_Olambda__612,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_ua(A,option(B)),Uu) = none(B) ).
% ATP.lambda_612
tff(fact_6371_ATP_Olambda__613,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_ss(A,B),Uu) = undefined(B) ).
% ATP.lambda_613
tff(fact_6372_ATP_Olambda__614,axiom,
! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
<=> $false ) ).
% ATP.lambda_614
tff(fact_6373_ATP_Olambda__615,axiom,
! [Uu: nat] :
( aa(nat,$o,aTP_Lamp_ol(nat,$o),Uu)
<=> $false ) ).
% ATP.lambda_615
tff(fact_6374_ATP_Olambda__616,axiom,
! [A: $tType,Uu: A] :
( aa(A,$o,aTP_Lamp_cy(A,$o),Uu)
<=> $false ) ).
% ATP.lambda_616
tff(fact_6375_ATP_Olambda__617,axiom,
! [A: $tType,Uu: fun(A,$o)] :
( aa(fun(A,$o),$o,aTP_Lamp_qi(fun(A,$o),$o),Uu)
<=> $true ) ).
% ATP.lambda_617
tff(fact_6376_ATP_Olambda__618,axiom,
! [C: $tType,Uu: C] :
( aa(C,$o,aTP_Lamp_xu(C,$o),Uu)
<=> $true ) ).
% ATP.lambda_618
tff(fact_6377_ATP_Olambda__619,axiom,
! [A: $tType,Uu: A] :
( aa(A,$o,aTP_Lamp_ci(A,$o),Uu)
<=> $true ) ).
% ATP.lambda_619
tff(fact_6378_ATP_Olambda__620,axiom,
! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_wh(A,fun(nat,nat)),Uu) = suc ).
% ATP.lambda_620
% Type constructors (630)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_1,axiom,
bounded_lattice_top(assn) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_2,axiom,
! [A18: $tType] :
( bounded_lattice_top(A18)
=> bounded_lattice_top(option(A18)) ) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_3,axiom,
! [A18: $tType] : bounded_lattice_top(filter(A18)) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_4,axiom,
bounded_lattice_top($o) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_5,axiom,
! [A18: $tType] : bounded_lattice_top(set(A18)) ).
tff(tcon_fun___Lattices_Obounded__lattice__top_6,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounded_lattice_top(fun(A18,A19)) ) ).
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A18: $tType,A19: $tType] :
( comple6319245703460814977attice(A19)
=> condit1219197933456340205attice(fun(A18,A19)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A18: $tType,A19: $tType] :
( comple592849572758109894attice(A19)
=> comple592849572758109894attice(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounde4967611905675639751up_bot(fun(A18,A19)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A18: $tType,A19: $tType] :
( comple6319245703460814977attice(A19)
=> comple6319245703460814977attice(fun(A18,A19)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A18: $tType,A19: $tType] :
( boolea8198339166811842893lgebra(A19)
=> boolea8198339166811842893lgebra(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounded_lattice_bot(fun(A18,A19)) ) ).
tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A18: $tType,A19: $tType] :
( comple6319245703460814977attice(A19)
=> comple9053668089753744459l_ccpo(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A18: $tType,A19: $tType] :
( semilattice_sup(A19)
=> semilattice_sup(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A18: $tType,A19: $tType] :
( semilattice_inf(A19)
=> semilattice_inf(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A18: $tType,A19: $tType] :
( distrib_lattice(A19)
=> distrib_lattice(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounded_lattice(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Oorder__top,axiom,
! [A18: $tType,A19: $tType] :
( order_top(A19)
=> order_top(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Oorder__bot,axiom,
! [A18: $tType,A19: $tType] :
( order_bot(A19)
=> order_bot(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A18: $tType,A19: $tType] :
( preorder(A19)
=> preorder(fun(A18,A19)) ) ).
tff(tcon_fun___Finite__Set_Ofinite,axiom,
! [A18: $tType,A19: $tType] :
( ( finite_finite(A18)
& finite_finite(A19) )
=> finite_finite(fun(A18,A19)) ) ).
