TPTP Problem File: ITP218_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP218_2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_List_Assn 00019_000444
% 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 : 0062_VEBT_List_Assn_00019_000444 [Des22]
% Status : Theorem
% Rating : 0.50 v8.1.0
% Syntax : Number of formulae : 8556 (1901 unt;1287 typ; 0 def)
% Number of atoms : 19927 (6949 equ)
% Maximal formula atoms : 22 ( 2 avg)
% Number of connectives : 14738 (2080 ~; 231 |;1364 &)
% (1440 <=>;9623 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Maximal term depth : 15 ( 2 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 1203 ( 883 >; 320 *; 0 +; 0 <<)
% Number of predicates : 265 ( 262 usr; 2 prp; 0-7 aty)
% Number of functors : 1013 (1013 usr; 22 con; 0-7 aty)
% Number of variables : 27918 (25486 !; 505 ?;27918 :)
% (1927 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 16:49:10.143
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
tff(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
tff(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
tff(ty_t_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_HOL_Obool,type,
bool: $tType ).
tff(ty_t_Set_Oset,type,
set: $tType > $tType ).
tff(ty_t_Rat_Orat,type,
rat: $tType ).
tff(ty_t_Num_Onum,type,
num: $tType ).
tff(ty_t_Nat_Onat,type,
nat: $tType ).
tff(ty_t_Int_Oint,type,
int: $tType ).
tff(ty_t_itself,type,
itself: $tType > $tType ).
tff(ty_t_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tf_b,type,
b: $tType ).
tff(ty_tf_a,type,
a: $tType ).
% Explicit typings (1262)
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_Ominus,type,
minus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
tff(sy_cl_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_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_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_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
tff(sy_cl_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_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aa____,type,
aTP_Lamp_aa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ab____,type,
aTP_Lamp_ab:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__af____,type,
aTP_Lamp_af:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ag____,type,
aTP_Lamp_ag:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ah____,type,
aTP_Lamp_ah:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__aj____,type,
aTP_Lamp_aj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ak____,type,
aTP_Lamp_ak:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__al____,type,
aTP_Lamp_al:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__am____,type,
aTP_Lamp_am:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__an____,type,
aTP_Lamp_an:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ao____,type,
aTP_Lamp_ao:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ap____,type,
aTP_Lamp_ap:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bb____,type,
aTP_Lamp_bb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bc____,type,
aTP_Lamp_bc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bd____,type,
aTP_Lamp_bd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__be____,type,
aTP_Lamp_be:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bf____,type,
aTP_Lamp_bf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bg____,type,
aTP_Lamp_bg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bh____,type,
aTP_Lamp_bh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bi____,type,
aTP_Lamp_bi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bj____,type,
aTP_Lamp_bj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bk____,type,
aTP_Lamp_bk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bl____,type,
aTP_Lamp_bl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bm____,type,
aTP_Lamp_bm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bn____,type,
aTP_Lamp_bn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bo____,type,
aTP_Lamp_bo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bp____,type,
aTP_Lamp_bp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bq____,type,
aTP_Lamp_bq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__br____,type,
aTP_Lamp_br:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bs____,type,
aTP_Lamp_bs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bt____,type,
aTP_Lamp_bt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bu____,type,
aTP_Lamp_bu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bv____,type,
aTP_Lamp_bv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bw____,type,
aTP_Lamp_bw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cb____,type,
aTP_Lamp_cb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cc____,type,
aTP_Lamp_cc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cd____,type,
aTP_Lamp_cd:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cj____,type,
aTP_Lamp_cj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ck____,type,
aTP_Lamp_ck:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cl____,type,
aTP_Lamp_cl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cm____,type,
aTP_Lamp_cm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cn____,type,
aTP_Lamp_cn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__co____,type,
aTP_Lamp_co:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ds____,type,
aTP_Lamp_ds:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dt____,type,
aTP_Lamp_dt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__du____,type,
aTP_Lamp_du:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dv____,type,
aTP_Lamp_dv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dw____,type,
aTP_Lamp_dw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dx____,type,
aTP_Lamp_dx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dy____,type,
aTP_Lamp_dy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dz____,type,
aTP_Lamp_dz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ea____,type,
aTP_Lamp_ea:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eb____,type,
aTP_Lamp_eb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ec____,type,
aTP_Lamp_ec:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ed____,type,
aTP_Lamp_ed:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ee____,type,
aTP_Lamp_ee:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ef____,type,
aTP_Lamp_ef:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eg____,type,
aTP_Lamp_eg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eh____,type,
aTP_Lamp_eh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ei____,type,
aTP_Lamp_ei:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ej____,type,
aTP_Lamp_ej:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ek____,type,
aTP_Lamp_ek:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__el____,type,
aTP_Lamp_el:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__em____,type,
aTP_Lamp_em:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__en____,type,
aTP_Lamp_en:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eo____,type,
aTP_Lamp_eo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eq____,type,
aTP_Lamp_eq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__er____,type,
aTP_Lamp_er:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__es____,type,
aTP_Lamp_es:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__et____,type,
aTP_Lamp_et:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eu____,type,
aTP_Lamp_eu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ew____,type,
aTP_Lamp_ew:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ex____,type,
aTP_Lamp_ex:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ez____,type,
aTP_Lamp_ez:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fa____,type,
aTP_Lamp_fa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fb____,type,
aTP_Lamp_fb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fc____,type,
aTP_Lamp_fc:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fi____,type,
aTP_Lamp_fi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fj____,type,
aTP_Lamp_fj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fk____,type,
aTP_Lamp_fk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fl____,type,
aTP_Lamp_fl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fm____,type,
aTP_Lamp_fm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fn____,type,
aTP_Lamp_fn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fo____,type,
aTP_Lamp_fo:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fs____,type,
aTP_Lamp_fs:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ft____,type,
aTP_Lamp_ft:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fu____,type,
aTP_Lamp_fu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fv____,type,
aTP_Lamp_fv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fw____,type,
aTP_Lamp_fw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fx____,type,
aTP_Lamp_fx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fy____,type,
aTP_Lamp_fy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fz____,type,
aTP_Lamp_fz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ga____,type,
aTP_Lamp_ga:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gi____,type,
aTP_Lamp_gi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gj____,type,
aTP_Lamp_gj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gk____,type,
aTP_Lamp_gk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gl____,type,
aTP_Lamp_gl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gm____,type,
aTP_Lamp_gm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gn____,type,
aTP_Lamp_gn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__go____,type,
aTP_Lamp_go:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gp____,type,
aTP_Lamp_gp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gq____,type,
aTP_Lamp_gq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gr____,type,
aTP_Lamp_gr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gs____,type,
aTP_Lamp_gs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gt____,type,
aTP_Lamp_gt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gu____,type,
aTP_Lamp_gu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gv____,type,
aTP_Lamp_gv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gw____,type,
aTP_Lamp_gw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gx____,type,
aTP_Lamp_gx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gy____,type,
aTP_Lamp_gy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hl____,type,
aTP_Lamp_hl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hm____,type,
aTP_Lamp_hm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ht____,type,
aTP_Lamp_ht:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hu____,type,
aTP_Lamp_hu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hy____,type,
aTP_Lamp_hy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hz____,type,
aTP_Lamp_hz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ia____,type,
aTP_Lamp_ia:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ic____,type,
aTP_Lamp_ic:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__id____,type,
aTP_Lamp_id:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ie____,type,
aTP_Lamp_ie:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__if____,type,
aTP_Lamp_if:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ig____,type,
aTP_Lamp_ig:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__il____,type,
aTP_Lamp_il:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__im____,type,
aTP_Lamp_im:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__in____,type,
aTP_Lamp_in:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__io____,type,
aTP_Lamp_io:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ip____,type,
aTP_Lamp_ip:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iq____,type,
aTP_Lamp_iq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ir____,type,
aTP_Lamp_ir:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__is____,type,
aTP_Lamp_is:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__it____,type,
aTP_Lamp_it:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iu____,type,
aTP_Lamp_iu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__iw____,type,
aTP_Lamp_iw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ix____,type,
aTP_Lamp_ix:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iy____,type,
aTP_Lamp_iy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iz____,type,
aTP_Lamp_iz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jb____,type,
aTP_Lamp_jb:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kg____,type,
aTP_Lamp_kg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kh____,type,
aTP_Lamp_kh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ki____,type,
aTP_Lamp_ki:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kn____,type,
aTP_Lamp_kn:
!>[A: $tType,B: $tType] : 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kt____,type,
aTP_Lamp_kt:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lj____,type,
aTP_Lamp_lj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lk____,type,
aTP_Lamp_lk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ll____,type,
aTP_Lamp_ll:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lm____,type,
aTP_Lamp_lm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lo____,type,
aTP_Lamp_lo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lp____,type,
aTP_Lamp_lp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lq____,type,
aTP_Lamp_lq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lr____,type,
aTP_Lamp_lr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lv____,type,
aTP_Lamp_lv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lw____,type,
aTP_Lamp_lw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lx____,type,
aTP_Lamp_lx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ly____,type,
aTP_Lamp_ly:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lz____,type,
aTP_Lamp_lz:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mc____,type,
aTP_Lamp_mc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__md____,type,
aTP_Lamp_md:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mi____,type,
aTP_Lamp_mi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mj____,type,
aTP_Lamp_mj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mk____,type,
aTP_Lamp_mk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mm____,type,
aTP_Lamp_mm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mn____,type,
aTP_Lamp_mn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mo____,type,
aTP_Lamp_mo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mp____,type,
aTP_Lamp_mp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mq____,type,
aTP_Lamp_mq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mr____,type,
aTP_Lamp_mr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ms____,type,
aTP_Lamp_ms:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mv____,type,
aTP_Lamp_mv:
!>[A: $tType,B: $tType] : 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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ns____,type,
aTP_Lamp_ns:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nt____,type,
aTP_Lamp_nt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nz____,type,
aTP_Lamp_nz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oa____,type,
aTP_Lamp_oa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ob____,type,
aTP_Lamp_ob:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oc____,type,
aTP_Lamp_oc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oe____,type,
aTP_Lamp_oe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__of____,type,
aTP_Lamp_of:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__og____,type,
aTP_Lamp_og:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oh____,type,
aTP_Lamp_oh:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ol____,type,
aTP_Lamp_ol:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__om____,type,
aTP_Lamp_om:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__on____,type,
aTP_Lamp_on:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oo____,type,
aTP_Lamp_oo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__or____,type,
aTP_Lamp_or:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__os____,type,
aTP_Lamp_os:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ot____,type,
aTP_Lamp_ot:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ou____,type,
aTP_Lamp_ou:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ov____,type,
aTP_Lamp_ov:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pt____,type,
aTP_Lamp_pt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pv____,type,
aTP_Lamp_pv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pw____,type,
aTP_Lamp_pw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__px____,type,
aTP_Lamp_px:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__py____,type,
aTP_Lamp_py:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qa____,type,
aTP_Lamp_qa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qb____,type,
aTP_Lamp_qb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qc____,type,
aTP_Lamp_qc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qd____,type,
aTP_Lamp_qd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qe____,type,
aTP_Lamp_qe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qf____,type,
aTP_Lamp_qf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qg____,type,
aTP_Lamp_qg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qh____,type,
aTP_Lamp_qh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qi____,type,
aTP_Lamp_qi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ql____,type,
aTP_Lamp_ql:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qm____,type,
aTP_Lamp_qm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qn____,type,
aTP_Lamp_qn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qo____,type,
aTP_Lamp_qo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qp____,type,
aTP_Lamp_qp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qq____,type,
aTP_Lamp_qq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qr____,type,
aTP_Lamp_qr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qu____,type,
aTP_Lamp_qu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qv____,type,
aTP_Lamp_qv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qw____,type,
aTP_Lamp_qw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qx____,type,
aTP_Lamp_qx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qy____,type,
aTP_Lamp_qy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qz____,type,
aTP_Lamp_qz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ra____,type,
aTP_Lamp_ra:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rb____,type,
aTP_Lamp_rb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rc____,type,
aTP_Lamp_rc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rd____,type,
aTP_Lamp_rd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__re____,type,
aTP_Lamp_re:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rf____,type,
aTP_Lamp_rf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rg____,type,
aTP_Lamp_rg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rh____,type,
aTP_Lamp_rh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ri____,type,
aTP_Lamp_ri:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rj____,type,
aTP_Lamp_rj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rk____,type,
aTP_Lamp_rk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rl____,type,
aTP_Lamp_rl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rm____,type,
aTP_Lamp_rm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rn____,type,
aTP_Lamp_rn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ro____,type,
aTP_Lamp_ro:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rp____,type,
aTP_Lamp_rp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rq____,type,
aTP_Lamp_rq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rr____,type,
aTP_Lamp_rr:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__se____,type,
aTP_Lamp_se:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sg____,type,
aTP_Lamp_sg:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tb____,type,
aTP_Lamp_tb:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tg____,type,
aTP_Lamp_tg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__th____,type,
aTP_Lamp_th:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ti____,type,
aTP_Lamp_ti:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tj____,type,
aTP_Lamp_tj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tk____,type,
aTP_Lamp_tk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tl____,type,
aTP_Lamp_tl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tm____,type,
aTP_Lamp_tm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tn____,type,
aTP_Lamp_tn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__to____,type,
aTP_Lamp_to:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tp____,type,
aTP_Lamp_tp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tq____,type,
aTP_Lamp_tq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tr____,type,
aTP_Lamp_tr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ts____,type,
aTP_Lamp_ts:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tt____,type,
aTP_Lamp_tt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tu____,type,
aTP_Lamp_tu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tv____,type,
aTP_Lamp_tv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tw____,type,
aTP_Lamp_tw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tx____,type,
aTP_Lamp_tx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ty____,type,
aTP_Lamp_ty:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tz____,type,
aTP_Lamp_tz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ua____,type,
aTP_Lamp_ua:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ub____,type,
aTP_Lamp_ub:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uc____,type,
aTP_Lamp_uc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ud____,type,
aTP_Lamp_ud:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ue____,type,
aTP_Lamp_ue:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__va____,type,
aTP_Lamp_va:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).
tff(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).
tff(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).
tff(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn) ).
tff(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
tff(sy_c_Assertions_Oentails,type,
entails: fun(assn,fun(assn,bool)) ).
tff(sy_c_Assertions_Oentailst,type,
entailst: ( assn * assn ) > $o ).
tff(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( fun(A,assn) > assn ) ).
tff(sy_c_Assertions_Oin__range,type,
in_range: fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).
tff(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
tff(sy_c_Assertions_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)),bool) ).
tff(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
tff(sy_c_Assertions_Oprecise,type,
precise:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,assn)) > $o ) ).
tff(sy_c_Assertions_Oproper,type,
proper: fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool) ).
tff(sy_c_Assertions_Opure__assn,type,
pure_assn: bool > assn ).
tff(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( bool > fun(product_prod(A,set(B)),bool) ) ).
tff(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : fun(product_prod(bool,product_prod(A,set(B))),fun(product_prod(bool,product_prod(A,set(B))),bool)) ).
tff(sy_c_Assertions_OrelH,type,
relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).
tff(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).
tff(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array(A) * list(A) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) ).
tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref(A) * A ) > assn ) ).
tff(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref(A) * A ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) ).
tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).
tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Owand__assn,type,
wand_assn: ( assn * assn ) > assn ).
tff(sy_c_Assertions_Owand__raw,type,
wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).
tff(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Automation_OFI,type,
fi: ( list(product_prod(assn,assn)) * assn * assn * assn * assn * assn ) > $o ).
tff(sy_c_Automation_OFI__QUERY,type,
fI_QUERY: ( assn * assn * assn ) > $o ).
tff(sy_c_Automation_OFI__RESULT,type,
fI_RESULT: ( list(product_prod(assn,assn)) * assn * assn * assn ) > $o ).
tff(sy_c_Automation_OSLN,type,
sln: assn ).
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_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_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).
tff(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(set(A),fun(set(B),bool)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set(list(A)) * A ) > set(list(A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( fun(list(A),B) * A * list(A) ) > B ) ).
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_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__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * 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__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).
tff(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).
tff(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).
tff(sy_c_Binomial_Obinomial,type,
binomial: ( nat * nat ) > nat ).
tff(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > fun(nat,A) ) ).
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 * nat ) > bool ) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself(A) * nat ) > bool ) ).
tff(sy_c_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__axioms,type,
boolea6902313364301356556axioms:
!>[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_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * fun(A,fun(A,A)) ) > $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,bool)) * fun(A,bool) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,A) ) > A ) ).
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,bool)) * set(A) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : fun(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),fun(fun(A,B),bool))) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit8047198070973881523_above:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit8119078960628432327_below:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).
tff(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * 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_Oeventually,type,
eventually:
!>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( fun(fun(A,bool),bool) > filter(A) ) ).
tff(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).
tff(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( set(A) > filter(A) ) ).
tff(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).
tff(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(filter(A),fun(filter(B),bool)) ) ).
tff(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( set(B) > nat ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : fun(set(A),bool) ).
tff(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) * B ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).
tff(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).
tff(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).
tff(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).
tff(sy_c_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_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] : fun(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] : ( A > A ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : fun(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,bool) > A ) ).
tff(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
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_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).
tff(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( list(char) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > heap_Time_Heap(product_unit) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set(A) * nat ) > A ) ).
tff(sy_c_Int_Onat,type,
nat2: int > nat ).
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,type,
semilattice:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
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,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices__Big_Olinorder__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,bool) * B ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Lattices__Big_Osemilattice__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,bool)) * fun(A,fun(A,bool)) ) > $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)) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).
tff(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).
tff(sy_c_List_Oall__interval__int,type,
all_interval_int: ( fun(int,bool) * int * int ) > $o ).
tff(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( fun(nat,bool) * nat * nat ) > $o ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).
tff(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).
tff(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list(A) * fun(A,list(B)) ) > list(B) ) ).
tff(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( fun(A,bool) * set(A) ) > $o ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : fun(list(list(A)),list(A)) ).
tff(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( list(A) > set(A) ) ).
tff(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( list(A) > fun(A,nat) ) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : fun(list(A),bool) ).
tff(sy_c_List_Odistinct__adj,type,
distinct_adj:
!>[A: $tType] : fun(list(A),bool) ).
tff(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : fun(nat,fun(list(A),list(A))) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).
tff(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).
tff(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).
tff(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).
tff(sy_c_List_Ofind,type,
find:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),option(A))) ).
tff(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))) ).
tff(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).
tff(sy_c_List_Ofolding__insort__key__axioms,type,
foldin3648464208017769352axioms:
!>[B: $tType,A: $tType] : ( ( set(B) * fun(B,A) ) > $o ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : fun(fun(B,fun(A,B)),fun(B,fun(list(A),B))) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))) ).
tff(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > fun(list(A),nat) ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : fun(A,fun(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)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).
tff(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(B,A),fun(B,fun(list(B),list(B)))) ) ).
tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(B,A),fun(set(B),list(B))) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( fun(B,A) > fun(B,fun(list(B),list(B))) ) ).
tff(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(list(B),list(B))) ).
tff(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : fun(set(A),list(A)) ).
tff(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : fun(B,fun(fun(A,fun(list(A),B)),fun(list(A),B))) ).
tff(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : fun(list(A),A) ).
tff(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),bool)) ).
tff(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool))) ).
tff(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(list(A),list(Aa))) ).
tff(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : fun(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C))) ).
tff(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : fun(list(A),set(A)) ).
tff(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( fun(A,nat) > fun(list(A),nat) ) ).
tff(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),bool)) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : fun(list(A),fun(nat,fun(A,list(A)))) ).
tff(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).
tff(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).
tff(sy_c_List_Olists,type,
lists:
!>[A: $tType] : fun(set(A),set(list(A))) ).
tff(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : fun(list(set(A)),set(list(A))) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( fun(A,bool) > fun(list(A),bool) ) ).
tff(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).
tff(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : fun(fun(A,B),fun(list(A),list(B))) ).
tff(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) * list(B) ) > list(B) ) ).
tff(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(list(A),list(B))),fun(product_prod(fun(A,B),product_prod(list(A),list(B))),bool)) ).
tff(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( fun(A,list(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_Omember,type,
member:
!>[A: $tType] : ( ( list(A) * A ) > $o ) ).
tff(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).
tff(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( list(A) > fun(nat,A) ) ).
tff(sy_c_List_Onths,type,
nths:
!>[A: $tType] : fun(list(A),fun(set(nat),list(A))) ).
tff(sy_c_List_Onull,type,
null:
!>[A: $tType] : fun(list(A),bool) ).
tff(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),fun(list(A),bool)) ) ).
tff(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),fun(list(A),bool)) ) ).
tff(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),product_prod(list(A),list(A)))) ).
tff(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : fun(list(A),fun(list(B),list(product_prod(A,B)))) ).
tff(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : fun(list(list(A)),list(list(A))) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : fun(nat,fun(A,list(A))) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : fun(nat,fun(list(A),list(A))) ).
tff(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : fun(set(A),fun(set(list(A)),set(list(A)))) ).
tff(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : fun(list(A),fun(list(A),set(list(A)))) ).
tff(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).
tff(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : fun(list(A),list(list(A))) ).
tff(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : fun(fun(A,fun(A,bool)),fun(list(A),bool)) ).
tff(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_List_Otake,type,
take:
!>[A: $tType] : fun(nat,fun(list(A),list(A))) ).
tff(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).
tff(sy_c_List_Othose,type,
those:
!>[A: $tType] : fun(list(option(A)),option(list(A))) ).
tff(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).
tff(sy_c_List_Ounion,type,
union:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Oupt,type,
upt: ( nat * nat ) > list(nat) ).
tff(sy_c_List_Oupto,type,
upto: ( int * int ) > list(int) ).
tff(sy_c_List_Oupto__aux,type,
upto_aux: ( int * int * list(int) ) > list(int) ).
tff(sy_c_List_Oupto__rel,type,
upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : fun(list(A),fun(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_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).
tff(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).
tff(sy_c_Misc_OCODE__ABORT,type,
cODE_ABORT:
!>[A: $tType] : ( fun(product_unit,A) > A ) ).
tff(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( fun(A,bool) > option(A) ) ).
tff(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod(bool,A),product_prod(bool,B))) ) ).
tff(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( set(A) > option(set(A)) ) ).
tff(sy_c_Misc_Ofilter__rev,type,
filter_rev:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).
tff(sy_c_Misc_Ofilter__rev__aux,type,
filter_rev_aux:
!>[A: $tType] : ( list(A) > fun(fun(A,bool),fun(list(A),list(A))) ) ).
tff(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).
tff(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).
tff(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * 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,bool)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool)) ).
tff(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).
tff(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( fun(B,option(A)) * set(B) * A ) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),bool)) ).
tff(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).
tff(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).
tff(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),bool)) ).
tff(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( fun(A,bool) * product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool)) ).
tff(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).
tff(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),bool) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).
tff(sy_c_Misc_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))),bool)) ).
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) * multiset(A) ) > multiset(B) ) ).
tff(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset(A) * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).
tff(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
linord6283353356039996273ltiset:
!>[A: $tType] : ( multiset(A) > list(A) ) ).
tff(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( list(A) > multiset(A) ) ).
tff(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( set(B) > multiset(B) ) ).
tff(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( multiset(A) > fun(A,nat) ) ).
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] : ( multiset(A) > set(A) ) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list(A) * list(A) ) > option(bool) ) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).
tff(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset(A) * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( fun(A,nat) * multiset(A) ) > fun(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_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Num_OBitM,type,
bitM: num > num ).
tff(sy_c_Num_Oinc,type,
inc: num > num ).
tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_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_Opred__numeral,type,
pred_numeral: num > nat ).
tff(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Num_Osqr,type,
sqr: num > num ).
tff(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : option(A) ).
tff(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : fun(A,option(A)) ).
tff(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).
tff(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * option(A) ) > option(Aa) ) ).
tff(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(option(A),fun(option(B),bool)) ) ).
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_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_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,bool)) > fun(fun(A,bool),A) ) ).
tff(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,B),bool) ) ).
tff(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).
tff(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).
tff(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $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_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : fun(fun(set(C),B),fun(set(fun(A,C)),fun(A,B))) ).
tff(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,bool)),fun(fun(C,A),fun(fun(C,B),bool))) ).
tff(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),A) ) > $o ) ).
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_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(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_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).
tff(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).
tff(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).
tff(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
tff(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).
tff(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).
tff(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).
tff(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).
tff(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(A,bool) ) ).
tff(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * set(A) ) > set(B) ) ).
tff(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(B) ) ).
tff(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( set(product_prod(A,A)) > $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_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > bool ) ).
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(bool,A) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > set(A) ) ).
tff(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Set_Obind,type,
bind2:
!>[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),bool)) ).
tff(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( fun(A,bool) * set(A) ) > set(A) ) ).
tff(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).
tff(sy_c_Set_Oinsert,type,
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,bool)) * set(A) ) > $o ) ).
tff(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : fun(A,fun(set(A),set(A))) ).
tff(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : fun(fun(A,B),fun(set(B),set(A))) ).
tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).
tff(sy_c_Set__Interval_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] : ( A > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > set(A) ) ).
tff(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).
tff(sy_c_Syntax__Match_Oac__operator,type,
syntax_ac_operator:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : fun(set(product_prod(A,A)),set(product_prod(A,A))) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_VEBT__List__Assn_Olist__assn,type,
vEBT_List_list_assn:
!>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) * list(C) ) > assn ) ).
tff(sy_c_VEBT__List__Assn_Olist__assn__rel,type,
vEBT_L4249061453398456502sn_rel:
!>[A: $tType,C: $tType] : fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),bool)) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).
tff(sy_c_Wellfounded_Oless__than,type,
less_than: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(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,bool)) > $o ) ).
tff(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: ( bool * bool ) > bool ).
tff(sy_c_fdisj,type,
fdisj: ( bool * bool ) > bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member2:
!>[A: $tType] : ( ( A * set(A) ) > bool ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_P,type,
p: fun(a,fun(b,assn)) ).
tff(sy_v_l_H,type,
l: list(b) ).
% Relevant facts (6687)
tff(fact_0_pure__assn__eq__conv,axiom,
! [P: bool,Q: bool] :
( ( pure_assn(P) = pure_assn(Q) )
<=> ( pp(P)
<=> pp(Q) ) ) ).
% pure_assn_eq_conv
tff(fact_1_list__assn_Osimps_I1_J,axiom,
! [C: $tType,A: $tType,P: fun(A,fun(C,assn))] : vEBT_List_list_assn(A,C,P,nil(A),nil(C)) = one_one(assn) ).
% list_assn.simps(1)
tff(fact_2_is__pure__assn__pure,axiom,
! [P: bool] : is_pure_assn(pure_assn(P)) ).
% is_pure_assn_pure
tff(fact_3_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : ~ list_ex1(A,P,nil(A)) ).
% list_ex1_simps(1)
tff(fact_4_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : bind(B,A,nil(B),F) = nil(A) ).
% bind_simps(1)
tff(fact_5_member__rec_I2_J,axiom,
! [A: $tType,Y: A] : ~ member(A,nil(A),Y) ).
% member_rec(2)
tff(fact_6_is__pure__assnE,axiom,
! [A3: assn] :
( is_pure_assn(A3)
=> ~ ! [P2: bool] : A3 != pure_assn(P2) ) ).
% is_pure_assnE
tff(fact_7_is__pure__assn__def,axiom,
! [A3: assn] :
( is_pure_assn(A3)
<=> ? [P3: bool] : A3 = pure_assn(P3) ) ).
% is_pure_assn_def
tff(fact_8_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] : aa(list(A),nat,gen_length(A,N),nil(A)) = N ).
% gen_length_code(1)
tff(fact_9_list__assn_Osimps_I4_J,axiom,
! [C: $tType,A: $tType,Uu: fun(A,fun(C,assn)),V: C,Va: list(C)] : vEBT_List_list_assn(A,C,Uu,nil(A),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V),Va)) = bot_bot(assn) ).
% list_assn.simps(4)
tff(fact_10_list__assn_Osimps_I3_J,axiom,
! [C: $tType,A: $tType,Uu: fun(A,fun(C,assn)),V: A,Va: list(A)] : vEBT_List_list_assn(A,C,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(C)) = bot_bot(assn) ).
% list_assn.simps(3)
tff(fact_11_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list(A),Y21: A,Y22: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y22) )
<=> ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
tff(fact_12_pure__assn__eq__emp__iff,axiom,
! [P: bool] :
( ( pure_assn(P) = one_one(assn) )
<=> pp(P) ) ).
% pure_assn_eq_emp_iff
tff(fact_13_pure__true,axiom,
pure_assn(fTrue) = one_one(assn) ).
% pure_true
tff(fact_14_pure__assn__eq__false__iff,axiom,
! [P: bool] :
( ( pure_assn(P) = bot_bot(assn) )
<=> ~ pp(P) ) ).
% pure_assn_eq_false_iff
tff(fact_15_pure__false,axiom,
pure_assn(fFalse) = bot_bot(assn) ).
% pure_false
tff(fact_16_assn__basic__inequalities_I3_J,axiom,
bot_bot(assn) != one_one(assn) ).
% assn_basic_inequalities(3)
tff(fact_17_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A] :
( member(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Y)
<=> ( ( X = Y )
| member(A,Xs,Y) ) ) ).
% member_rec(1)
tff(fact_18_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn(one_one(assn)) ).
% is_pure_assn_basic_simps(2)
tff(fact_19_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn(bot_bot(assn)) ).
% is_pure_assn_basic_simps(1)
tff(fact_20_transpose_Ocases,axiom,
! [A: $tType,X: list(list(A))] :
( ( X != nil(list(A)) )
=> ( ! [Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)
=> ~ ! [X2: A,Xs2: list(A),Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss) ) ) ).
% transpose.cases
tff(fact_21_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) != Xs ).
% not_Cons_self2
tff(fact_22_list__nonempty__induct,axiom,
! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
( ( Xs != nil(A) )
=> ( ! [X2: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2))) ) )
=> pp(aa(list(A),bool,P,Xs)) ) ) ) ).
% list_nonempty_induct
tff(fact_23_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [X2: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),nil(B)))
=> ( ! [Y2: B,Ys2: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% list_induct2'
tff(fact_24_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y3: A,Ys3: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) ).
% neq_Nil_conv
tff(fact_25_remdups__adj_Ocases,axiom,
! [A: $tType,X: list(A)] :
( ( X != nil(A) )
=> ( ! [X2: A] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) ) ) ).
% remdups_adj.cases
tff(fact_26_min__list_Ocases,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: list(A)] :
( ! [X2: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ( X = nil(A) ) ) ) ).
% min_list.cases
tff(fact_27_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [X212: A,X222: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X222) ) ).
% list.exhaust
tff(fact_28_list_OdiscI,axiom,
! [A: $tType,List: list(A),X21: A,X22: list(A)] :
( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22) )
=> ( List != nil(A) ) ) ).
% list.discI
tff(fact_29_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22) ).
% list.distinct(1)
tff(fact_30_bot__apply,axiom,
! [D: $tType,C: $tType] :
( bot(C)
=> ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).
% bot_apply
tff(fact_31_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),R: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( ! [Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),nil(B)))
=> ( ! [Ys2: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),Ys2))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,X2),Y2))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))) ) )
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( ~ pp(aa(B,bool,aa(A,fun(B,bool),R,X2),Y2))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_32_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: fun(A,bool)] :
( pp(member2(A,A3,collect(A,P)))
<=> pp(aa(A,bool,P,A3)) ) ).
% mem_Collect_eq
tff(fact_33_Collect__mem__eq,axiom,
! [A: $tType,A4: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A4)) = A4 ).
% Collect_mem_eq
tff(fact_34_Collect__cong,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) )
=> ( collect(A,P) = collect(A,Q) ) ) ).
% Collect_cong
tff(fact_35_ext,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
% ext
tff(fact_36_list__induct__first2,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [X2: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))
=> ( ! [X1: A,X23: A,Xs2: list(A)] :
( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,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),X23),Xs2)))) )
=> pp(aa(list(A),bool,P,Xs)) ) ) ) ).
% list_induct_first2
tff(fact_37_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),W1: list(A),W2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [E: A,W12: list(A),W22: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W12),W22))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),W12)),W22)) )
=> ( ! [E: B,W13: list(A),W23: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),W23))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E),W23))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W1),W2)) ) ) ) ).
% list_2pre_induct
tff(fact_38_neq__NilE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [X2: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) ) ).
% neq_NilE
tff(fact_39_quicksort_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A)] :
( ( X != nil(A) )
=> ~ ! [X2: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) ) ) ).
% quicksort.cases
tff(fact_40_list__assn_Oelims,axiom,
! [C: $tType,A: $tType,X: fun(A,fun(C,assn)),Xa: list(A),Xb: list(C),Y: assn] :
( ( vEBT_List_list_assn(A,C,X,Xa,Xb) = Y )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(C) )
=> ( Y != one_one(assn) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [C2: C,Cs: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),C2),Cs) )
=> ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(C,assn,aa(A,fun(C,assn),X,A5),C2)),vEBT_List_list_assn(A,C,X,As,Cs)) ) ) )
=> ( ( ? [V2: A,Va2: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)
=> ( ( Xb = nil(C) )
=> ( Y != bot_bot(assn) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V2: C,Va2: list(C)] : Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va2)
=> ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).
% list_assn.elims
tff(fact_41_mult__1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).
% mult_1
tff(fact_42_mult_Oright__neutral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).
% mult.right_neutral
tff(fact_43_merge__pure__star,axiom,
! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).
% merge_pure_star
tff(fact_44_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_45_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_46_is__pure__assn__starI,axiom,
! [A3: assn,B2: assn] :
( is_pure_assn(A3)
=> ( is_pure_assn(B2)
=> is_pure_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)) ) ) ).
% is_pure_assn_starI
tff(fact_47_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_48_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),R) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),R)) ).
% assn_times_assoc
tff(fact_49_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_50_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_51_mult_Oassoc,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% mult.assoc
tff(fact_52_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).
% mult.comm_neutral
tff(fact_53_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).
% comm_monoid_mult_class.mult_1
tff(fact_54_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_55_list__assn_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,P: fun(A,fun(C,assn)),A3: A,As2: list(A),C3: C,Cs2: list(C)] : vEBT_List_list_assn(A,C,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),C3),Cs2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(C,assn,aa(A,fun(C,assn),P,A3),C3)),vEBT_List_list_assn(A,C,P,As2,Cs2)) ).
% list_assn.simps(2)
tff(fact_56_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
% one_reorient
tff(fact_57_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).
% bot_fun_def
tff(fact_58_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list(A),N2: A,R2: 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),N2),R2) )
=> ( ( N1 = N2 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
tff(fact_59_norm__assertion__simps_I1_J,axiom,
! [A3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),A3) = A3 ).
% norm_assertion_simps(1)
tff(fact_60_norm__assertion__simps_I2_J,axiom,
! [A3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),one_one(assn)) = A3 ).
% norm_assertion_simps(2)
tff(fact_61_product__lists_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(list(A)),list(list(A)),product_lists(A),nil(list(A))) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).
% product_lists.simps(1)
tff(fact_62_sngr__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P4: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P4,X)),sngr_assn(A,P4,Y)) = bot_bot(assn) ) ).
% sngr_same_false
tff(fact_63_subseqs_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(list(A)),subseqs(A),nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).
% subseqs.simps(1)
tff(fact_64_insert__Nil,axiom,
! [A: $tType,X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).
% insert_Nil
tff(fact_65_snga__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P4: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P4,X)),snga_assn(A,P4,Y)) = bot_bot(assn) ) ).
% snga_same_false
tff(fact_66_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,As: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( Y != revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) ) ) ) ) ).
% revg.elims
tff(fact_67_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B] : list_collect_set(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F,A3) ).
% list_collect_set_simps(2)
tff(fact_68_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: fun(A,B),Xa: list(A),Xb: list(B),Y: list(B)] :
( ( map_tailrec_rev(A,B,X,Xa,Xb) = Y )
=> ( ( ( Xa = nil(A) )
=> ( Y != Xb ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( Y != map_tailrec_rev(A,B,X,As,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X,A5)),Xb)) ) ) ) ) ).
% map_tailrec_rev.elims
tff(fact_69_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),X: A] : arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = X ) ).
% arg_min_list.simps(1)
tff(fact_70_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),nil(A))) ).
% ord.lexordp_eq_simps(3)
tff(fact_71_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Xs: list(A),Y: A,Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),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)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs),Ys)) ) ) ) ).
% ord.lexordp_eq_simps(4)
tff(fact_72_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),nil(A)),Ys)) ).
% ord.lexordp_eq_simps(1)
tff(fact_73_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs),nil(A)))
<=> ( Xs = nil(A) ) ) ).
% ord.lexordp_eq_simps(2)
tff(fact_74_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,set(A))] : list_collect_set(B,A,F,nil(B)) = bot_bot(set(A)) ).
% list_collect_set_simps(1)
tff(fact_75_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs),Xs)) ).
% ord.lexordp_eq_refl
tff(fact_76_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp_eq(A,Less) = lexordp_eq(A,Less) ).
% ord.lexordp_eq.cong
tff(fact_77_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),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))) ) ).
% ord.lexordp_eq.Cons
tff(fact_78_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs),Ys))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),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))) ) ) ) ).
% ord.lexordp_eq.Cons_eq
tff(fact_79_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),nil(A)),Ys)) ).
% ord.lexordp_eq.Nil
tff(fact_80_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A3: A,As2: list(A),Bs: list(B)] : map_tailrec_rev(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),Bs) = map_tailrec_rev(A,B,F,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F,A3)),Bs)) ).
% map_tailrec_rev.simps(2)
tff(fact_81_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Bs: list(B)] : map_tailrec_rev(A,B,F,nil(A),Bs) = Bs ).