tff(tcon_fun___Lattices_Olattice,axiom,
! [A18: $tType,A19: $tType] :
( lattice(A19)
=> lattice(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A18: $tType,A19: $tType] :
( order(A19)
=> order(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Otop,axiom,
! [A18: $tType,A19: $tType] :
( top(A19)
=> top(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A18: $tType,A19: $tType] :
( ord(A19)
=> ord(fun(A18,A19)) ) ).
tff(tcon_fun___Orderings_Obot,axiom,
! [A18: $tType,A19: $tType] :
( bot(A19)
=> bot(fun(A18,A19)) ) ).
tff(tcon_fun___Groups_Ouminus,axiom,
! [A18: $tType,A19: $tType] :
( uminus(A19)
=> uminus(fun(A18,A19)) ) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder(int) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_7,axiom,
condit1219197933456340205attice(int) ).
tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations(int) ).
tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat(int) ).
tff(tcon_Int_Oint___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___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative(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_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor(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_Onormalization__semidom,axiom,
normal8620421768224518004emidom(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring(int) ).
tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_8,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_9,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Lattices_Odistrib__lattice_10,axiom,
distrib_lattice(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(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___Rings_Osemidom__modulo,axiom,
semidom_modulo(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide(int) ).
tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral(int) ).
tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn(int) ).
tff(tcon_Int_Oint___Orderings_Opreorder_11,axiom,
preorder(int) ).
tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top(int) ).
tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot(int) ).
tff(tcon_Int_Oint___Lattices_Olattice_12,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_13,axiom,
order(int) ).
tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral(int) ).
tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0(int) ).
tff(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(tcon_Int_Oint___Orderings_Oord_14,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_15,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(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___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_16,axiom,
condit6923001295902523014norder(nat) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_24,axiom,
normal6328177297339901930cative(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_25,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_26,axiom,
semiri6575147826004484403cancel(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_27,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_28,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_29,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_30,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_31,axiom,
linord2810124833399127020strict(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_33,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_34,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_35,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_37,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
linord8928482502909563296strict(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_41,axiom,
semiri6843258321239162965malize(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
ordere2520102378445227354miring(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_48,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_49,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring_50,axiom,
linordered_semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_51,axiom,
ordered_semiring_0(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_52,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_53,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_54,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_55,axiom,
distrib_lattice(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_56,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_57,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_58,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_59,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring_60,axiom,
ordered_semiring(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_61,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_62,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_63,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_64,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_65,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
order_bot(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
no_top(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_85,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Rings_Omult__zero_86,axiom,
mult_zero(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_87,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_88,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_89,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Orderings_Obot_90,axiom,
bot(nat) ).
tff(tcon_Nat_Onat___Groups_Otimes_91,axiom,
times(nat) ).
tff(tcon_Nat_Onat___Power_Opower_92,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_93,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_94,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_95,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_96,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_97,axiom,
heap(nat) ).
tff(tcon_Num_Onum___Orderings_Opreorder_98,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_99,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_100,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_101,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Otimes_102,axiom,
times(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_103,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_104,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_105,axiom,
semiri6575147826004484403cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_106,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_107,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_108,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_109,axiom,
linord2810124833399127020strict(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_110,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_111,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_112,axiom,
linord715952674999750819strict(rat) ).
tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_113,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_114,axiom,
linord181362715937106298miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_115,axiom,
linord8928482502909563296strict(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_116,axiom,
semiri3467727345109120633visors(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_117,axiom,
ordere6658533253407199908up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_118,axiom,
ordere166539214618696060dd_abs(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_119,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_120,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_121,axiom,
cancel2418104881723323429up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_122,axiom,
ring_15535105094025558882visors(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_123,axiom,
cancel1802427076303600483id_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_124,axiom,
linord4710134922213307826strict(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_125,axiom,
comm_s4317794764714335236cancel(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_126,axiom,
ordere2520102378445227354miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_127,axiom,
linord6961819062388156250ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_128,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_129,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring_130,axiom,
linordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_131,axiom,
ordered_semiring_0(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_132,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_133,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_134,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_135,axiom,
distrib_lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_136,axiom,
ab_semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_137,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_138,axiom,
ab_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring_139,axiom,
ordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_140,axiom,
ordered_ring_abs(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_141,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_142,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_143,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_144,axiom,
comm_semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_145,axiom,
comm_semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__mult_146,axiom,
semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom__divide_147,axiom,
semidom_divide(rat) ).