% map_tailrec_rev.simps(1)
tff(fact_82_revg_Osimps_I2_J,axiom,
! [A: $tType,A3: A,As2: list(A),B2: list(A)] : revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),B2) = revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),B2)) ).
% revg.simps(2)
tff(fact_83_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list(A)] : revg(A,nil(A),B2) = B2 ).
% revg.simps(1)
tff(fact_84_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),A1),A22))
=> ( ( A1 != nil(A) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys2: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X2),Y2)) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X2),Y2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs2),Ys2)) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
tff(fact_85_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),A1),A22))
<=> ( ? [Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = Ys3 ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y3),X4))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs3),Ys3)) ) ) ) ).
% ord.lexordp_eq.simps
tff(fact_86_assn__aci_I10_J,axiom,
! [A3: assn,B2: assn,C3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),C3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C3)),B2) ).
% assn_aci(10)
tff(fact_87_star__aci_I3_J,axiom,
! [A3: assn,B2: assn,C3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C3)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C3)) ).
% star_aci(3)
tff(fact_88_star__aci_I2_J,axiom,
! [A3: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),A3) ).
% star_aci(2)
tff(fact_89_star__assoc,axiom,
! [A3: assn,B2: assn,C3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),C3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C3)) ).
% star_assoc
tff(fact_90_and__extract__pure__left__iff,axiom,
! [B2: bool,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn(B2)),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),Q)),pure_assn(B2)) ).
% and_extract_pure_left_iff
tff(fact_91_and__extract__pure__right__iff,axiom,
! [P: assn,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),pure_assn(B2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),one_one(assn)),P)),pure_assn(B2)) ).
% and_extract_pure_right_iff
tff(fact_92_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X: fun(C,B),A3: C] : list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),nil(C)))) = aa(B,set(A),F,aa(C,B,X,A3)) ).
% list_collect_set_map_simps(2)
tff(fact_93_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),A1),A22))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,X2),Y2))
=> ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs2),Ys2)) ) ) ) ) ) ).
% listrelp.cases
tff(fact_94_listrelp_Osimps,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),A1),A22))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs3: list(A),Ys3: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
& pp(aa(B,bool,aa(A,fun(B,bool),R3,X4),Y3))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs3),Ys3)) ) ) ) ).
% listrelp.simps
tff(fact_95_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X: A] : aa(list(A),list(A),mergesort_by_rel(A,R),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_96_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A)] : ~ ord_lexordp_eq(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),nil(A)) ) ).
% lexordp_eq_simps(3)
tff(fact_97_entails__solve__finalize_I2_J,axiom,
! [M: list(product_prod(assn,assn))] : fI_RESULT(M,one_one(assn),one_one(assn),one_one(assn)) ).
% entails_solve_finalize(2)
tff(fact_98_frame__inference__finalize,axiom,
! [M: list(product_prod(assn,assn)),F2: assn] : fI_RESULT(M,F2,one_one(assn),F2) ).
% frame_inference_finalize
tff(fact_99_mult_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% mult.right_assoc
tff(fact_100_mult_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2) ) ).
% mult.right_commute
tff(fact_101_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: list(A)] :
( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),A3) = nil(B) )
<=> ( A3 = nil(A) ) ) ).
% list.map_disc_iff
tff(fact_102_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] :
( ( nil(A) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) )
<=> ( Xs = nil(B) ) ) ).
% Nil_is_map_conv
tff(fact_103_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = nil(A) )
<=> ( Xs = nil(B) ) ) ).
% map_is_Nil_conv
tff(fact_104_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] :
( ord_lexordp_eq(A,Xs,nil(A))
<=> ( Xs = nil(A) ) ) ) ).
% lexordp_eq_simps(2)
tff(fact_105_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ys: list(A)] : ord_lexordp_eq(A,nil(A),Ys) ) ).
% lexordp_eq_simps(1)
tff(fact_106_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] : aa(list(A),list(A),mergesort_by_rel(A,R),nil(A)) = nil(A) ).
% mergesort_by_rel_simps(1)
tff(fact_107_merge__pure__and,axiom,
! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).
% merge_pure_and
tff(fact_108_and__extract__pure__right__ctx__iff,axiom,
! [P: assn,Q: assn,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn(B2))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),pure_assn(B2)) ).
% and_extract_pure_right_ctx_iff
tff(fact_109_and__extract__pure__left__ctx__iff,axiom,
! [P: assn,B2: bool,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2))),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),pure_assn(B2)) ).
% and_extract_pure_left_ctx_iff
tff(fact_110_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),nil(C))) = bot_bot(set(A)) ).
% list_collect_set_map_simps(1)
tff(fact_111_assn__aci_I4_J,axiom,
! [X: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),Y) ).
% assn_aci(4)
tff(fact_112_assn__aci_I3_J,axiom,
! [X: assn,Y: assn,Z2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z2)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),Z2)) ).
% assn_aci(3)
tff(fact_113_assn__aci_I1_J,axiom,
! [X: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),Y) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),X) ).
% assn_aci(1)
tff(fact_114_norm__assertion__simps_I31_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),X) = X ).
% norm_assertion_simps(31)
tff(fact_115_norm__assertion__simps_I14_J,axiom,
! [X: assn,Y: assn,Z2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),Y)),Z2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Y),Z2)) ).
% norm_assertion_simps(14)
tff(fact_116_lexordp__eq__refl,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] : ord_lexordp_eq(A,Xs,Xs) ) ).
% lexordp_eq_refl
tff(fact_117_lexordp__eq__trans,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A),Zs: list(A)] :
( ord_lexordp_eq(A,Xs,Ys)
=> ( ord_lexordp_eq(A,Ys,Zs)
=> ord_lexordp_eq(A,Xs,Zs) ) ) ) ).
% lexordp_eq_trans
tff(fact_118_lexordp__eq__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp_eq(A,Xs,Ys)
| ord_lexordp_eq(A,Ys,Xs) ) ) ).
% lexordp_eq_linear
tff(fact_119_lexordp__eq__antisym,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp_eq(A,Xs,Ys)
=> ( ord_lexordp_eq(A,Ys,Xs)
=> ( Xs = Ys ) ) ) ) ).
% lexordp_eq_antisym
tff(fact_120_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set(A)] :
( pp(member2(A,X,S))
=> ( S != bot_bot(set(A)) ) ) ).
% memb_imp_not_empty
tff(fact_121_set__notEmptyE,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ~ ! [X2: A] : ~ pp(member2(A,X2,S)) ) ).
% set_notEmptyE
tff(fact_122_map__eq__consE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fa: A,Fl: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ls) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
=> ~ ! [A5: B,L2: list(B)] :
( ( Ls = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),L2) )
=> ( ( aa(B,A,F,A5) = Fa )
=> ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L2) != Fl ) ) ) ) ).
% map_eq_consE
tff(fact_123_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list(A),F: fun(B,A),Ww: list(B),A3: B] :
( ( W = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ww) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A3)),W) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww)) ) ) ).
% map_consI(1)
tff(fact_124_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
<=> ? [Z3: B,Zs2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
& ( aa(B,A,F,Z3) = Y )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs2) = Ys ) ) ) ).
% map_eq_Cons_conv
tff(fact_125_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),F: fun(B,A),Ys: list(B)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
<=> ? [Z3: B,Zs2: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
& ( X = aa(B,A,F,Z3) )
& ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs2) ) ) ) ).
% Cons_eq_map_conv
tff(fact_126_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
=> ? [Z4: B,Zs3: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z4),Zs3) )
& ( aa(B,A,F,Z4) = Y )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs3) = Ys ) ) ) ).
% map_eq_Cons_D
tff(fact_127_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),F: fun(B,A),Ys: list(B)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
=> ? [Z4: B,Zs3: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z4),Zs3) )
& ( X = aa(B,A,F,Z4) )
& ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs3) ) ) ) ).
% Cons_eq_map_D
tff(fact_128_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X21: A,X22: list(A)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F,X21)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X22)) ).
% list.simps(9)
tff(fact_129_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),nil(A)) = nil(B) ).
% list.simps(8)
tff(fact_130_norm__assertion__simps_I9_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),bot_bot(assn)),X) = bot_bot(assn) ).
% norm_assertion_simps(9)
tff(fact_131_norm__assertion__simps_I10_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),bot_bot(assn)) = bot_bot(assn) ).
% norm_assertion_simps(10)
tff(fact_132_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ys: list(A)] : ord_lexordp_eq(A,nil(A),Ys) ) ).
% lexordp_eq.Nil
tff(fact_133_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R3,X),Y))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs),Ys))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),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))) ) ) ).
% listrelp.Cons
tff(fact_134_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),nil(A)),nil(B))) ).
% listrelp.Nil
tff(fact_135_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_136_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_137_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_138_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_139_inf__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ).
% inf_apply
tff(fact_140_inf__right__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_right_idem
tff(fact_141_inf_Oright__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).
% inf.right_idem
tff(fact_142_inf__left__idem,axiom,
! [A: $tType] :
( semilattice_inf(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),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_left_idem
tff(fact_143_inf_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).
% inf.left_idem
tff(fact_144_inf__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),X) = X ) ).
% inf_idem
tff(fact_145_inf_Oidem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),A3) = A3 ) ).
% inf.idem
tff(fact_146_empty__Collect__eq,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( bot_bot(set(A)) = collect(A,P) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P,X4)) ) ).
% empty_Collect_eq
tff(fact_147_empty__iff,axiom,
! [A: $tType,C3: A] : ~ pp(member2(A,C3,bot_bot(set(A)))) ).
% empty_iff
tff(fact_148_all__not__in__conv,axiom,
! [A: $tType,A4: set(A)] :
( ! [X4: A] : ~ pp(member2(A,X4,A4))
<=> ( A4 = bot_bot(set(A)) ) ) ).
% all_not_in_conv
tff(fact_149_Collect__empty__eq,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( collect(A,P) = bot_bot(set(A)) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P,X4)) ) ).
% Collect_empty_eq
tff(fact_150_Int__emptyI,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ! [X2: A] :
( pp(member2(A,X2,A4))
=> ~ pp(member2(A,X2,B3)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).
% Int_emptyI
tff(fact_151_disjoint__iff,axiom,
! [A: $tType,A4: set(A),B3: 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)) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> ~ pp(member2(A,X4,B3)) ) ) ).
% disjoint_iff
tff(fact_152_Int__empty__left,axiom,
! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B3) = bot_bot(set(A)) ).
% Int_empty_left
tff(fact_153_Int__empty__right,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),bot_bot(set(A))) = bot_bot(set(A)) ).
% Int_empty_right
tff(fact_154_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set(A),B3: 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)) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> ! [Xa2: A] :
( pp(member2(A,Xa2,B3))
=> ( X4 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_155_disjointI,axiom,
! [A: $tType,A3: set(A),B2: set(A)] :
( ! [X2: A] :
( pp(member2(A,X2,A3))
=> ~ pp(member2(A,X2,B2)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ).
% disjointI
tff(fact_156_emptyE,axiom,
! [A: $tType,A3: A] : ~ pp(member2(A,A3,bot_bot(set(A)))) ).
% emptyE
tff(fact_157_equals0D,axiom,
! [A: $tType,A4: set(A),A3: A] :
( ( A4 = bot_bot(set(A)) )
=> ~ pp(member2(A,A3,A4)) ) ).
% equals0D
tff(fact_158_equals0I,axiom,
! [A: $tType,A4: set(A)] :
( ! [Y2: A] : ~ pp(member2(A,Y2,A4))
=> ( A4 = bot_bot(set(A)) ) ) ).
% equals0I
tff(fact_159_ex__in__conv,axiom,
! [A: $tType,A4: set(A)] :
( ? [X4: A] : pp(member2(A,X4,A4))
<=> ( A4 != bot_bot(set(A)) ) ) ).
% ex_in_conv
tff(fact_160_bot__set__def,axiom,
! [A: $tType] : bot_bot(set(A)) = collect(A,bot_bot(fun(A,bool))) ).
% bot_set_def
tff(fact_161_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_sup_aci(4)
tff(fact_162_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: 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),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_sup_aci(3)
tff(fact_163_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).
% inf_sup_aci(2)
tff(fact_164_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).
% inf_sup_aci(1)
tff(fact_165_inf_Oassoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ).
% inf.assoc
tff(fact_166_inf__assoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).
% inf_assoc
tff(fact_167_inf_Ocommute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),A3) ) ).
% inf.commute
tff(fact_168_inf__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).
% inf_commute
tff(fact_169_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.inf1
tff(fact_170_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.inf2
tff(fact_171_inf_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ).
% inf.left_commute
tff(fact_172_inf__left__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: 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),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_left_commute
tff(fact_173_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).
% inf_fun_def
tff(fact_174_FI__finalize,axiom,
! [M2: list(product_prod(assn,assn)),P4: assn,Up: assn,Q2: assn,Uq: assn,F: assn] :
( fI_RESULT(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P4),Up),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),Uq),F)
=> fi(M2,P4,Q2,Up,Uq,F) ) ).
% FI_finalize
tff(fact_175_Set_Ois__empty__def,axiom,
! [A: $tType,A4: set(A)] :
( is_empty(A,A4)
<=> ( A4 = bot_bot(set(A)) ) ) ).
% Set.is_empty_def
tff(fact_176_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] : map(A,B) = map_tailrec(A,B) ).
% map_eq_map_tailrec
tff(fact_177_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X: fun(C,B),A3: C,L: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,aa(C,B,X,A3))),list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),L))) ).
% list_collect_set_map_simps(3)
tff(fact_178_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_179_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( ord_lexordp_eq(A,A1,A22)
=> ( ( A1 != nil(A) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys2: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2)) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2))
=> ~ ord_lexordp_eq(A,Xs2,Ys2) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
tff(fact_180_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( ord_lexordp_eq(A,A1,A22)
<=> ( ? [Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = Ys3 ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
& ord_lexordp_eq(A,Xs3,Ys3) ) ) ) ) ).
% lexordp_eq.simps
tff(fact_181_bot__empty__eq,axiom,
! [A: $tType,X3: A] :
( pp(aa(A,bool,bot_bot(fun(A,bool)),X3))
<=> pp(member2(A,X3,bot_bot(set(A)))) ) ).
% bot_empty_eq
tff(fact_182_Collect__empty__eq__bot,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( collect(A,P) = bot_bot(set(A)) )
<=> ( P = bot_bot(fun(A,bool)) ) ) ).
% Collect_empty_eq_bot
tff(fact_183_mult_Osafe__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [X: A,Y: A,A3: A,B2: A] :
( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ) ) ).
% mult.safe_commute
tff(fact_184_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).
% sup.idem
tff(fact_185_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).
% sup_idem
tff(fact_186_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).
% sup.left_idem
tff(fact_187_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).
% sup_left_idem
tff(fact_188_sup_Oright__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).
% sup.right_idem
tff(fact_189_sup__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ).
% sup_apply
tff(fact_190_Int__iff,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
<=> ( pp(member2(A,C3,A4))
& pp(member2(A,C3,B3)) ) ) ).
% Int_iff
tff(fact_191_IntI,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,A4))
=> ( pp(member2(A,C3,B3))
=> pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).
% IntI
tff(fact_192_UnCI,axiom,
! [A: $tType,C3: A,B3: set(A),A4: set(A)] :
( ( ~ pp(member2(A,C3,B3))
=> pp(member2(A,C3,A4)) )
=> pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ) ).
% UnCI
tff(fact_193_Un__iff,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
<=> ( pp(member2(A,C3,A4))
| pp(member2(A,C3,B3)) ) ) ).
% Un_iff
tff(fact_194_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).
% sup_bot.right_neutral
tff(fact_195_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
<=> ( ( A3 = bot_bot(A) )
& ( B2 = bot_bot(A) ) ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_196_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).
% sup_bot.left_neutral
tff(fact_197_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
<=> ( ( A3 = bot_bot(A) )
& ( B2 = bot_bot(A) ) ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_198_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_199_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_200_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_201_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_202_sup__inf__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = X ) ).
% sup_inf_absorb
tff(fact_203_inf__sup__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = X ) ).
% inf_sup_absorb
tff(fact_204_Un__empty,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = bot_bot(set(A)) )
<=> ( ( A4 = bot_bot(set(A)) )
& ( B3 = bot_bot(set(A)) ) ) ) ).
% Un_empty
tff(fact_205_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set(A),T: 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),T)),S) = S ).
% Un_Int_eq(1)
tff(fact_206_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set(A),T: 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),T)),T) = T ).
% Un_Int_eq(2)
tff(fact_207_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set(A),T: 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),T)) = S ).
% Un_Int_eq(3)
tff(fact_208_Un__Int__eq_I4_J,axiom,
! [A: $tType,T: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T)) = T ).
% Un_Int_eq(4)
tff(fact_209_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set(A),T: 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),T)),S) = S ).
% Int_Un_eq(1)
tff(fact_210_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set(A),T: 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),T)),T) = T ).
% Int_Un_eq(2)
tff(fact_211_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set(A),T: 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),T)) = S ).
% Int_Un_eq(3)
tff(fact_212_Int__Un__eq_I4_J,axiom,
! [A: $tType,T: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T)) = T ).
% Int_Un_eq(4)
tff(fact_213_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A),Y: A,Ys: list(A)] :
( ord_lexordp_eq(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))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
& ord_lexordp_eq(A,Xs,Ys) ) ) ) ) ).
% lexordp_eq_simps(4)
tff(fact_214_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B,L: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),L)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,A3)),list_collect_set(B,A,F,L)) ).
% list_collect_set_simps(3)
tff(fact_215_Int__left__commute,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4)) ).
% Int_left_commute
tff(fact_216_Un__Int__distrib2,axiom,
! [A: $tType,B3: set(A),C4: set(A),A4: 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)),B3),C4)),A4) = 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)),B3),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A4)) ).
% Un_Int_distrib2
tff(fact_217_Int__left__absorb,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).
% Int_left_absorb
tff(fact_218_Int__Un__distrib2,axiom,
! [A: $tType,B3: set(A),C4: set(A),A4: 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)),B3),C4)),A4) = 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)),B3),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A4)) ).
% Int_Un_distrib2
tff(fact_219_Un__Int__distrib,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C4)) ).
% Un_Int_distrib
tff(fact_220_Int__Un__distrib,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4)) ).
% Int_Un_distrib
tff(fact_221_Un__Int__crazy,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A4)) = 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)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A4)) ).
% Un_Int_crazy
tff(fact_222_Int__commute,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A4) ).
% Int_commute
tff(fact_223_Int__absorb,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),A4) = A4 ).
% Int_absorb
tff(fact_224_Int__assoc,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) ).
% Int_assoc
tff(fact_225_IntD2,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> pp(member2(A,C3,B3)) ) ).
% IntD2
tff(fact_226_IntD1,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> pp(member2(A,C3,A4)) ) ).
% IntD1
tff(fact_227_IntE,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> ~ ( pp(member2(A,C3,A4))
=> ~ pp(member2(A,C3,B3)) ) ) ).
% IntE
tff(fact_228_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).
% inf_sup_aci(8)
tff(fact_229_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% inf_sup_aci(7)
tff(fact_230_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).
% inf_sup_aci(6)
tff(fact_231_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).
% inf_sup_aci(5)
tff(fact_232_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).
% sup.assoc
tff(fact_233_sup__assoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).
% sup_assoc
tff(fact_234_less__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% less_supI1
tff(fact_235_less__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% less_supI2
tff(fact_236_sup_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb3
tff(fact_237_sup_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb4
tff(fact_238_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),A3) ) ).
% sup.commute
tff(fact_239_sup__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).
% sup_commute
tff(fact_240_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_241_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_242_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).
% sup.left_commute
tff(fact_243_sup__left__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% sup_left_commute
tff(fact_244_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).
% sup.strict_boundedE
tff(fact_245_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
tff(fact_246_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI1
tff(fact_247_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI2
tff(fact_248_UnE,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
=> ( ~ pp(member2(A,C3,A4))
=> pp(member2(A,C3,B3)) ) ) ).
% UnE
tff(fact_249_UnI1,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,A4))
=> pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ) ).
% UnI1
tff(fact_250_UnI2,axiom,
! [A: $tType,C3: A,B3: set(A),A4: set(A)] :
( pp(member2(A,C3,B3))
=> pp(member2(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ) ).
% UnI2
tff(fact_251_bex__Un,axiom,
! [A: $tType,A4: set(A),B3: set(A),P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
& pp(aa(A,bool,P,X4)) )
<=> ( ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,P,X4)) )
| ? [X4: A] :
( pp(member2(A,X4,B3))
& pp(aa(A,bool,P,X4)) ) ) ) ).
% bex_Un
tff(fact_252_ball__Un,axiom,
! [A: $tType,A4: set(A),B3: set(A),P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
=> pp(aa(A,bool,P,X4)) )
<=> ( ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,P,X4)) )
& ! [X4: A] :
( pp(member2(A,X4,B3))
=> pp(aa(A,bool,P,X4)) ) ) ) ).
% ball_Un
tff(fact_253_Un__assoc,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) ).
% Un_assoc
tff(fact_254_Un__absorb,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),A4) = A4 ).
% Un_absorb
tff(fact_255_Un__commute,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A4) ).
% Un_commute
tff(fact_256_Un__left__absorb,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) ).
% Un_left_absorb
tff(fact_257_Un__left__commute,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C4)) ).
% Un_left_commute
tff(fact_258_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).
% sup_fun_def
tff(fact_259_lt__ex,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] :
? [Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ).
% lt_ex
tff(fact_260_gt__ex,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] :
? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_1)) ) ).
% gt_ex
tff(fact_261_dense,axiom,
! [A: $tType] :
( dense_order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ? [Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),Y)) ) ) ) ).
% dense
tff(fact_262_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
% less_imp_neq
tff(fact_263_order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% order.asym
tff(fact_264_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% ord_eq_less_trans
tff(fact_265_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( ( B2 = C3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% ord_less_eq_trans
tff(fact_266_less__induct,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [X2: A] :
( ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X2))
=> pp(aa(A,bool,P,Y4)) )
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% less_induct
tff(fact_267_antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv3
tff(fact_268_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ( X != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_cases
tff(fact_269_dual__order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% dual_order.asym
tff(fact_270_dual__order_Oirrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).
% dual_order.irrefl
tff(fact_271_exists__least__iff,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool)] :
( ? [X_12: A] : pp(aa(A,bool,P,X_12))
<=> ? [N3: A] :
( pp(aa(A,bool,P,N3))
& ! [M3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M3),N3))
=> ~ pp(aa(A,bool,P,M3)) ) ) ) ) ).
% exists_least_iff
tff(fact_272_linorder__less__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B4))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B4)) )
=> ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A5),A5))
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ) ).
% linorder_less_wlog
tff(fact_273_order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% order.strict_trans
tff(fact_274_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_275_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).
% dual_order.strict_trans
tff(fact_276_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
tff(fact_277_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_278_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_neqE
tff(fact_279_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_asym
tff(fact_280_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_neq_iff
tff(fact_281_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% order_less_asym'
tff(fact_282_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_less_trans
tff(fact_283_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% ord_eq_less_subst
tff(fact_284_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( ( aa(A,B,F,B2) = C3 )
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,A3)),C3)) ) ) ) ) ).
% ord_less_eq_subst
tff(fact_285_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).
% order_less_irrefl
tff(fact_286_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% order_less_subst1
tff(fact_287_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C3: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,B2)),C3))
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,X2)),aa(A,C,F,Y2))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C3)) ) ) ) ) ).
% order_less_subst2
tff(fact_288_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_not_sym
tff(fact_289_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,P: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> pp(P) ) ) ) ).
% order_less_imp_triv
tff(fact_290_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_less_linear
tff(fact_291_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
tff(fact_292_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
tff(fact_293_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_imp_not_less
tff(fact_294_syntax__fo__nomatch__def,axiom,
! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax7388354845996824322omatch(A,B,Pat,Obj) ).
% syntax_fo_nomatch_def
tff(fact_295_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_296_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).
% boolean_algebra.disj_conj_distrib2
tff(fact_297_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).
% boolean_algebra.conj_disj_distrib2
tff(fact_298_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% boolean_algebra.disj_conj_distrib
tff(fact_299_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% boolean_algebra.conj_disj_distrib
tff(fact_300_sup__inf__distrib2,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).
% sup_inf_distrib2
tff(fact_301_sup__inf__distrib1,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% sup_inf_distrib1
tff(fact_302_inf__sup__distrib2,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).
% inf_sup_distrib2
tff(fact_303_inf__sup__distrib1,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_sup_distrib1
tff(fact_304_distrib__imp2,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Z4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ) ) ).
% distrib_imp2
tff(fact_305_distrib__imp1,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Z4))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ) ) ).
% distrib_imp1
tff(fact_306_Un__empty__right,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),bot_bot(set(A))) = A4 ).
% Un_empty_right
tff(fact_307_Un__empty__left,axiom,
! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B3) = B3 ).
% Un_empty_left
tff(fact_308_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( ( A3 != bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).
% bot.not_eq_extremum
tff(fact_309_bot_Oextremum__strict,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).
% bot.extremum_strict
tff(fact_310_not__psubset__empty,axiom,
! [A: $tType,A4: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),bot_bot(set(A)))) ).
% not_psubset_empty
tff(fact_311_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).
% inf.strict_coboundedI2
tff(fact_312_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).
% inf.strict_coboundedI1
tff(fact_313_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
tff(fact_314_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% inf.strict_boundedE
tff(fact_315_inf_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb4
tff(fact_316_inf_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb3
tff(fact_317_less__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% less_infI2
tff(fact_318_less__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% less_infI1
tff(fact_319_FI__p__nomatch,axiom,
! [M2: list(product_prod(assn,assn)),Ps: assn,Qs: assn,Q2: assn,P4: assn,Up: assn,Uq: assn,F: assn] :
( fi(M2,Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P4),Up),Uq,F)
=> fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps),P4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F) ) ).
% FI_p_nomatch
tff(fact_320_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( ord_lexordp_eq(A,Xs,Ys)
=> ord_lexordp_eq(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)) ) ) ) ) ).
% lexordp_eq.Cons_eq
tff(fact_321_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ord_lexordp_eq(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)) ) ) ).
% lexordp_eq.Cons
tff(fact_322_assn__aci_I11_J,axiom,
! [X: assn,Y: assn,A3: assn,B2: assn] :
( syntax7388354845996824322omatch(assn,assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),X),Y),A3)
=> ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),A3) ) ) ).
% assn_aci(11)
tff(fact_323_less__1__mult,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M2: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M2),N))) ) ) ) ).
% less_1_mult
tff(fact_324_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).
% less_numeral_extra(4)
tff(fact_325_FI__q__nomatch,axiom,
! [M2: list(product_prod(assn,assn)),Up: assn,Qs: assn,Q2: assn,Uq: assn,F: assn] :
( fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Up),Qs,sln,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),Uq),F)
=> fi(M2,sln,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F) ) ).
% FI_q_nomatch
tff(fact_326_map__tailrec__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),As2: list(A)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map_tailrec(A,B),F),As2) = aa(list(B),list(B),rev(B),map_tailrec_rev(A,B,F,As2,nil(B))) ).
% map_tailrec_def
tff(fact_327_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 ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ) )
=> ~ ! [X1: A,L1: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X23))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X23))
=> ( ( ( X1 = X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,L22)) ) )
& ( ( X1 != X23 )
=> ( Y = 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),X1),L1),L22)) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
tff(fact_328_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X: fun(C,B),L: list(C),L3: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),L))),list_collect_set(B,A,F,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),X),L3))) ).
% list_collect_set_map_simps(4)
tff(fact_329_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = nil(A) )
<=> ( A4 = bot_bot(set(A)) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_330_lexordp__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A),P: fun(list(A),fun(list(A),bool))] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> ( ! [Y2: A,Ys2: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
=> ( ! [X2: A,Xs2: list(A),Y2: A,Ys2: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) )
=> ( ! [X2: A,Xs2: list(A),Ys2: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys2))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2))) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys)) ) ) ) ) ) ).
% lexordp_induct
tff(fact_331_lexordp__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> ( ( ( Xs = nil(A) )
=> ! [Y2: A,Ys4: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys4) )
=> ( ! [X2: A] :
( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4)
=> ! [Y2: A] :
( ? [Ys4: list(A)] : Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys4)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2)) ) )
=> ~ ! [X2: A,Xs4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4) )
=> ! [Ys4: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys4) )
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs4),Ys4)) ) ) ) ) ) ) ).
% lexordp_cases
tff(fact_332_lexordp_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
<=> ( ? [Y3: A,Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys3)) ) ) ) ) ).
% lexordp.simps
tff(fact_333_same__append__eq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: 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),Zs) )
<=> ( Ys = Zs ) ) ).
% same_append_eq
tff(fact_334_append__same__eq,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xs) )
<=> ( Ys = Zs ) ) ).
% append_same_eq
tff(fact_335_append__assoc,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs)) ).
% append_assoc
tff(fact_336_append_Oassoc,axiom,
! [A: $tType,A3: list(A),B2: list(A),C3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),B2)),C3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C3)) ).
% append.assoc
tff(fact_337_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),rev(A),Ys) )
<=> ( Xs = Ys ) ) ).
% rev_is_rev_conv
tff(fact_338_rev__rev__ident,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xs)) = Xs ).
% rev_rev_ident
tff(fact_339_merge__pure__or,axiom,
! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fdisj(A3,B2)) ).
% merge_pure_or
tff(fact_340_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
% append_is_Nil_conv
tff(fact_341_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
% Nil_is_append_conv
tff(fact_342_self__append__conv2,axiom,
! [A: $tType,Y: list(A),Xs: list(A)] :
( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
<=> ( Xs = nil(A) ) ) ).
% self_append_conv2
tff(fact_343_append__self__conv2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
<=> ( Xs = nil(A) ) ) ).
% append_self_conv2
tff(fact_344_self__append__conv,axiom,
! [A: $tType,Y: list(A),Ys: list(A)] :
( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
<=> ( Ys = nil(A) ) ) ).
% self_append_conv
tff(fact_345_append__self__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
<=> ( Ys = nil(A) ) ) ).
% append_self_conv
tff(fact_346_append__Nil2,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),nil(A)) = Xs ).
% append_Nil2
tff(fact_347_append_Oright__neutral,axiom,
! [A: $tType,A3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),nil(A)) = A3 ).
% append.right_neutral
tff(fact_348_empty__append__eq__id,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X3) = X3 ).
% empty_append_eq_id
tff(fact_349_map__append,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys)) ).
% map_append
tff(fact_350_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),rev(A),Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
% rev_is_Nil_conv
tff(fact_351_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),rev(A),Xs) )
<=> ( Xs = nil(A) ) ) ).
% Nil_is_rev_conv
tff(fact_352_rev__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),rev(A),Xs)) ).
% rev_append
tff(fact_353_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),nil(A)),Ys))
<=> ( Ys != nil(A) ) ) ) ).
% lexordp_simps(1)
tff(fact_354_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),nil(A))) ) ).
% lexordp_simps(2)
tff(fact_355_append1__eq__conv,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),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))) )
<=> ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
tff(fact_356_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list(A),E2: A,L23: list(A),E3: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( E2 = E3 )
& ( L23 = nil(A) ) ) ) ).
% list_e_eq_lel(2)
tff(fact_357_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L12: list(A),E2: A,L23: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L23)) )
<=> ( ( L12 = nil(A) )
& ( E2 = E3 )
& ( L23 = nil(A) ) ) ) ).
% list_e_eq_lel(1)
tff(fact_358_list__se__match_I4_J,axiom,
! [A: $tType,L23: list(A),A3: A,L12: list(A)] :
( ( L23 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
<=> ( ( L12 = nil(A) )
& ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(4)
tff(fact_359_list__se__match_I3_J,axiom,
! [A: $tType,L12: list(A),A3: A,L23: list(A)] :
( ( L12 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
<=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
& ( L23 = nil(A) ) ) ) ) ).
% list_se_match(3)
tff(fact_360_list__se__match_I2_J,axiom,
! [A: $tType,L23: list(A),L12: list(A),A3: A] :
( ( L23 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(2)
tff(fact_361_list__se__match_I1_J,axiom,
! [A: $tType,L12: list(A),L23: list(A),A3: A] :
( ( L12 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
<=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
& ( L23 = nil(A) ) ) ) ) ).
% list_se_match(1)
tff(fact_362_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list(A),E1: A,E22: A,L23: list(A),E12: A,E23: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),nil(A))) )
<=> ( ( L12 = nil(A) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L23 = nil(A) ) ) ) ).
% list_ee_eq_leel(2)
tff(fact_363_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E12: A,E23: A,L12: list(A),E1: A,E22: A,L23: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),L23))) )
<=> ( ( L12 = nil(A) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L23 = nil(A) ) ) ) ).
% list_ee_eq_leel(1)
tff(fact_364_rev__singleton__conv,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
<=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).
% rev_singleton_conv
tff(fact_365_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = aa(list(A),list(A),rev(A),Xs) )
<=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = Xs ) ) ).
% singleton_rev_conv
tff(fact_366_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A),Y: A,Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys)) ) ) ) ) ).
% lexordp_simps(3)
tff(fact_367_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = nil(A) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
tff(fact_368_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B),L3: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,L)),list_collect_set(B,A,F,L3)) ).
% list_collect_set_simps(4)
tff(fact_369_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list(B),F: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),F) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F,X)),bind(B,A,Xs,F)) ).
% bind_simps(2)
tff(fact_370_revg__fun,axiom,
! [A: $tType,A3: list(A),B2: list(A)] : revg(A,A3,B2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),A3)),B2) ).
% revg_fun
tff(fact_371_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] :
( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
<=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) ) ) ).
% rev_eq_Cons_iff
tff(fact_372_psubsetD,axiom,
! [A: $tType,A4: set(A),B3: set(A),C3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> ( pp(member2(A,C3,A4))
=> pp(member2(A,C3,B3)) ) ) ).
% psubsetD
tff(fact_373_psubset__trans,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),C4)) ) ) ).
% psubset_trans
tff(fact_374_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),Ts: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts) )
<=> ? [Us: list(A)] :
( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys) = Ts ) )
| ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = Zs )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ts) ) ) ) ) ).
% append_eq_append_conv2
tff(fact_375_append__eq__appendI,axiom,
! [A: $tType,Xs: list(A),Xs1: list(A),Zs: list(A),Ys: list(A),Us2: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs1) = Zs )
=> ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Us2) )
=> ( 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),Zs),Us2) ) ) ) ).
% append_eq_appendI
tff(fact_376_rev__swap,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),rev(A),Xs) = Ys )
<=> ( Xs = aa(list(A),list(A),rev(A),Ys) ) ) ).
% rev_swap
tff(fact_377_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( order(A)
=> ! [Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ).
% lexordp_irreflexive'
tff(fact_378_lexordp__append__leftI,axiom,
! [A: $tType] :
( ord(A)
=> ! [Us2: list(A),Vs: list(A),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us2),Vs))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ) ).
% lexordp_append_leftI
tff(fact_379_lexordp__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
| ( Xs = Ys )
| pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Xs)) ) ) ).
% lexordp_linear
tff(fact_380_lexordp__trans,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A),Zs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Zs))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Zs)) ) ) ) ).
% lexordp_trans
tff(fact_381_lexordp__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Xs)) ) ) ).
% lexordp_antisym
tff(fact_382_lexordp__append__leftD,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A),Us2: list(A),Vs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
=> ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),A5))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us2),Vs)) ) ) ) ).
% lexordp_append_leftD
tff(fact_383_lexordp__append__rightI,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ys: list(A),Xs: list(A)] :
( ( Ys != nil(A) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))) ) ) ).
% lexordp_append_rightI
tff(fact_384_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
% rev.simps(2)
tff(fact_385_lexordp__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
<=> ( ? [X4: A,Vs2: list(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),X4),Vs2))
| ? [Us: list(A),A6: A,B5: A,Vs2: list(A),Ws: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),B5))
& ( 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),A6),Vs2)) )
& ( 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),B5),Ws)) ) ) ) ) ) ).
% lexordp_iff
tff(fact_386_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Us2: list(A),Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),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),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ) ) ).
% lexordp_append_left_rightI
tff(fact_387_rev_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).
% rev.simps(1)
tff(fact_388_rev__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),rev(B),Xs)) ).
% rev_map
tff(fact_389_append__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = 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),Ys)) ).
% append_Cons
tff(fact_390_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list(A),Ys: list(A),Xs: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) = Ys )
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Zs) )
=> ( 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),Zs) ) ) ) ).
% Cons_eq_appendI
tff(fact_391_list__match__lel__lel,axiom,
! [A: $tType,C1: list(A),Qs: A,C22: list(A),C12: list(A),Qs2: A,C23: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) )
=> ( ! [C21: list(A)] :
( ( C1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
=> ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list(A)] :
( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
=> ( C22 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) ) ) ) ) ) ).
% list_match_lel_lel
tff(fact_392_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = Ys )
=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).
% eq_Nil_appendI
tff(fact_393_append_Oleft__neutral,axiom,
! [A: $tType,A3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A3) = A3 ).
% append.left_neutral
tff(fact_394_append__Nil,axiom,
! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ).
% append_Nil
tff(fact_395_map__tailrec__rev,axiom,
! [B: $tType,A: $tType,F: fun(B,A),As2: list(B),Bs: list(A)] : map_tailrec_rev(B,A,F,As2,Bs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),As2))),Bs) ).
% map_tailrec_rev
tff(fact_396_Misc_Omap__eq__append__conv,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
<=> ? [L4: list(B),L5: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L5) )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L4) = Fl )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L5) = Fl2 ) ) ) ).
% Misc.map_eq_append_conv
tff(fact_397_Misc_Oappend__eq__map__conv,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ls) )
<=> ? [L4: list(B),L5: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L5) )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L4) = Fl )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L5) = Fl2 ) ) ) ).