tff(tcon_Rat_Orat___Num_Osemiring__numeral_148,axiom,
semiring_numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__add_149,axiom,
semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__less__one_150,axiom,
zero_less_one(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring_151,axiom,
comm_semiring(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_152,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_153,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_154,axiom,
zero_neq_one(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring_155,axiom,
ordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_156,axiom,
idom_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_157,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_158,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__mult_159,axiom,
monoid_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_160,axiom,
comm_ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__add_161,axiom,
monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_162,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_163,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__top_164,axiom,
no_top(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__bot_165,axiom,
no_bot(rat) ).
tff(tcon_Rat_Orat___Lattices_Olattice_166,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_167,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Rings_Omult__zero_168,axiom,
mult_zero(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_169,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_170,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_171,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_172,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_173,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_174,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_175,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_176,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Otimes_177,axiom,
times(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Rat_Orat___Power_Opower_178,axiom,
power(rat) ).
tff(tcon_Rat_Orat___Num_Onumeral_179,axiom,
numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Ozero_180,axiom,
zero(rat) ).
tff(tcon_Rat_Orat___Rings_Oring_181,axiom,
ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom_182,axiom,
idom(rat) ).
tff(tcon_Rat_Orat___Groups_Oone_183,axiom,
one(rat) ).
tff(tcon_Rat_Orat___Rings_Odvd_184,axiom,
dvd(rat) ).
tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_185,axiom,
! [A18: $tType] : condit1219197933456340205attice(set(A18)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_186,axiom,
! [A18: $tType] : comple592849572758109894attice(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_187,axiom,
! [A18: $tType] : bounde4967611905675639751up_bot(set(A18)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_188,axiom,
! [A18: $tType] : comple6319245703460814977attice(set(A18)) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_189,axiom,
! [A18: $tType] : boolea8198339166811842893lgebra(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_190,axiom,
! [A18: $tType] : bounded_lattice_bot(set(A18)) ).
tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_191,axiom,
! [A18: $tType] : comple9053668089753744459l_ccpo(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_192,axiom,
! [A18: $tType] : semilattice_sup(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_193,axiom,
! [A18: $tType] : semilattice_inf(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Odistrib__lattice_194,axiom,
! [A18: $tType] : distrib_lattice(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_195,axiom,
! [A18: $tType] : bounded_lattice(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder__top_196,axiom,
! [A18: $tType] : order_top(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder__bot_197,axiom,
! [A18: $tType] : order_bot(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Opreorder_198,axiom,
! [A18: $tType] : preorder(set(A18)) ).
tff(tcon_Set_Oset___Finite__Set_Ofinite_199,axiom,
! [A18: $tType] :
( finite_finite(A18)
=> finite_finite(set(A18)) ) ).
tff(tcon_Set_Oset___Lattices_Olattice_200,axiom,
! [A18: $tType] : lattice(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder_201,axiom,
! [A18: $tType] : order(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Otop_202,axiom,
! [A18: $tType] : top(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oord_203,axiom,
! [A18: $tType] : ord(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Obot_204,axiom,
! [A18: $tType] : bot(set(A18)) ).
tff(tcon_Set_Oset___Groups_Ouminus_205,axiom,
! [A18: $tType] : uminus(set(A18)) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_206,axiom,
condit1219197933456340205attice($o) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_207,axiom,
comple592849572758109894attice($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_208,axiom,
bounde4967611905675639751up_bot($o) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_209,axiom,
comple6319245703460814977attice($o) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_210,axiom,
boolea8198339166811842893lgebra($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_211,axiom,
bounded_lattice_bot($o) ).
tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_212,axiom,
comple9053668089753744459l_ccpo($o) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_213,axiom,
semilattice_sup($o) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_214,axiom,
semilattice_inf($o) ).
tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_215,axiom,
distrib_lattice($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_216,axiom,
bounded_lattice($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder__top_217,axiom,
order_top($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder__bot_218,axiom,
order_bot($o) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_219,axiom,
preorder($o) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_220,axiom,
linorder($o) ).
tff(tcon_HOL_Obool___Finite__Set_Ofinite_221,axiom,
finite_finite($o) ).