% Misc.append_eq_map_conv
tff(fact_398_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
=> ~ ! [L2: list(B),L6: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
=> ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L2) = Fl )
=> ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L6) != Fl2 ) ) ) ) ).
% map_eq_appendE
tff(fact_399_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ls) )
=> ~ ! [L2: list(B),L6: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
=> ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L2) = Fl )
=> ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L6) != Fl2 ) ) ) ) ).
% append_eq_mapE
tff(fact_400_List_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
<=> ? [Us: list(B),Vs2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs2) )
& ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Us) )
& ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Vs2) ) ) ) ).
% List.map_eq_append_conv
tff(fact_401_List_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F: fun(B,A),Xs: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) )
<=> ? [Us: list(B),Vs2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs2) )
& ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Us) )
& ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Vs2) ) ) ) ).
% List.append_eq_map_conv
tff(fact_402_star__or__dist2,axiom,
! [C4: assn,A4: assn,B3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),A4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),B3)) ).
% star_or_dist2
tff(fact_403_star__or__dist1,axiom,
! [A4: assn,B3: assn,C4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)),C4) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),C4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C4)) ).
% star_or_dist1
tff(fact_404_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),U: list(A),V: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),U),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V))) ).
% ord.lexordp_eq_pref
tff(fact_405_lexordp__irreflexive,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] :
( ! [X2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).
% lexordp_irreflexive
tff(fact_406_norm__assertion__simps_I6_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),bot_bot(assn)) = X ).
% norm_assertion_simps(6)
tff(fact_407_norm__assertion__simps_I5_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),bot_bot(assn)),X) = X ).
% norm_assertion_simps(5)
tff(fact_408_lexordp__eq__pref,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: list(A),V: list(A)] : ord_lexordp_eq(A,U,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)) ) ).
% lexordp_eq_pref
tff(fact_409_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = aa(set(A),list(A),linord4507533701916653071of_set(A),B3) )
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( A4 = B3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_410_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L23: list(A)] : merge(A,nil(A),L23) = L23 ) ).
% merge.simps(1)
tff(fact_411_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
<=> ( ord_lexordp_eq(A,Xs,Ys)
& ~ ord_lexordp_eq(A,Ys,Xs) ) ) ) ).
% lexordp_conv_lexordp_eq
tff(fact_412_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp_eq(A,Xs,Ys)
<=> ( ( Xs = Ys )
| pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys)) ) ) ) ).
% lexordp_eq_conv_lexord
tff(fact_413_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> ord_lexordp_eq(A,Xs,Ys) ) ) ).
% lexordp_into_lexordp_eq
tff(fact_414_rev__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [X2: A,Xs2: list(A)] :
( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))) )
=> pp(aa(list(A),bool,P,Xs)) ) ) ).
% rev_induct
tff(fact_415_rev__exhaust,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ~ ! [Ys2: list(A),Y2: A] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))) ) ).
% rev_exhaust
tff(fact_416_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
<=> ( ( ( Ys = nil(A) )
& ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = Zs ) )
| ? [Ys5: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) = Ys )
& ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs) ) ) ) ) ).
% Cons_eq_append_conv
tff(fact_417_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
<=> ( ( ( Ys = nil(A) )
& ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
| ? [Ys5: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys5) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Zs) = Xs ) ) ) ) ).
% append_eq_Cons_conv
tff(fact_418_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
( ( Xs != nil(A) )
=> ( ! [X2: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))))) ) )
=> pp(aa(list(A),bool,P,Xs)) ) ) ) ).
% rev_nonempty_induct
tff(fact_419_neq__Nil__revE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [Ll: list(A),E: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),nil(A))) ) ).
% neq_Nil_revE
tff(fact_420_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [X2: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),nil(B)))
=> ( ! [Y2: B,Ys2: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),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)))))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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))))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% rev_induct2'
tff(fact_421_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
<=> ? [Xs3: list(A),X4: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))) ) ).
% neq_Nil_rev_conv
tff(fact_422_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),bool))] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ! [X2: A,Y2: B] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))))
=> ( ! [X2: A,Xs2: list(A),Y2: B] :
( ( Xs2 != nil(A) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))) )
=> ( ! [X2: A,Y2: B,Ys2: list(B)] :
( ( Ys2 != nil(B) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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))))) )
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> ( ( Xs2 != nil(A) )
=> ( ( Ys2 != nil(B) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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))))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
tff(fact_423_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
=> ( ( ( Ys = nil(A) )
=> ( Zs != 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),Zs) != Xs ) ) ) ) ).
% list_Cons_eq_append_cases
tff(fact_424_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
=> ( ( ( Ys = nil(A) )
=> ( Zs != 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),Zs) != Xs ) ) ) ) ).
% list_append_eq_Cons_cases
tff(fact_425_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list(A),L: list(A),F: fun(B,A),Ww: list(B),A3: 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),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ww)),L) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A3)),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),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww))),L) ) ) ).
% map_consI(2)
tff(fact_426_SLN__left,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),P) = P ).
% SLN_left
tff(fact_427_SLN__right,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),sln) = P ).
% SLN_right
tff(fact_428_lexordp_OCons,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ).
% lexordp.Cons
tff(fact_429_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ) ) ).
% lexordp.Cons_eq
tff(fact_430_lexordp_ONil,axiom,
! [A: $tType] :
( ord(A)
=> ! [Y: A,Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ).
% lexordp.Nil
tff(fact_431_SLN__def,axiom,
sln = one_one(assn) ).
% SLN_def
tff(fact_432_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X12: A,X24: A,L12: list(A),L23: list(A)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X24))
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),L23))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X24))
=> ( ( ( X12 = X24 )
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,L23)) ) )
& ( ( X12 != X24 )
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),L23)) ) ) ) ) ) ) ).
% merge.simps(3)
tff(fact_433_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [V: A,Va: list(A)] : merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ) ).
% merge.simps(2)
tff(fact_434_lexordp_Ocases,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
=> ( ( ( A1 = nil(A) )
=> ! [Y2: A,Ys2: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys2: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2)) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys2)) ) ) ) ) ) ) ) ) ).
% lexordp.cases
tff(fact_435_finite__Int,axiom,
! [A: $tType,F2: set(A),G2: set(A)] :
( ( pp(aa(set(A),bool,finite_finite2(A),F2))
| pp(aa(set(A),bool,finite_finite2(A),G2)) )
=> pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F2),G2))) ) ).
% finite_Int
tff(fact_436_ex__min__if__finite,axiom,
! [A: $tType] :
( order(A)
=> ! [S: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member2(A,X2,S))
& ~ ? [Xa3: A] :
( pp(member2(A,Xa3,S))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa3),X2)) ) ) ) ) ) ).
% ex_min_if_finite
tff(fact_437_infinite__growing,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X5: set(A)] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> ? [Xa3: A] :
( pp(member2(A,Xa3,X5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Xa3)) ) )
=> ~ pp(aa(set(A),bool,finite_finite2(A),X5)) ) ) ) ).
% infinite_growing
tff(fact_438_FI__init,axiom,
! [P: assn,Q: assn,F2: assn] :
( fi(nil(product_prod(assn,assn)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),Q),sln,sln,F2)
=> fI_QUERY(P,Q,F2) ) ).
% FI_init
tff(fact_439_finite_OemptyI,axiom,
! [A: $tType] : pp(aa(set(A),bool,finite_finite2(A),bot_bot(set(A)))) ).
% finite.emptyI
tff(fact_440_infinite__imp__nonempty,axiom,
! [A: $tType,S: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),S))
=> ( S != bot_bot(set(A)) ) ) ).
% infinite_imp_nonempty
tff(fact_441_finite__transitivity__chain,axiom,
! [A: $tType,A4: set(A),R: fun(A,fun(A,bool))] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ! [X2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),R,X2),X2))
=> ( ! [X2: A,Y2: A,Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R,Y2),Z4))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Z4)) ) )
=> ( ! [X2: A] :
( pp(member2(A,X2,A4))
=> ? [Y4: A] :
( pp(member2(A,Y4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Y4)) ) )
=> ( A4 = bot_bot(set(A)) ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_442_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)) )
=> ! [L2: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))) )
=> ( Y != L2 ) ) )
=> ( ! [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) )
=> ! [L2: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),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)),L2),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),Ls2: 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),Ls2)) )
=> ( 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),Ls2) ) ) ) ) ) ) ) ) ).
% merge_list.elims
tff(fact_443_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: fun(B,list(A)),X: B,Xs: list(B)] : maps(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F,X)),maps(B,A,F,Xs)) ).
% maps_simps(1)
tff(fact_444_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rotate1(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),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
% rotate1.simps(2)
tff(fact_445_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list(list(A)),Ys: list(A),Zs: list(A)] :
( ( aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
<=> ( ( ( Xss2 = nil(list(A)) )
=> ( ( Ys = nil(A) )
& ( Zs = nil(A) ) ) )
& ( ( Xss2 != nil(list(A)) )
=> ? [Xss1: list(list(A)),Xs3: list(A),Xs5: list(A),Xss22: list(list(A))] :
( ( Xss2 = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),Xs5)),Xss22)) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss1)),Xs3) )
& ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),aa(list(list(A)),list(A),concat(A),Xss22)) ) ) ) ) ) ).
% concat_eq_append_conv
tff(fact_446_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),rotate1(A),Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
% rotate1_is_Nil_conv
tff(fact_447_concat__append,axiom,
! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xs)),aa(list(list(A)),list(A),concat(A),Ys)) ).
% concat_append
tff(fact_448_assn__aci_I8_J,axiom,
! [X: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),Y)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),Y) ).
% assn_aci(8)
tff(fact_449_assn__aci_I7_J,axiom,
! [X: assn,Y: assn,Z2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z2)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),Z2)) ).
% assn_aci(7)
tff(fact_450_assn__aci_I5_J,axiom,
! [X: assn,Y: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),Y) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),X) ).
% assn_aci(5)
tff(fact_451_norm__assertion__simps_I32_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),X) = X ).
% norm_assertion_simps(32)
tff(fact_452_norm__assertion__simps_I15_J,axiom,
! [X: assn,Y: assn,Z2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),Y)),Z2) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Y),Z2)) ).
% norm_assertion_simps(15)
tff(fact_453_concat__map__maps,axiom,
! [A: $tType,B: $tType,F: fun(B,list(A)),Xs: list(B)] : aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F),Xs)) = maps(B,A,F,Xs) ).
% concat_map_maps
tff(fact_454_maps__def,axiom,
! [B: $tType,A: $tType,F: fun(A,list(B)),Xs: list(A)] : maps(A,B,F,Xs) = aa(list(list(B)),list(B),concat(B),aa(list(A),list(list(B)),aa(fun(A,list(B)),fun(list(A),list(list(B))),map(A,list(B)),F),Xs)) ).
% maps_def
tff(fact_455_map__concat,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(list(B)),list(B),concat(B),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),Xs)) ).
% map_concat
tff(fact_456_rev__concat,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(A),list(A),rev(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),rev(A)),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) ).
% rev_concat
tff(fact_457_List_Obind__def,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,list(B))] : bind(A,B,Xs,F) = aa(list(list(B)),list(B),concat(B),aa(list(A),list(list(B)),aa(fun(A,list(B)),fun(list(A),list(list(B))),map(A,list(B)),F),Xs)) ).
% List.bind_def
tff(fact_458_rotate1_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).
% rotate1.simps(1)
tff(fact_459_rotate1__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),rotate1(B),Xs)) ).
% rotate1_map
tff(fact_460_concat_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(list(A)),list(A),concat(A),nil(list(A))) = nil(A) ).
% concat.simps(1)
tff(fact_461_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list(A),Xs: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),aa(list(list(A)),list(A),concat(A),Xs)) ).
% concat.simps(2)
tff(fact_462_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
% maps_simps(2)
tff(fact_463_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_464_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_465_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_466_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_467_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: 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),Ls))) = 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),Ls) ) ).
% merge_list.simps(5)
tff(fact_468_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list(list(A)),Ys: list(A),Zs: list(A)] :
( ( aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
=> ( ( Xss2 != nil(list(A)) )
=> ? [Xss12: list(list(A)),Xs2: list(A),Xs4: list(A),Xss23: list(list(A))] :
( ( Xss2 = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs4)),Xss23)) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss12)),Xs2) )
& ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(list(A)),list(A),concat(A),Xss23)) ) ) ) ) ).
% concat_eq_appendD
tff(fact_469_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ~ ? [X3: A] :
( pp(member2(A,X3,S))
& pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S)))) ) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_470_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> pp(member2(A,lattic7623131987881927897min_on(A,B,F,S),S)) ) ) ) ).
% arg_min_if_finite(1)
tff(fact_471_Sup__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,A4)),lattic5882676163264333800up_fin(A,B3)) ) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_472_Inf__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) ) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_473_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),nil(A)),Uu) = nil(product_prod(A,B)) ).
% product.simps(1)
tff(fact_474_butlast__snoc,axiom,
! [A: $tType,Xs: list(A),X: A] : aa(list(A),list(A),butlast(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)))) = Xs ).
% butlast_snoc
tff(fact_475_last__snoc,axiom,
! [A: $tType,Xs: list(A),X: A] : last(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)))) = X ).
% last_snoc
tff(fact_476_Max__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),lattic643756798349783984er_Max(A,A4)),X))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).
% Max_less_iff
tff(fact_477_concat__conv__foldr,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(list(A)),list(A),concat(A),Xss2) = aa(list(A),list(A),aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),foldr(list(A),list(A)),append(A)),Xss2),nil(A)) ).
% concat_conv_foldr
tff(fact_478_Min__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),lattic643756798350308766er_Min(A,A4)))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).
% Min_gr_iff
tff(fact_479_SuccI,axiom,
! [A: $tType,Kl: list(A),K: A,Kl2: set(list(A))] :
( pp(member2(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Kl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),nil(A))),Kl2))
=> pp(member2(A,K,bNF_Greatest_Succ(A,Kl2,Kl))) ) ).
% SuccI
tff(fact_480_foldr__append,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A3: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),Xs),aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),Ys),A3)) ).
% foldr_append
tff(fact_481_last__appendL,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( Ys = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Xs) ) ) ).
% last_appendL
tff(fact_482_last__appendR,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( Ys != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Ys) ) ) ).
% last_appendR
tff(fact_483_inf__Sup__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A4: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,A3,A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),lattic5882676163264333800up_fin(A,A4)) = A3 ) ) ) ) ).
% inf_Sup_absorb
tff(fact_484_append__butlast__last__id,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),butlast(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,Xs)),nil(A))) = Xs ) ) ).
% append_butlast_last_id
tff(fact_485_butlast_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).
% butlast.simps(1)
tff(fact_486_map__butlast,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),butlast(B),Xs)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ).
% map_butlast
tff(fact_487_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: 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))) = Ys )
<=> ( ( Ys != nil(A) )
& ( aa(list(A),list(A),butlast(A),Ys) = Xs )
& ( last(A,Ys) = X ) ) ) ).
% snoc_eq_iff_butlast
tff(fact_488_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) )
& ( aa(list(A),list(A),butlast(A),Ys) = Xs )
& ( last(A,Ys) = X ) ) ) ).
% snoc_eq_iff_butlast'
tff(fact_489_Min__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> pp(member2(A,lattic643756798350308766er_Min(A,A4),A4)) ) ) ) ).
% Min_in
tff(fact_490_Max__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> pp(member2(A,lattic643756798349783984er_Max(A,A4),A4)) ) ) ) ).
% Max_in
tff(fact_491_last_Osimps,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( ( Xs = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = X ) )
& ( ( Xs != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = last(A,Xs) ) ) ) ).
% last.simps
tff(fact_492_last__ConsL,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( Xs = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = X ) ) ).
% last_ConsL
tff(fact_493_last__ConsR,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( Xs != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = last(A,Xs) ) ) ).
% last_ConsR
tff(fact_494_last__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,B)] :
( ( Xs != nil(A) )
=> ( last(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)) = aa(A,B,F,last(A,Xs)) ) ) ).
% last_map
tff(fact_495_last__append,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( ( Ys = nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Xs) ) )
& ( ( Ys != nil(A) )
=> ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Ys) ) ) ) ).
% last_append
tff(fact_496_longest__common__suffix,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
? [Ss: list(A),Xs4: list(A),Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),Ss) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),Ss) )
& ( ( Xs4 = nil(A) )
| ( Ys4 = nil(A) )
| ( last(A,Xs4) != last(A,Ys4) ) ) ) ).
% longest_common_suffix
tff(fact_497_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).
% butlast.simps(2)
tff(fact_498_butlast__append,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( ( Ys = nil(A) )
=> ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),butlast(A),Xs) ) )
& ( ( Ys != nil(A) )
=> ( aa(list(A),list(A),butlast(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),aa(list(A),list(A),butlast(A),Ys)) ) ) ) ).
% butlast_append
tff(fact_499_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ).
% Inf_fin.in_idem
tff(fact_500_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),foldr(A,list(A)),cons(A)),Xs),nil(A)) = Xs ).
% Misc.foldr_Cons
tff(fact_501_Min__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),lattic643756798350308766er_Min(A,A4)),X))
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).
% Min_less_iff
tff(fact_502_Max__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),lattic643756798349783984er_Max(A,A4)))
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).
% Max_gr_iff
tff(fact_503_butlast__eq__consE,axiom,
! [A: $tType,L: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),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_504_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),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_505_SuccD,axiom,
! [A: $tType,K: A,Kl2: set(list(A)),Kl: list(A)] :
( pp(member2(A,K,bNF_Greatest_Succ(A,Kl2,Kl)))
=> pp(member2(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Kl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),nil(A))),Kl2)) ) ).
% SuccD
tff(fact_506_shift__def,axiom,
! [B: $tType,A: $tType,Lab: fun(list(A),B),K: A,X3: list(A)] : bNF_Greatest_shift(A,B,Lab,K,X3) = aa(list(A),B,Lab,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),X3)) ).
% shift_def
tff(fact_507_empty__Shift,axiom,
! [A: $tType,Kl2: set(list(A)),K: A] :
( pp(member2(list(A),nil(A),Kl2))
=> ( pp(member2(A,K,bNF_Greatest_Succ(A,Kl2,nil(A))))
=> pp(member2(list(A),nil(A),bNF_Greatest_Shift(A,Kl2,K))) ) ) ).
% empty_Shift
tff(fact_508_Succ__Shift,axiom,
! [A: $tType,Kl2: set(list(A)),K: A,Kl: list(A)] : bNF_Greatest_Succ(A,bNF_Greatest_Shift(A,Kl2,K),Kl) = bNF_Greatest_Succ(A,Kl2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),Kl)) ).
% Succ_Shift
tff(fact_509_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,A4)) ) ) ) ) ).
% Sup_fin.insert
tff(fact_510_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).
% Inf_fin.insert
tff(fact_511_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),lattic5882676163264333800up_fin(A,A4))) ) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_512_Sup__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic5882676163264333800up_fin(A,B3)),lattic5882676163264333800up_fin(A,A4)) = lattic5882676163264333800up_fin(A,A4) ) ) ) ) ) ).
% Sup_fin.subset
tff(fact_513_Inf__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ) ).
% Inf_fin.subset
tff(fact_514_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,A4)) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_515_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,lattic5882676163264333800up_fin(A,A4),A4)) ) ) ) ) ).
% Sup_fin.closed
tff(fact_516_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).
% order_refl
tff(fact_517_dual__order_Orefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).
% dual_order.refl
tff(fact_518_subsetI,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ! [X2: A] :
( pp(member2(A,X2,A4))
=> pp(member2(A,X2,B3)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% subsetI
tff(fact_519_subset__antisym,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( A4 = B3 ) ) ) ).
% subset_antisym
tff(fact_520_insertCI,axiom,
! [A: $tType,A3: A,B3: set(A),B2: A] :
( ( ~ pp(member2(A,A3,B3))
=> ( A3 = B2 ) )
=> pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).
% insertCI
tff(fact_521_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A4: set(A)] :
( pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A4)))
<=> ( ( A3 = B2 )
| pp(member2(A,A3,A4)) ) ) ).
% insert_iff
tff(fact_522_insert__absorb2,axiom,
! [A: $tType,X: A,A4: 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),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4) ).
% insert_absorb2
tff(fact_523_inf_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% inf.bounded_iff
tff(fact_524_le__inf__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).
% le_inf_iff
tff(fact_525_le__sup__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% le_sup_iff
tff(fact_526_sup_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).
% sup.bounded_iff
tff(fact_527_empty__subsetI,axiom,
! [A: $tType,A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A4)) ).
% empty_subsetI
tff(fact_528_subset__empty,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),bot_bot(set(A))))
<=> ( A4 = bot_bot(set(A)) ) ) ).
% subset_empty
tff(fact_529_singletonI,axiom,
! [A: $tType,A3: A] : pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ).
% singletonI
tff(fact_530_insert__subset,axiom,
! [A: $tType,X: A,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),B3))
<=> ( pp(member2(A,X,B3))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ) ).
% insert_subset
tff(fact_531_Int__subset__iff,axiom,
! [A: $tType,C4: set(A),A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A4))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B3)) ) ) ).
% Int_subset_iff
tff(fact_532_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% Int_insert_right_if1
tff(fact_533_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
( ~ pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ).
% Int_insert_right_if0
tff(fact_534_insert__inter__insert,axiom,
! [A: $tType,A3: A,A4: set(A),B3: 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),A3),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).
% insert_inter_insert
tff(fact_535_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C4: set(A),B3: set(A)] :
( pp(member2(A,A3,C4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) ) ) ).
% Int_insert_left_if1
tff(fact_536_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C4: set(A),B3: set(A)] :
( ~ pp(member2(A,A3,C4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4) ) ) ).
% Int_insert_left_if0
tff(fact_537_Un__subset__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C4))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4)) ) ) ).
% Un_subset_iff
tff(fact_538_Un__insert__left,axiom,
! [A: $tType,A3: A,B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) ).
% Un_insert_left
tff(fact_539_Un__insert__right,axiom,
! [A: $tType,A4: set(A),A3: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ).
% Un_insert_right
tff(fact_540_psubsetI,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != B3 )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3)) ) ) ).
% psubsetI
tff(fact_541_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A4: set(A)] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) )
<=> ( ( A3 = B2 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).
% singleton_insert_inj_eq
tff(fact_542_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A4: set(A),B2: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
<=> ( ( A3 = B2 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).
% singleton_insert_inj_eq'
tff(fact_543_disjoint__insert_I2_J,axiom,
! [A: $tType,A4: set(A),B2: A,B3: set(A)] :
( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) )
<=> ( ~ pp(member2(A,B2,A4))
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ) ).
% disjoint_insert(2)
tff(fact_544_disjoint__insert_I1_J,axiom,
! [A: $tType,B3: set(A),A3: A,A4: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = bot_bot(set(A)) )
<=> ( ~ pp(member2(A,A3,B3))
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A4) = bot_bot(set(A)) ) ) ) ).
% disjoint_insert(1)
tff(fact_545_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A4: set(A),B3: 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),A3),A4)),B3) )
<=> ( ~ pp(member2(A,A3,B3))
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ) ).
% insert_disjoint(2)
tff(fact_546_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A4: set(A),B3: 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),A3),A4)),B3) = bot_bot(set(A)) )
<=> ( ~ pp(member2(A,A3,B3))
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ) ).
% insert_disjoint(1)
tff(fact_547_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_548_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_549_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_550_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).
% Inf_fin.singleton
tff(fact_551_Max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798349783984er_Max(A,A4)),X))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).
% Max.bounded_iff
tff(fact_552_Min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A4)))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).
% Min.bounded_iff
tff(fact_553_in__mono,axiom,
! [A: $tType,A4: set(A),B3: set(A),X: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(member2(A,X,A4))
=> pp(member2(A,X,B3)) ) ) ).
% in_mono
tff(fact_554_insertE,axiom,
! [A: $tType,A3: A,B2: A,A4: set(A)] :
( pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A4)))
=> ( ( A3 != B2 )
=> pp(member2(A,A3,A4)) ) ) ).
% insertE
tff(fact_555_subsetD,axiom,
! [A: $tType,A4: set(A),B3: set(A),C3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(member2(A,C3,A4))
=> pp(member2(A,C3,B3)) ) ) ).
% subsetD
tff(fact_556_insertI1,axiom,
! [A: $tType,A3: A,B3: set(A)] : pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3))) ).
% insertI1
tff(fact_557_insertI2,axiom,
! [A: $tType,A3: A,B3: set(A),B2: A] :
( pp(member2(A,A3,B3))
=> pp(member2(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).
% insertI2
tff(fact_558_equalityE,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( A4 = B3 )
=> ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ) ).
% equalityE
tff(fact_559_subset__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(member2(A,X4,B3)) ) ) ).
% subset_eq
tff(fact_560_equalityD1,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( A4 = B3 )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% equalityD1
tff(fact_561_equalityD2,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( A4 = B3 )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ).
% equalityD2
tff(fact_562_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set(A)] :
( pp(member2(A,X,A4))
=> ~ ! [B6: set(A)] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B6) )
=> pp(member2(A,X,B6)) ) ) ).
% Set.set_insert
tff(fact_563_subset__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> ! [T2: A] :
( pp(member2(A,T2,A4))
=> pp(member2(A,T2,B3)) ) ) ).
% subset_iff
tff(fact_564_insert__mono,axiom,
! [A: $tType,C4: set(A),D2: set(A),A3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),D2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),C4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),D2))) ) ).
% insert_mono
tff(fact_565_subset__refl,axiom,
! [A: $tType,A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),A4)) ).
% subset_refl
tff(fact_566_Collect__mono,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))) ) ).
% Collect_mono
tff(fact_567_insert__ident,axiom,
! [A: $tType,X: A,A4: set(A),B3: set(A)] :
( ~ pp(member2(A,X,A4))
=> ( ~ pp(member2(A,X,B3))
=> ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3) )
<=> ( A4 = B3 ) ) ) ) ).
% insert_ident
tff(fact_568_subset__trans,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4)) ) ) ).
% subset_trans
tff(fact_569_insert__absorb,axiom,
! [A: $tType,A3: A,A4: set(A)] :
( pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = A4 ) ) ).
% insert_absorb
tff(fact_570_insert__eq__iff,axiom,
! [A: $tType,A3: A,A4: set(A),B2: A,B3: set(A)] :
( ~ pp(member2(A,A3,A4))
=> ( ~ pp(member2(A,B2,B3))
=> ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3) )
<=> ( ( ( A3 = B2 )
=> ( A4 = B3 ) )
& ( ( A3 != B2 )
=> ? [C5: set(A)] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),C5) )
& ~ pp(member2(A,B2,C5))
& ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),C5) )
& ~ pp(member2(A,A3,C5)) ) ) ) ) ) ) ).
% insert_eq_iff
tff(fact_571_set__eq__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( A4 = B3 )
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ) ).
% set_eq_subset
tff(fact_572_subset__insert,axiom,
! [A: $tType,X: A,A4: set(A),B3: set(A)] :
( ~ pp(member2(A,X,A4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ) ).
% subset_insert
tff(fact_573_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: 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),A4)) = 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),A4)) ).
% insert_commute
tff(fact_574_subset__insertI,axiom,
! [A: $tType,B3: set(A),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3))) ).
% subset_insertI
tff(fact_575_subset__insertI2,axiom,
! [A: $tType,A4: set(A),B3: set(A),B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).
% subset_insertI2
tff(fact_576_Collect__mono__iff,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)))
<=> ! [X4: A] :
( pp(aa(A,bool,P,X4))
=> pp(aa(A,bool,Q,X4)) ) ) ).
% Collect_mono_iff
tff(fact_577_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A4: set(A)] :
( pp(member2(A,A3,A4))
=> ? [B6: set(A)] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B6) )
& ~ pp(member2(A,A3,B6)) ) ) ).
% mk_disjoint_insert
tff(fact_578_nle__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& ( B2 != A3 ) ) ) ) ).
% nle_le
tff(fact_579_le__cases3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X)) )
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).
% le_cases3
tff(fact_580_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ( X = Y )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).
% order_class.order_eq_iff
tff(fact_581_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% ord_eq_le_trans
tff(fact_582_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( B2 = C3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% ord_le_eq_trans
tff(fact_583_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( X = Y ) ) ) ) ).
% order_antisym
tff(fact_584_order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% order.trans
tff(fact_585_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).
% order_trans
tff(fact_586_linorder__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B4))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B4)) )
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ).
% linorder_wlog
tff(fact_587_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% dual_order.eq_iff
tff(fact_588_dual__order_Oantisym,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
tff(fact_589_dual__order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).
% dual_order.trans
tff(fact_590_antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( A3 = B2 ) ) ) ) ).
% antisym
tff(fact_591_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))) ) ) ).
% le_funD
tff(fact_592_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))) ) ) ).
% le_funE
tff(fact_593_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)))
=> pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G)) ) ) ).
% le_funI
tff(fact_594_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
<=> ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,G,X4))) ) ) ).
% le_fun_def
tff(fact_595_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% Orderings.order_eq_iff
tff(fact_596_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% order_subst1
tff(fact_597_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C3: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,B2)),C3))
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,X2)),aa(A,C,F,Y2))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,A3)),C3)) ) ) ) ) ).
% order_subst2
tff(fact_598_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( A3 = B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> ( ( C3 = D3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),D3)) ) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_599_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( ( X = Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% order_eq_refl
tff(fact_600_subset__Collect__conv,axiom,
! [A: $tType,S: set(A),P: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),collect(A,P)))
<=> ! [X4: A] :
( pp(member2(A,X4,S))
=> pp(aa(A,bool,P,X4)) ) ) ).
% subset_Collect_conv
tff(fact_601_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_linear
tff(fact_602_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% ord_eq_le_subst
tff(fact_603_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( aa(A,B,F,B2) = C3 )
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,A3)),C3)) ) ) ) ) ).
% ord_le_eq_subst
tff(fact_604_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_le_cases
tff(fact_605_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% order_antisym_conv
tff(fact_606_subset__singletonD,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
=> ( ( A4 = bot_bot(set(A)) )
| ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ).
% subset_singletonD
tff(fact_607_subset__singleton__iff,axiom,
! [A: $tType,X5: set(A),A3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))
<=> ( ( X5 = bot_bot(set(A)) )
| ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).
% subset_singleton_iff
tff(fact_608_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [S: set(B),P: fun(set(B),bool),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( pp(aa(set(B),bool,P,bot_bot(set(B))))
=> ( ! [X2: B,S2: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),S2))
=> ( ! [Y4: B] :
( pp(member2(B,Y4,S2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,Y4)),aa(B,A,F,X2))) )
=> ( pp(aa(set(B),bool,P,S2))
=> pp(aa(set(B),bool,P,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X2),S2))) ) ) )
=> pp(aa(set(B),bool,P,S)) ) ) ) ) ).
% finite_ranking_induct
tff(fact_609_finite__subset__induct_H,axiom,
! [A: $tType,F2: set(A),A4: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F2),A4))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [A5: A,F3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( pp(member2(A,A5,A4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F3),A4))
=> ( ~ pp(member2(A,A5,F3))
=> ( pp(aa(set(A),bool,P,F3))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F3))) ) ) ) ) )
=> pp(aa(set(A),bool,P,F2)) ) ) ) ) ).
% finite_subset_induct'
tff(fact_610_finite__subset__induct,axiom,
! [A: $tType,F2: set(A),A4: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F2),A4))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [A5: A,F3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( pp(member2(A,A5,A4))
=> ( ~ pp(member2(A,A5,F3))
=> ( pp(aa(set(A),bool,P,F3))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F3))) ) ) ) )
=> pp(aa(set(A),bool,P,F2)) ) ) ) ) ).
% finite_subset_induct
tff(fact_611_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
=> ( A3 = B2 ) ) ).
% singleton_inject
tff(fact_612_insert__not__empty,axiom,
! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) != bot_bot(set(A)) ).
% insert_not_empty
tff(fact_613_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C3: A,D3: A] :
( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D3),bot_bot(set(A)))) )
<=> ( ( ( A3 = C3 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
tff(fact_614_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( pp(member2(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))
<=> ( B2 = A3 ) ) ).
% singleton_iff
tff(fact_615_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( pp(member2(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))
=> ( B2 = A3 ) ) ).
% singletonD
tff(fact_616_leD,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% leD
tff(fact_617_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% leI
tff(fact_618_nless__le,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
| ( A3 = B2 ) ) ) ) ).
% nless_le
tff(fact_619_antisym__conv1,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv1
tff(fact_620_antisym__conv2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv2
tff(fact_621_dense__ge,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y: A] :
( ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).
% dense_ge
tff(fact_622_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,Z2: A] :
( ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).
% dense_le
tff(fact_623_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).
% less_le_not_le
tff(fact_624_not__le__imp__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% not_le_imp_less
tff(fact_625_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
tff(fact_626_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
tff(fact_627_order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% order.strict_trans1
tff(fact_628_order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% order.strict_trans2
tff(fact_629_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% order.strict_iff_not
tff(fact_630_dense__ge__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
=> ( ! [W3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),W3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W3),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% dense_ge_bounded
tff(fact_631_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ! [W3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W3),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W3),Z2)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% dense_le_bounded
tff(fact_632_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
tff(fact_633_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
tff(fact_634_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).
% dual_order.strict_trans1
tff(fact_635_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).
% dual_order.strict_trans2
tff(fact_636_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% dual_order.strict_iff_not
tff(fact_637_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% order.strict_implies_order
tff(fact_638_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% dual_order.strict_implies_order
tff(fact_639_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y ) ) ) ) ).
% order_le_less
tff(fact_640_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& ( X != Y ) ) ) ) ).
% order_less_le
tff(fact_641_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_not_le
tff(fact_642_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_not_less
tff(fact_643_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% order_less_imp_le
tff(fact_644_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( A3 != B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% order_le_neq_trans
tff(fact_645_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% order_neq_le_trans
tff(fact_646_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_le_less_trans
tff(fact_647_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_less_le_trans
tff(fact_648_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% order_le_less_subst1
tff(fact_649_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C3: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,B2)),C3))
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,X2)),aa(A,C,F,Y2))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C3)) ) ) ) ) ).
% order_le_less_subst2
tff(fact_650_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C3: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C3))) ) ) ) ) ).
% order_less_le_subst1
tff(fact_651_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C3: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,B2)),C3))
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,X2)),aa(A,C,F,Y2))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C3)) ) ) ) ) ).
% order_less_le_subst2
tff(fact_652_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_le_less_linear
tff(fact_653_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_654_Int__insert__right,axiom,
! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
( ( pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) )
& ( ~ pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ) ) ) ).
% Int_insert_right
tff(fact_655_Int__insert__left,axiom,
! [A: $tType,A3: A,C4: set(A),B3: set(A)] :
( ( pp(member2(A,A3,C4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) ) )
& ( ~ pp(member2(A,A3,C4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4) ) ) ) ).
% Int_insert_left
tff(fact_656_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).
% le_numeral_extra(4)
tff(fact_657_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_uniqueI
tff(fact_658_bot_Oextremum__unique,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
<=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_unique
tff(fact_659_bot_Oextremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A3)) ) ).
% bot.extremum
tff(fact_660_inf_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).
% inf.coboundedI2
tff(fact_661_inf_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).
% inf.coboundedI1
tff(fact_662_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb_iff2
tff(fact_663_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb_iff1
tff(fact_664_inf_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2)) ) ).
% inf.cobounded2
tff(fact_665_inf_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3)) ) ).
% inf.cobounded1
tff(fact_666_inf_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).
% inf.order_iff
tff(fact_667_inf__greatest,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))) ) ) ) ).
% inf_greatest
tff(fact_668_inf_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))) ) ) ) ).
% inf.boundedI
tff(fact_669_inf_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% inf.boundedE
tff(fact_670_inf__absorb2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).
% inf_absorb2
tff(fact_671_inf__absorb1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% inf_absorb1
tff(fact_672_inf_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb2
tff(fact_673_inf_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb1
tff(fact_674_le__iff__inf,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% le_iff_inf
tff(fact_675_inf__unique,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X2: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y2)),X2))
=> ( ! [X2: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y2)),Y2))
=> ( ! [X2: A,Y2: A,Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,Y2),Z4))) ) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% inf_unique
tff(fact_676_inf_OorderI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% inf.orderI
tff(fact_677_inf_OorderE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).
% inf.orderE
tff(fact_678_le__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% le_infI2
tff(fact_679_le__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% le_infI1
tff(fact_680_inf__mono,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C3),D3))) ) ) ) ).
% inf_mono
tff(fact_681_le__infI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).
% le_infI
tff(fact_682_le__infE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).
% le_infE
tff(fact_683_inf__le2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).
% inf_le2
tff(fact_684_inf__le1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).
% inf_le1
tff(fact_685_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).
% inf_sup_ord(1)
tff(fact_686_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).
% inf_sup_ord(2)
tff(fact_687_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% inf_sup_ord(4)
tff(fact_688_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% inf_sup_ord(3)
tff(fact_689_le__supE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).
% le_supE
tff(fact_690_le__supI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).
% le_supI
tff(fact_691_sup__ge1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% sup_ge1
tff(fact_692_sup__ge2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% sup_ge2
tff(fact_693_le__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% le_supI1
tff(fact_694_le__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% le_supI2
tff(fact_695_sup_Omono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,A3: A,D3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).
% sup.mono
tff(fact_696_sup__mono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3))) ) ) ) ).
% sup_mono
tff(fact_697_sup__least,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X)) ) ) ) ).
% sup_least
tff(fact_698_le__iff__sup,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% le_iff_sup
tff(fact_699_sup_OorderE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).
% sup.orderE
tff(fact_700_sup_OorderI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% sup.orderI
tff(fact_701_sup__unique,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X2: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,X2),Y2)))
=> ( ! [X2: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F,X2),Y2)))
=> ( ! [X2: A,Y2: A,Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,Y2),Z4)),X2)) ) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% sup_unique
tff(fact_702_sup_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb1
tff(fact_703_sup_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb2
tff(fact_704_sup__absorb1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).