tff(tcon_HOL_Obool___Lattices_Olattice_222,axiom,
lattice($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder_223,axiom,
order($o) ).
tff(tcon_HOL_Obool___Orderings_Otop_224,axiom,
top($o) ).
tff(tcon_HOL_Obool___Orderings_Oord_225,axiom,
ord($o) ).
tff(tcon_HOL_Obool___Orderings_Obot_226,axiom,
bot($o) ).
tff(tcon_HOL_Obool___Groups_Ouminus_227,axiom,
uminus($o) ).
tff(tcon_HOL_Obool___Heap_Oheap_228,axiom,
heap($o) ).
tff(tcon_Heap_Oref___Heap_Oheap_229,axiom,
! [A18: $tType] : heap(ref(A18)) ).
tff(tcon_List_Olist___Heap_Oheap_230,axiom,
! [A18: $tType] :
( heap(A18)
=> heap(list(A18)) ) ).
tff(tcon_Heap_Oarray___Heap_Oheap_231,axiom,
! [A18: $tType] : heap(array(A18)) ).
tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_232,axiom,
! [A18: $tType,A19: $tType] :
( ( finite_finite(A18)
& finite_finite(A19) )
=> finite_finite(sum_sum(A18,A19)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_233,axiom,
! [A18: $tType,A19: $tType] :
( ( heap(A18)
& heap(A19) )
=> heap(sum_sum(A18,A19)) ) ).
tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_234,axiom,
! [A18: $tType] : condit1219197933456340205attice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_235,axiom,
! [A18: $tType] : bounde4967611905675639751up_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_236,axiom,
! [A18: $tType] : comple6319245703460814977attice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_237,axiom,
! [A18: $tType] : bounded_lattice_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_238,axiom,
! [A18: $tType] : comple9053668089753744459l_ccpo(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_239,axiom,
! [A18: $tType] : semilattice_sup(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_240,axiom,
! [A18: $tType] : semilattice_inf(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_241,axiom,
! [A18: $tType] : distrib_lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_242,axiom,
! [A18: $tType] : bounded_lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__top_243,axiom,
! [A18: $tType] : order_top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_244,axiom,
! [A18: $tType] : order_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Opreorder_245,axiom,
! [A18: $tType] : preorder(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Olattice_246,axiom,
! [A18: $tType] : lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder_247,axiom,
! [A18: $tType] : order(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Otop_248,axiom,
! [A18: $tType] : top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oord_249,axiom,
! [A18: $tType] : ord(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Obot_250,axiom,
! [A18: $tType] : bot(filter(A18)) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_251,axiom,
! [A18: $tType] :
( comple5582772986160207858norder(A18)
=> condit6923001295902523014norder(option(A18)) ) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_252,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> condit1219197933456340205attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_253,axiom,
! [A18: $tType] :
( comple592849572758109894attice(A18)
=> comple592849572758109894attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_254,axiom,
! [A18: $tType] :
( lattice(A18)
=> bounde4967611905675639751up_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A18: $tType] :
( comple5582772986160207858norder(A18)
=> comple5582772986160207858norder(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_255,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> comple6319245703460814977attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_256,axiom,
! [A18: $tType] :
( lattice(A18)
=> bounded_lattice_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_257,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> comple9053668089753744459l_ccpo(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_258,axiom,
! [A18: $tType] :
( semilattice_sup(A18)
=> semilattice_sup(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_259,axiom,
! [A18: $tType] :
( semilattice_inf(A18)
=> semilattice_inf(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_260,axiom,
! [A18: $tType] :
( distrib_lattice(A18)
=> distrib_lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice_261,axiom,
! [A18: $tType] :
( bounded_lattice_top(A18)
=> bounded_lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Owellorder_262,axiom,
! [A18: $tType] :
( wellorder(A18)
=> wellorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__top_263,axiom,
! [A18: $tType] :
( order_top(A18)
=> order_top(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__bot_264,axiom,
! [A18: $tType] :
( order(A18)
=> order_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Opreorder_265,axiom,
! [A18: $tType] :
( preorder(A18)
=> preorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Olinorder_266,axiom,
! [A18: $tType] :
( linorder(A18)
=> linorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Finite__Set_Ofinite_267,axiom,
! [A18: $tType] :
( finite_finite(A18)
=> finite_finite(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Olattice_268,axiom,
! [A18: $tType] :
( lattice(A18)
=> lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder_269,axiom,
! [A18: $tType] :
( order(A18)
=> order(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Otop_270,axiom,
! [A18: $tType] :
( order_top(A18)
=> top(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oord_271,axiom,
! [A18: $tType] :
( preorder(A18)
=> ord(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Obot_272,axiom,
! [A18: $tType] :
( order(A18)
=> bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_273,axiom,
! [A18: $tType] :
( heap(A18)
=> heap(option(A18)) ) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_274,axiom,
bounde4967611905675639751up_bot(assn) ).
tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_275,axiom,
boolea8198339166811842893lgebra(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_276,axiom,
bounded_lattice_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_277,axiom,
semilattice_sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_278,axiom,
semilattice_inf(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_279,axiom,
distrib_lattice(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_280,axiom,
bounded_lattice(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_281,axiom,
ab_semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_282,axiom,
comm_monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_283,axiom,
semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__top_284,axiom,
order_top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_285,axiom,
order_bot(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Opreorder_286,axiom,
preorder(assn) ).
tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_287,axiom,
monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Olattice_288,axiom,
lattice(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder_289,axiom,
order(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Otop_290,axiom,
top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oord_291,axiom,
ord(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Obot_292,axiom,
bot(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ouminus_293,axiom,
uminus(assn) ).
tff(tcon_Assertions_Oassn___Groups_Otimes_294,axiom,
times(assn) ).
tff(tcon_Assertions_Oassn___Power_Opower_295,axiom,
power(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oone_296,axiom,
one(assn) ).
tff(tcon_Assertions_Oassn___Rings_Odvd_297,axiom,
dvd(assn) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_298,axiom,
heap(typerep) ).
tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_299,axiom,
! [A18: $tType] :
( preorder(A18)
=> ordere6658533253407199908up_add(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_300,axiom,
! [A18: $tType] : cancel2418104881723323429up_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_301,axiom,
! [A18: $tType] : cancel1802427076303600483id_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_302,axiom,
! [A18: $tType] : cancel_semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_303,axiom,
! [A18: $tType] : comm_monoid_diff(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_304,axiom,
! [A18: $tType] : ab_semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_305,axiom,
! [A18: $tType] : comm_monoid_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_306,axiom,
! [A18: $tType] : semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Orderings_Opreorder_307,axiom,
! [A18: $tType] :
( preorder(A18)
=> preorder(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_308,axiom,
! [A18: $tType] : monoid_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Orderings_Oorder_309,axiom,
! [A18: $tType] :
( preorder(A18)
=> order(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Orderings_Oord_310,axiom,
! [A18: $tType] :
( preorder(A18)
=> ord(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ozero_311,axiom,
! [A18: $tType] : zero(multiset(A18)) ).
tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_312,axiom,
! [A18: $tType,A19: $tType] :
( ( finite_finite(A18)
& finite_finite(A19) )
=> finite_finite(product_prod(A18,A19)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_313,axiom,
! [A18: $tType,A19: $tType] :
( ( heap(A18)
& heap(A19) )
=> heap(product_prod(A18,A19)) ) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_314,axiom,
condit6923001295902523014norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_315,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_316,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_317,axiom,
bounde4967611905675639751up_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_318,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_319,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_320,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_321,axiom,
bounded_lattice_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_322,axiom,
comple9053668089753744459l_ccpo(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_323,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_324,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_325,axiom,
distrib_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_326,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_327,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_328,axiom,
order_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_329,axiom,
order_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_330,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_331,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_332,axiom,
finite_finite(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_333,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_334,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Otop_335,axiom,
top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_336,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Obot_337,axiom,
bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_338,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_339,axiom,
heap(product_unit) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_340,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_341,axiom,
semiri1453513574482234551roduct(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_342,axiom,
euclid5411537665997757685th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_343,axiom,
euclid8789492081693882211th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_344,axiom,
ordere1937475149494474687imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_345,axiom,
euclid3128863361964157862miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_346,axiom,
euclid4440199948858584721cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_347,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_348,axiom,
euclid8851590272496341667cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_349,axiom,
semiri6575147826004484403cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_350,axiom,
strict9044650504122735259up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_351,axiom,
ordere580206878836729694up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_352,axiom,
ordere2412721322843649153imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_353,axiom,
bit_se359711467146920520ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_354,axiom,
linord2810124833399127020strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_355,axiom,
strict7427464778891057005id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_356,axiom,
ordere8940638589300402666id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_357,axiom,
euclid3725896446679973847miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_358,axiom,
linord715952674999750819strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_359,axiom,
linord4140545234300271783up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_360,axiom,
bit_ri3973907225187159222ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_361,axiom,
linord181362715937106298miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_362,axiom,
linord8928482502909563296strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_363,axiom,
semiri3467727345109120633visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_364,axiom,
ordere6658533253407199908up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_365,axiom,
ordere166539214618696060dd_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_366,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_367,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_368,axiom,
cancel2418104881723323429up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_369,axiom,
ring_15535105094025558882visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_370,axiom,
cancel1802427076303600483id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_371,axiom,
linord4710134922213307826strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_372,axiom,
comm_s4317794764714335236cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_373,axiom,
bit_semiring_bits(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_374,axiom,