% sup_absorb1
tff(fact_705_sup__absorb2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% sup_absorb2
tff(fact_706_sup_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).
% sup.boundedE
tff(fact_707_sup_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3)) ) ) ) ).
% sup.boundedI
tff(fact_708_sup_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).
% sup.order_iff
tff(fact_709_sup_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).
% sup.cobounded1
tff(fact_710_sup_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).
% sup.cobounded2
tff(fact_711_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb_iff1
tff(fact_712_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb_iff2
tff(fact_713_sup_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI1
tff(fact_714_sup_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI2
tff(fact_715_Int__Collect__mono,axiom,
! [A: $tType,A4: set(A),B3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ! [X2: A] :
( pp(member2(A,X2,A4))
=> ( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),collect(A,Q)))) ) ) ).
% Int_Collect_mono
tff(fact_716_Int__greatest,axiom,
! [A: $tType,C4: set(A),A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).
% Int_greatest
tff(fact_717_Int__absorb2,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = A4 ) ) ).
% Int_absorb2
tff(fact_718_Int__absorb1,axiom,
! [A: $tType,B3: set(A),A4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = B3 ) ) ).
% Int_absorb1
tff(fact_719_Int__lower2,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),B3)) ).
% Int_lower2
tff(fact_720_Int__lower1,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),A4)) ).
% Int_lower1
tff(fact_721_Int__mono,axiom,
! [A: $tType,A4: set(A),C4: set(A),B3: set(A),D2: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),D2))) ) ) ).
% Int_mono
tff(fact_722_inter__eq__subsetI,axiom,
! [A: $tType,S: set(A),S3: set(A),A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),S3))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S3) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S) ) ) ) ).
% inter_eq_subsetI
tff(fact_723_Un__mono,axiom,
! [A: $tType,A4: set(A),C4: set(A),B3: set(A),D2: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),D2))) ) ) ).
% Un_mono
tff(fact_724_Un__least,axiom,
! [A: $tType,A4: set(A),C4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C4)) ) ) ).
% Un_least
tff(fact_725_Un__upper1,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ).
% Un_upper1
tff(fact_726_Un__upper2,axiom,
! [A: $tType,B3: set(A),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))) ).
% Un_upper2
tff(fact_727_Un__absorb1,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = B3 ) ) ).
% Un_absorb1
tff(fact_728_Un__absorb2,axiom,
! [A: $tType,B3: set(A),A4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = A4 ) ) ).
% Un_absorb2
tff(fact_729_subset__UnE,axiom,
! [A: $tType,C4: set(A),A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
=> ~ ! [A7: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),A4))
=> ! [B7: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),B3))
=> ( C4 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B7) ) ) ) ) ).
% subset_UnE
tff(fact_730_subset__Un__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = B3 ) ) ).
% subset_Un_eq
tff(fact_731_subset__iff__psubset__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
| ( A4 = B3 ) ) ) ).
% subset_iff_psubset_eq
tff(fact_732_subset__psubset__trans,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),C4)) ) ) ).
% subset_psubset_trans
tff(fact_733_subset__not__subset__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
& ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ) ).
% subset_not_subset_eq
tff(fact_734_psubset__subset__trans,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),C4)) ) ) ).
% psubset_subset_trans
tff(fact_735_psubset__imp__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% psubset_imp_subset
tff(fact_736_psubset__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
& ( A4 != B3 ) ) ) ).
% psubset_eq
tff(fact_737_psubsetE,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ) ).
% psubsetE
tff(fact_738_less__eq__assn__def,axiom,
! [A3: assn,B2: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
<=> ( A3 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B2) ) ) ).
% less_eq_assn_def
tff(fact_739_Max__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: set(A),N4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M),N4))
=> ( ( M != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798349783984er_Max(A,M)),lattic643756798349783984er_Max(A,N4))) ) ) ) ) ).
% Max_mono
tff(fact_740_Max_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798349783984er_Max(A,A4)),lattic643756798349783984er_Max(A,B3))) ) ) ) ) ).
% Max.subset_imp
tff(fact_741_Min__antimono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: set(A),N4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M),N4))
=> ( ( M != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798350308766er_Min(A,N4)),lattic643756798350308766er_Min(A,M))) ) ) ) ) ).
% Min_antimono
tff(fact_742_Min_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798350308766er_Min(A,B3)),lattic643756798350308766er_Min(A,A4))) ) ) ) ) ).
% Min.subset_imp
tff(fact_743_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))) ) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_744_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),lattic5882676163264333800up_fin(A,B3))) ) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_745_infinite__finite__induct,axiom,
! [A: $tType,P: fun(set(A),bool),A4: set(A)] :
( ! [A8: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A8))
=> pp(aa(set(A),bool,P,A8)) )
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [X2: A,F3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( ~ pp(member2(A,X2,F3))
=> ( pp(aa(set(A),bool,P,F3))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F3))) ) ) )
=> pp(aa(set(A),bool,P,A4)) ) ) ) ).
% infinite_finite_induct
tff(fact_746_finite__ne__induct,axiom,
! [A: $tType,F2: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( ( F2 != bot_bot(set(A)) )
=> ( ! [X2: A] : pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))
=> ( ! [X2: A,F3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( ( F3 != bot_bot(set(A)) )
=> ( ~ pp(member2(A,X2,F3))
=> ( pp(aa(set(A),bool,P,F3))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F3))) ) ) ) )
=> pp(aa(set(A),bool,P,F2)) ) ) ) ) ).
% finite_ne_induct
tff(fact_747_finite__induct,axiom,
! [A: $tType,F2: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [X2: A,F3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( ~ pp(member2(A,X2,F3))
=> ( pp(aa(set(A),bool,P,F3))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),F3))) ) ) )
=> pp(aa(set(A),bool,P,F2)) ) ) ) ).
% finite_induct
tff(fact_748_finite_Osimps,axiom,
! [A: $tType,A3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A3))
<=> ( ( A3 = bot_bot(set(A)) )
| ? [A9: set(A),A6: A] :
( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),A9) )
& pp(aa(set(A),bool,finite_finite2(A),A9)) ) ) ) ).
% finite.simps
tff(fact_749_finite_Ocases,axiom,
! [A: $tType,A3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A3))
=> ( ( A3 != bot_bot(set(A)) )
=> ~ ! [A8: set(A)] :
( ? [A5: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),A8)
=> ~ pp(aa(set(A),bool,finite_finite2(A),A8)) ) ) ) ).
% finite.cases
tff(fact_750_insert__is__Un,axiom,
! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = 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),A3),bot_bot(set(A)))),A4) ).
% insert_is_Un
tff(fact_751_Un__singleton__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A),X: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
<=> ( ( ( A4 = bot_bot(set(A)) )
& ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
| ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B3 = bot_bot(set(A)) ) )
| ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).
% Un_singleton_iff
tff(fact_752_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set(A),B3: 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)),A4),B3) )
<=> ( ( ( A4 = bot_bot(set(A)) )
& ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
| ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B3 = bot_bot(set(A)) ) )
| ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
& ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).
% singleton_Un_iff
tff(fact_753_finite__has__minimal,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member2(A,X2,A4))
& ! [Xa3: A] :
( pp(member2(A,Xa3,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa3),X2))
=> ( X2 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_754_finite__has__maximal,axiom,
! [A: $tType] :
( order(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member2(A,X2,A4))
& ! [Xa3: A] :
( pp(member2(A,Xa3,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3))
=> ( X2 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_755_distrib__inf__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)))) ) ).
% distrib_inf_le
tff(fact_756_distrib__sup__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)))) ) ).
% distrib_sup_le
tff(fact_757_disjoint__mono,axiom,
! [A: $tType,A3: set(A),A10: set(A),B2: set(A),B8: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A10))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B2),B8))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A10),B8) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ) ) ).
% disjoint_mono
tff(fact_758_Un__Int__assoc__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A4)) ) ).
% Un_Int_assoc_eq
tff(fact_759_finite__linorder__max__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [B4: A,A8: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A8))
=> ( ! [X3: A] :
( pp(member2(A,X3,A8))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),B4)) )
=> ( pp(aa(set(A),bool,P,A8))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A8))) ) ) )
=> pp(aa(set(A),bool,P,A4)) ) ) ) ) ).
% finite_linorder_max_induct
tff(fact_760_finite__linorder__min__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [B4: A,A8: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A8))
=> ( ! [X3: A] :
( pp(member2(A,X3,A8))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X3)) )
=> ( pp(aa(set(A),bool,P,A8))
=> pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A8))) ) ) )
=> pp(aa(set(A),bool,P,A4)) ) ) ) ) ).
% finite_linorder_min_induct
tff(fact_761_ShiftD,axiom,
! [A: $tType,Kl: list(A),Kl2: set(list(A)),K: A] :
( pp(member2(list(A),Kl,bNF_Greatest_Shift(A,Kl2,K)))
=> pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),Kl),Kl2)) ) ).
% ShiftD
tff(fact_762_Max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798349783984er_Max(A,A4)),X)) ) ) ) ) ).
% Max.boundedI
tff(fact_763_Max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798349783984er_Max(A,A4)),X))
=> ! [A11: A] :
( pp(member2(A,A11,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A11),X)) ) ) ) ) ) ).
% Max.boundedE
tff(fact_764_eq__Max__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),M2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ( M2 = lattic643756798349783984er_Max(A,A4) )
<=> ( pp(member2(A,M2,A4))
& ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M2)) ) ) ) ) ) ) ).
% eq_Max_iff
tff(fact_765_Max__ge__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic643756798349783984er_Max(A,A4)))
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).
% Max_ge_iff
tff(fact_766_Max__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),M2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ( lattic643756798349783984er_Max(A,A4) = M2 )
<=> ( pp(member2(A,M2,A4))
& ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M2)) ) ) ) ) ) ) ).
% Max_eq_iff
tff(fact_767_Min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A4))) ) ) ) ) ).
% Min.boundedI
tff(fact_768_Min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic643756798350308766er_Min(A,A4)))
=> ! [A11: A] :
( pp(member2(A,A11,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A11)) ) ) ) ) ) ).
% Min.boundedE
tff(fact_769_eq__Min__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),M2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ( M2 = lattic643756798350308766er_Min(A,A4) )
<=> ( pp(member2(A,M2,A4))
& ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),X4)) ) ) ) ) ) ) ).
% eq_Min_iff
tff(fact_770_Min__le__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic643756798350308766er_Min(A,A4)),X))
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).
% Min_le_iff
tff(fact_771_Min__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),M2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ( lattic643756798350308766er_Min(A,A4) = M2 )
<=> ( pp(member2(A,M2,A4))
& ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),X4)) ) ) ) ) ) ) ).
% Min_eq_iff
tff(fact_772_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),X))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_773_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_774_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),X)) ) ) ) ) ).
% Sup_fin.boundedI
tff(fact_775_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A4)),X))
=> ! [A11: A] :
( pp(member2(A,A11,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A11),X)) ) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_776_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))) ) ) ) ) ).
% Inf_fin.boundedI
tff(fact_777_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)))
=> ! [A11: A] :
( pp(member2(A,A11,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A11)) ) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_778_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S: set(A),Y: A,F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ( pp(member2(A,Y,S))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S))),aa(A,B,F,Y))) ) ) ) ) ).
% arg_min_least
tff(fact_779_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_780_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A4),A4)) ) ) ) ) ).
% Inf_fin.closed
tff(fact_781_mergesort__def,axiom,
! [A: $tType] :
( ord(A)
=> ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% mergesort_def
tff(fact_782_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_783_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_784_subset__emptyI,axiom,
! [A: $tType,A4: set(A)] :
( ! [X2: A] : ~ pp(member2(A,X2,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),bot_bot(set(A)))) ) ).
% subset_emptyI
tff(fact_785_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),lattic643756798350308766er_Min(A,A4)),aa(set(A),list(A),linord4507533701916653071of_set(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),lattic643756798350308766er_Min(A,A4)),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set_nonempty
tff(fact_786_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,minus_minus(set(A),A4,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_787_Sup__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,A4) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,A4) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic5882676163264333800up_fin(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Sup_fin.remove
tff(fact_788_Inf__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Inf_fin.remove
tff(fact_789_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),minus_minus(set(A),A4,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_790_is__singletonE,axiom,
! [A: $tType,A4: set(A)] :
( is_singleton(A,A4)
=> ~ ! [X2: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).
% is_singletonE
tff(fact_791_minus__apply,axiom,
! [B: $tType,A: $tType] :
( minus(B)
=> ! [A4: fun(A,B),B3: fun(A,B),X: A] : aa(A,B,minus_minus(fun(A,B),A4,B3),X) = minus_minus(B,aa(A,B,A4,X),aa(A,B,B3,X)) ) ).
% minus_apply
tff(fact_792_Diff__idemp,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : minus_minus(set(A),minus_minus(set(A),A4,B3),B3) = minus_minus(set(A),A4,B3) ).
% Diff_idemp
tff(fact_793_Diff__iff,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,minus_minus(set(A),A4,B3)))
<=> ( pp(member2(A,C3,A4))
& ~ pp(member2(A,C3,B3)) ) ) ).
% Diff_iff
tff(fact_794_DiffI,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,A4))
=> ( ~ pp(member2(A,C3,B3))
=> pp(member2(A,C3,minus_minus(set(A),A4,B3))) ) ) ).
% DiffI
tff(fact_795_Diff__empty,axiom,
! [A: $tType,A4: set(A)] : minus_minus(set(A),A4,bot_bot(set(A))) = A4 ).
% Diff_empty
tff(fact_796_empty__Diff,axiom,
! [A: $tType,A4: set(A)] : minus_minus(set(A),bot_bot(set(A)),A4) = bot_bot(set(A)) ).
% empty_Diff
tff(fact_797_Diff__cancel,axiom,
! [A: $tType,A4: set(A)] : minus_minus(set(A),A4,A4) = bot_bot(set(A)) ).
% Diff_cancel
tff(fact_798_insert__Diff1,axiom,
! [A: $tType,X: A,B3: set(A),A4: set(A)] :
( pp(member2(A,X,B3))
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4),B3) = minus_minus(set(A),A4,B3) ) ) ).
% insert_Diff1
tff(fact_799_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set(A),B3: set(A)] :
( ~ pp(member2(A,X,A4))
=> ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)) = minus_minus(set(A),A4,B3) ) ) ).
% Diff_insert0
tff(fact_800_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),B3,A4)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A4) ).
% Un_Diff_cancel2
tff(fact_801_Un__Diff__cancel,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),minus_minus(set(A),B3,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) ).
% Un_Diff_cancel
tff(fact_802_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( minus_minus(set(A),A4,B3) = bot_bot(set(A)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% Diff_eq_empty_iff
tff(fact_803_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) ).
% insert_Diff_single
tff(fact_804_Diff__disjoint,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),minus_minus(set(A),B3,A4)) = bot_bot(set(A)) ).
% Diff_disjoint
tff(fact_805_less__assn__def,axiom,
! [A3: assn,B2: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less(assn),A3),B2))
<=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
& ( A3 != B2 ) ) ) ).
% less_assn_def
tff(fact_806_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F),G))
<=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
& ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G),F)) ) ) ) ).
% less_fun_def
tff(fact_807_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C3: A,B2: A] : minus_minus(A,minus_minus(A,A3,C3),B2) = minus_minus(A,minus_minus(A,A3,B2),C3) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_808_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C3,D3) )
=> ( ( A3 = B2 )
<=> ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_809_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( minus(B)
=> ! [A4: fun(A,B),B3: fun(A,B),X3: A] : aa(A,B,minus_minus(fun(A,B),A4,B3),X3) = minus_minus(B,aa(A,B,A4,X3),aa(A,B,B3,X3)) ) ).
% fun_diff_def
tff(fact_810_DiffD2,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,minus_minus(set(A),A4,B3)))
=> ~ pp(member2(A,C3,B3)) ) ).
% DiffD2
tff(fact_811_DiffD1,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,minus_minus(set(A),A4,B3)))
=> pp(member2(A,C3,A4)) ) ).
% DiffD1
tff(fact_812_DiffE,axiom,
! [A: $tType,C3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C3,minus_minus(set(A),A4,B3)))
=> ~ ( pp(member2(A,C3,A4))
=> pp(member2(A,C3,B3)) ) ) ).
% DiffE
tff(fact_813_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C3,D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3)) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_814_diff__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,C3)),minus_minus(A,B2,C3))) ) ) ).
% diff_right_mono
tff(fact_815_diff__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,C3,A3)),minus_minus(A,C3,B2))) ) ) ).
% diff_left_mono
tff(fact_816_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,C3)),minus_minus(A,B2,D3))) ) ) ) ).
% diff_mono
tff(fact_817_diff__strict__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,C3)),minus_minus(A,B2,C3))) ) ) ).
% diff_strict_right_mono
tff(fact_818_diff__strict__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,C3,A3)),minus_minus(A,C3,B2))) ) ) ).
% diff_strict_left_mono
tff(fact_819_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C3,D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3)) ) ) ) ).
% diff_eq_diff_less
tff(fact_820_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,C3)),minus_minus(A,B2,D3))) ) ) ) ).
% diff_strict_mono
tff(fact_821_left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),C3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% left_diff_distrib
tff(fact_822_right__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).
% right_diff_distrib
tff(fact_823_left__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,C3)),A3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) ) ).
% left_diff_distrib'
tff(fact_824_right__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).
% right_diff_distrib'
tff(fact_825_double__diff,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4))
=> ( minus_minus(set(A),B3,minus_minus(set(A),C4,A4)) = A4 ) ) ) ).
% double_diff
tff(fact_826_Diff__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),A4,B3)),A4)) ).
% Diff_subset
tff(fact_827_Diff__mono,axiom,
! [A: $tType,A4: set(A),C4: set(A),D2: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D2),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),A4,B3)),minus_minus(set(A),C4,D2))) ) ) ).
% Diff_mono
tff(fact_828_insert__Diff__if,axiom,
! [A: $tType,X: A,B3: set(A),A4: set(A)] :
( ( pp(member2(A,X,B3))
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4),B3) = minus_minus(set(A),A4,B3) ) )
& ( ~ pp(member2(A,X,B3))
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),minus_minus(set(A),A4,B3)) ) ) ) ).
% insert_Diff_if
tff(fact_829_Int__Diff,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),minus_minus(set(A),B3,C4)) ).
% Int_Diff
tff(fact_830_Diff__Int2,axiom,
! [A: $tType,A4: set(A),C4: set(A),B3: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4),B3) ).
% Diff_Int2
tff(fact_831_Diff__Diff__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : minus_minus(set(A),A4,minus_minus(set(A),A4,B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).
% Diff_Diff_Int
tff(fact_832_Diff__Int__distrib,axiom,
! [A: $tType,C4: set(A),A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),minus_minus(set(A),A4,B3)) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),B3)) ).
% Diff_Int_distrib
tff(fact_833_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),minus_minus(set(A),A4,B3)),C4) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) ).
% Diff_Int_distrib2
tff(fact_834_set__diff__diff__left,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : minus_minus(set(A),minus_minus(set(A),A4,B3),C4) = minus_minus(set(A),A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) ).