ordere2520102378445227354miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_375,axiom,
linord6961819062388156250ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_376,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_377,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_378,axiom,
linordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_379,axiom,
ordered_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_380,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_381,axiom,
ab_semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_382,axiom,
algebraic_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_383,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_384,axiom,
ab_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_385,axiom,
ordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_386,axiom,
ordered_ring_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_387,axiom,
semiring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_388,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_389,axiom,
semiring_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_390,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_391,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_392,axiom,
comm_semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_393,axiom,
comm_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_394,axiom,
semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_395,axiom,
semidom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_396,axiom,
semidom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_397,axiom,
semiring_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_398,axiom,
semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_399,axiom,
zero_less_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_400,axiom,
comm_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_401,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_402,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_403,axiom,
zero_neq_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_404,axiom,
ordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_405,axiom,
idom_abs_sgn(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_406,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_407,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_408,axiom,
monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_409,axiom,
comm_ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_410,axiom,
monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_411,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_412,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_413,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
mult_zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
comm_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
neg_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
ring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_420,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_421,axiom,
uminus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_422,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_423,axiom,
times(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Power_Opower_424,axiom,
power(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_425,axiom,
numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_426,axiom,
zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring_427,axiom,
ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_428,axiom,
idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oone_429,axiom,
one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_430,axiom,
dvd(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_431,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_432,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_433,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_434,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_435,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_436,axiom,
semiri6575147826004484403cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_437,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_438,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_439,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_440,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_441,axiom,
linord2810124833399127020strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_442,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_443,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_444,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_445,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_446,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_447,axiom,
linord8928482502909563296strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_448,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_449,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_450,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_451,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_452,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_453,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_454,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_455,axiom,
ordere2520102378445227354miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_456,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_457,axiom,
linordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_458,axiom,
ordered_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_459,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_460,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_461,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_462,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_463,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_464,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_465,axiom,
ordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_466,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_467,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_468,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_469,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_470,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_471,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_472,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_473,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_474,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_475,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_476,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_477,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_478,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_479,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_480,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_481,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_482,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_483,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_484,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_485,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_486,axiom,
mult_zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_487,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_488,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_489,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Otimes_490,axiom,
times(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_491,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_492,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_493,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_494,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_495,axiom,
dvd(code_natural) ).
% Helper facts (6)
tff(help_fAll_1_1_U,axiom,
! [A: $tType,P: fun(A,$o),X: A] :
( ~ aa(fun(A,$o),$o,fAll(A),P)
| aa(A,$o,P,X) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: $o] :
( (P)
| aa($o,$o,fNot,(P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: $o] :
( ~ aa($o,$o,fNot,(P))
| ~ (P) ) ).
tff(help_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: $tType,P: fun(A,$o)] :
( aa(A,$o,P,fChoice(A,P))
= ( ? [X9: A] : aa(A,$o,P,X9) ) ) ).
% Conjectures (1)
tff(conj_0,conjecture,
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,one_assn_raw)),h)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),h) = bot_bot(set(nat)) ) ) ).
%------------------------------------------------------------------------------