% set_diff_diff_left
tff(fact_835_Un__Diff,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A4,C4)),minus_minus(set(A),B3,C4)) ).
% Un_Diff
tff(fact_836_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
=> ? [B4: A] : pp(member2(A,B4,minus_minus(set(A),B3,A4))) ) ).
% psubset_imp_ex_mem
tff(fact_837_diff__shunt__var,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( minus_minus(A,X,Y) = bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% diff_shunt_var
tff(fact_838_subset__minus__empty,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( minus_minus(set(A),A4,B3) = bot_bot(set(A)) ) ) ).
% subset_minus_empty
tff(fact_839_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set(A)] :
( ~ pp(member2(A,X,A4))
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = A4 ) ) ).
% Diff_insert_absorb
tff(fact_840_Diff__insert2,axiom,
! [A: $tType,A4: set(A),A3: A,B3: set(A)] : minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = minus_minus(set(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),B3) ).
% Diff_insert2
tff(fact_841_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set(A)] :
( pp(member2(A,A3,A4))
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = A4 ) ) ).
% insert_Diff
tff(fact_842_Diff__insert,axiom,
! [A: $tType,A4: set(A),A3: A,B3: set(A)] : minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3)) = minus_minus(set(A),minus_minus(set(A),A4,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).
% Diff_insert
tff(fact_843_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A4: set(A)] :
( ( X != Y )
=> ( minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4),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),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))) ) ) ).
% insert_minus_eq
tff(fact_844_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X5: set(A)] :
( ~ pp(member2(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_845_subset__Diff__insert,axiom,
! [A: $tType,A4: set(A),B3: set(A),X: A,C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),minus_minus(set(A),B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),C4))))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),minus_minus(set(A),B3,C4)))
& ~ pp(member2(A,X,A4)) ) ) ).
% subset_Diff_insert
tff(fact_846_Int__Diff__disjoint,axiom,
! [A: $tType,A4: set(A),B3: 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)),A4),B3)),minus_minus(set(A),A4,B3)) = bot_bot(set(A)) ).
% Int_Diff_disjoint
tff(fact_847_Diff__triv,axiom,
! [A: $tType,A4: set(A),B3: 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)) )
=> ( minus_minus(set(A),A4,B3) = A4 ) ) ).
% Diff_triv
tff(fact_848_disjoint__alt__simp1,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( minus_minus(set(A),A4,B3) = A4 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).
% disjoint_alt_simp1
tff(fact_849_disjoint__alt__simp2,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( minus_minus(set(A),A4,B3) != A4 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp2
tff(fact_850_Diff__subset__conv,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),A4,B3)),C4))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4))) ) ).
% Diff_subset_conv
tff(fact_851_Diff__partition,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),minus_minus(set(A),B3,A4)) = B3 ) ) ).
% Diff_partition
tff(fact_852_Diff__Un,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : minus_minus(set(A),A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),minus_minus(set(A),A4,B3)),minus_minus(set(A),A4,C4)) ).
% Diff_Un
tff(fact_853_Diff__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] : minus_minus(set(A),A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A4,B3)),minus_minus(set(A),A4,C4)) ).
% Diff_Int
tff(fact_854_Int__Diff__Un,axiom,
! [A: $tType,A4: set(A),B3: 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)),A4),B3)),minus_minus(set(A),A4,B3)) = A4 ).
% Int_Diff_Un
tff(fact_855_Un__Diff__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),A4,B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = A4 ).
% Un_Diff_Int
tff(fact_856_is__singleton__the__elem,axiom,
! [A: $tType,A4: set(A)] :
( is_singleton(A,A4)
<=> ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A4)),bot_bot(set(A))) ) ) ).
% is_singleton_the_elem
tff(fact_857_is__singletonI_H,axiom,
! [A: $tType,A4: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] :
( pp(member2(A,X2,A4))
=> ( pp(member2(A,Y2,A4))
=> ( X2 = Y2 ) ) )
=> is_singleton(A,A4) ) ) ).
% is_singletonI'
tff(fact_858_infinite__remove,axiom,
! [A: $tType,S: set(A),A3: A] :
( ~ pp(aa(set(A),bool,finite_finite2(A),S))
=> ~ pp(aa(set(A),bool,finite_finite2(A),minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).
% infinite_remove
tff(fact_859_infinite__coinduct,axiom,
! [A: $tType,X5: fun(set(A),bool),A4: set(A)] :
( pp(aa(set(A),bool,X5,A4))
=> ( ! [A8: set(A)] :
( pp(aa(set(A),bool,X5,A8))
=> ? [X3: A] :
( pp(member2(A,X3,A8))
& ( pp(aa(set(A),bool,X5,minus_minus(set(A),A8,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))))
| ~ pp(aa(set(A),bool,finite_finite2(A),minus_minus(set(A),A8,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))))) ) ) )
=> ~ pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ).
% infinite_coinduct
tff(fact_860_finite__empty__induct,axiom,
! [A: $tType,A4: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,P,A4))
=> ( ! [A5: A,A8: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A8))
=> ( pp(member2(A,A5,A8))
=> ( pp(aa(set(A),bool,P,A8))
=> pp(aa(set(A),bool,P,minus_minus(set(A),A8,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),bot_bot(set(A)))))) ) ) )
=> pp(aa(set(A),bool,P,bot_bot(set(A)))) ) ) ) ).
% finite_empty_induct
tff(fact_861_Diff__single__insert,axiom,
! [A: $tType,A4: set(A),X: A,B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3))) ) ).
% Diff_single_insert
tff(fact_862_subset__insert__iff,axiom,
! [A: $tType,A4: set(A),X: A,B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
<=> ( ( pp(member2(A,X,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)) )
& ( ~ pp(member2(A,X,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ) ) ).
% subset_insert_iff
tff(fact_863_remove__subset,axiom,
! [A: $tType,X: A,S: set(A)] :
( pp(member2(A,X,S))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_864_disjoint__alt__simp3,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),minus_minus(set(A),A4,B3)),A4))
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp3
tff(fact_865_finite__remove__induct,axiom,
! [A: $tType,B3: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [A8: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A8))
=> ( ( A8 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),B3))
=> ( ! [X3: A] :
( pp(member2(A,X3,A8))
=> pp(aa(set(A),bool,P,minus_minus(set(A),A8,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))))) )
=> pp(aa(set(A),bool,P,A8)) ) ) ) )
=> pp(aa(set(A),bool,P,B3)) ) ) ) ).
% finite_remove_induct
tff(fact_866_remove__induct,axiom,
! [A: $tType,P: fun(set(A),bool),B3: set(A)] :
( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ( ~ pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(set(A),bool,P,B3)) )
=> ( ! [A8: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A8))
=> ( ( A8 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),B3))
=> ( ! [X3: A] :
( pp(member2(A,X3,A8))
=> pp(aa(set(A),bool,P,minus_minus(set(A),A8,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))))) )
=> pp(aa(set(A),bool,P,A8)) ) ) ) )
=> pp(aa(set(A),bool,P,B3)) ) ) ) ).
% remove_induct
tff(fact_867_finite__induct__select,axiom,
! [A: $tType,S: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [T3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T3),S))
=> ( pp(aa(set(A),bool,P,T3))
=> ? [X3: A] :
( pp(member2(A,X3,minus_minus(set(A),S,T3)))
& pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),T3))) ) ) )
=> pp(aa(set(A),bool,P,S)) ) ) ) ).
% finite_induct_select
tff(fact_868_psubset__insert__iff,axiom,
! [A: $tType,A4: set(A),X: A,B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
<=> ( ( pp(member2(A,X,B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3)) )
& ( ~ pp(member2(A,X,B3))
=> ( ( pp(member2(A,X,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)) )
& ( ~ pp(member2(A,X,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_869_is__singleton__def,axiom,
! [A: $tType,A4: set(A)] :
( is_singleton(A,A4)
<=> ? [X4: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).
% is_singleton_def
tff(fact_870_inf__period_I2_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,bool),D2: A,Q: fun(A,bool)] :
( ! [X2: A,K2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P,minus_minus(A,X2,aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))) )
=> ( ! [X2: A,K2: A] :
( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q,minus_minus(A,X2,aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))) )
=> ! [X3: A,K3: A] :
( ( pp(aa(A,bool,P,X3))
| pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2))))
| pp(aa(A,bool,Q,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2)))) ) ) ) ) ) ).
% inf_period(2)
tff(fact_871_inf__period_I1_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,bool),D2: A,Q: fun(A,bool)] :
( ! [X2: A,K2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P,minus_minus(A,X2,aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))) )
=> ( ! [X2: A,K2: A] :
( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q,minus_minus(A,X2,aa(A,A,aa(A,fun(A,A),times_times(A),K2),D2)))) )
=> ! [X3: A,K3: A] :
( ( pp(aa(A,bool,P,X3))
& pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2))))
& pp(aa(A,bool,Q,minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2)))) ) ) ) ) ) ).
% inf_period(1)
tff(fact_872_remove__def,axiom,
! [A: $tType,X: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),A4) = minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% remove_def
tff(fact_873_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_874_Min_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,A4) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,A4) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Min.remove
tff(fact_875_Min_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% Min.insert_remove
tff(fact_876_Max_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,A4) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,A4) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Max.remove
tff(fact_877_Max_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% Max.insert_remove
tff(fact_878_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).
% Inf_fin.semilattice_order_set_axioms
tff(fact_879_Min__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,A4)) ) ) ) ) ).
% Min_insert
tff(fact_880_min_Oidem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),A3) = A3 ) ).
% min.idem
tff(fact_881_min_Oleft__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).
% min.left_idem
tff(fact_882_min_Oright__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).
% min.right_idem
tff(fact_883_max_Oidem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).
% max.idem
tff(fact_884_max_Oleft__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).
% max.left_idem
tff(fact_885_max_Oright__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).
% max.right_idem
tff(fact_886_member__remove,axiom,
! [A: $tType,X: A,Y: A,A4: set(A)] :
( pp(member2(A,X,aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),A4)))
<=> ( pp(member2(A,X,A4))
& ( X != Y ) ) ) ).
% member_remove
tff(fact_887_min_Oabsorb1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb1
tff(fact_888_min_Oabsorb2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb2
tff(fact_889_min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% min.bounded_iff
tff(fact_890_min__arg__le_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [M2: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 ) ) ) ).
% min_arg_le(2)
tff(fact_891_min__arg__le_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [N: A,M2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N ) ) ) ).
% min_arg_le(1)
tff(fact_892_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: A,N: A] :
( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),M2)) ) ) ).
% min_eq_arg(2)
tff(fact_893_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: A,N: A] :
( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),N)) ) ) ).
% min_eq_arg(1)
tff(fact_894_max_Oabsorb1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb1
tff(fact_895_max_Oabsorb2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb2
tff(fact_896_max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).
% max.bounded_iff
tff(fact_897_min_Oabsorb3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb3
tff(fact_898_min_Oabsorb4,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb4
tff(fact_899_min__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).
% min_less_iff_conj
tff(fact_900_min__simps_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min_simps(2)
tff(fact_901_min__simps_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min_simps(1)
tff(fact_902_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% min_less_self_conv(2)
tff(fact_903_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% min_less_self_conv(1)
tff(fact_904_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: A,N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)),N))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = N ) ) ) ).
% min_arg_not_ge(2)
tff(fact_905_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M2: A,N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)),M2))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N) = M2 ) ) ) ).
% min_arg_not_ge(1)
tff(fact_906_max_Oabsorb3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb3
tff(fact_907_max_Oabsorb4,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb4
tff(fact_908_max__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).
% max_less_iff_conj
tff(fact_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_Max__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,A4)) ) ) ) ) ).
% Max_insert
tff(fact_918_max_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) ) ).
% max.assoc
tff(fact_919_min_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ).
% min.assoc
tff(fact_920_max_Ocommute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3) ) ).
% max.commute
tff(fact_921_min_Ocommute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3) ) ).
% min.commute
tff(fact_922_max_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) ) ).
% max.left_commute
tff(fact_923_max__min__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),A3)) ) ).
% max_min_distrib1
tff(fact_924_max__min__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3)) ) ).
% max_min_distrib2
tff(fact_925_min_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ).
% min.left_commute
tff(fact_926_min__max__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),A3)) ) ).
% min_max_distrib1
tff(fact_927_min__max__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C3)) ) ).
% min_max_distrib2
tff(fact_928_remove1__commute,axiom,
! [A: $tType,X: A,Y: A,Zs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),Zs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Zs)) ).
% remove1_commute
tff(fact_929_max_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A,A3: A,D3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).
% max.mono
tff(fact_930_max_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).
% max.orderE
tff(fact_931_max_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% max.orderI
tff(fact_932_max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).
% max.boundedE
tff(fact_933_max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3)) ) ) ) ).
% max.boundedI
tff(fact_934_max_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).
% max.order_iff
tff(fact_935_max_Ocobounded1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).
% max.cobounded1
tff(fact_936_max_Ocobounded2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).
% max.cobounded2
tff(fact_937_le__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) ) ) ) ).
% le_max_iff_disj
tff(fact_938_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb_iff1
tff(fact_939_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb_iff2
tff(fact_940_max_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI1
tff(fact_941_max_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI2
tff(fact_942_max__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ) ).
% max_def
tff(fact_943_max__absorb1,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).
% max_absorb1
tff(fact_944_max__absorb2,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).
% max_absorb2
tff(fact_945_less__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).
% less_max_iff_disj
tff(fact_946_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).
% max.strict_boundedE
tff(fact_947_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% max.strict_order_iff
tff(fact_948_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI1
tff(fact_949_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI2
tff(fact_950_min_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D3))) ) ) ) ).
% min.mono
tff(fact_951_min_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).
% min.orderE
tff(fact_952_min_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% min.orderI
tff(fact_953_min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).
% min.boundedE
tff(fact_954_min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))) ) ) ) ).
% min.boundedI
tff(fact_955_min_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).
% min.order_iff
tff(fact_956_min_Ocobounded1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3)) ) ).
% min.cobounded1
tff(fact_957_min_Ocobounded2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2)) ) ).
% min.cobounded2
tff(fact_958_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb_iff1
tff(fact_959_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb_iff2
tff(fact_960_min_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).
% min.coboundedI1
tff(fact_961_min_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).
% min.coboundedI2
tff(fact_962_min__le__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% min_le_iff_disj
tff(fact_963_min__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ) ).
% min_def
tff(fact_964_min__absorb1,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).
% min_absorb1
tff(fact_965_min__absorb2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).
% min_absorb2
tff(fact_966_min__less__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).
% min_less_iff_disj
tff(fact_967_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).
% min.strict_boundedE
tff(fact_968_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% min.strict_order_iff
tff(fact_969_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).
% min.strict_coboundedI1
tff(fact_970_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).
% min.strict_coboundedI2
tff(fact_971_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_972_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_973_sup__max,axiom,
! [A: $tType] :
( ( semilattice_sup(A)
& linorder(A) )
=> ( sup_sup(A) = ord_max(A) ) ) ).
% sup_max
tff(fact_974_inf__min,axiom,
! [A: $tType] :
( ( semilattice_inf(A)
& linorder(A) )
=> ( inf_inf(A) = ord_min(A) ) ) ).
% inf_min
tff(fact_975_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A)] :
( ( ( X = Y )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = Xs ) )
& ( ( X != Y )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)) ) ) ) ).
% remove1.simps(2)
tff(fact_976_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),nil(A)) = nil(A) ).
% remove1.simps(1)
tff(fact_977_Max_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,B3)),lattic643756798349783984er_Max(A,A4)) = lattic643756798349783984er_Max(A,A4) ) ) ) ) ) ).
% Max.subset
tff(fact_978_Max_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,lattic643756798349783984er_Max(A,A4),A4)) ) ) ) ) ).
% Max.closed
tff(fact_979_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,A4)) ) ) ) ) ) ).
% Max.insert_not_elem
tff(fact_980_Max_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,A4)),lattic643756798349783984er_Max(A,B3)) ) ) ) ) ) ) ).
% Max.union
tff(fact_981_Min_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,B3)),lattic643756798350308766er_Min(A,A4)) = lattic643756798350308766er_Min(A,A4) ) ) ) ) ) ).
% Min.subset
tff(fact_982_Min_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,lattic643756798350308766er_Min(A,A4),A4)) ) ) ) ) ).
% Min.closed
tff(fact_983_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,A4)) ) ) ) ) ) ).
% Min.insert_not_elem
tff(fact_984_Min_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,A4)),lattic643756798350308766er_Min(A,B3)) ) ) ) ) ) ) ).
% Min.union
tff(fact_985_complete__linorder__inf__min,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ( inf_inf(A) = ord_min(A) ) ) ).
% complete_linorder_inf_min
tff(fact_986_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A4: set(A),B3: set(A)] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4))) ) ) ) ) ).
% semilattice_order_set.subset_imp
tff(fact_987_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),coset(A,Xs)) = coset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)) ).
% remove_code(2)
tff(fact_988_mult__le__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).
% mult_le_cancel_left1
tff(fact_989_mult__le__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).
% mult_le_cancel_left2
tff(fact_990_mult__le__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).
% mult_le_cancel_right1
tff(fact_991_mult__le__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).
% mult_le_cancel_right2
tff(fact_992_mult__less__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).
% mult_less_cancel_left1
tff(fact_993_mult__less__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).
% mult_less_cancel_left2
tff(fact_994_le__zero__eq,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
<=> ( N = zero_zero(A) ) ) ) ).
% le_zero_eq
tff(fact_995_not__gr__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
<=> ( N = zero_zero(A) ) ) ) ).
% not_gr_zero
tff(fact_996_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% mult_zero_left
tff(fact_997_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% mult_zero_right
tff(fact_998_mult__eq__0__iff,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% mult_eq_0_iff
tff(fact_999_mult__cancel__left,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
<=> ( ( C3 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
tff(fact_1000_mult__cancel__right,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [A3: A,C3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
<=> ( ( C3 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
tff(fact_1001_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : minus_minus(A,A3,A3) = zero_zero(A) ) ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1002_diff__zero,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : minus_minus(A,A3,zero_zero(A)) = A3 ) ).
% diff_zero
tff(fact_1003_zero__diff,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A] : minus_minus(A,zero_zero(A),A3) = zero_zero(A) ) ).
% zero_diff
tff(fact_1004_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : minus_minus(A,A3,zero_zero(A)) = A3 ) ).
% diff_0_right
tff(fact_1005_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : minus_minus(A,A3,A3) = zero_zero(A) ) ).
% diff_self
tff(fact_1006_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),minus_minus(A,A3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% diff_ge_0_iff_ge
tff(fact_1007_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),minus_minus(A,A3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% zero_comp_diff_simps(1)
tff(fact_1008_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),minus_minus(A,A3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% diff_gt_0_iff_gt
tff(fact_1009_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),minus_minus(A,A3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% zero_comp_diff_simps(2)
tff(fact_1010_mult__cancel__right2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A3: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = C3 )
<=> ( ( C3 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_right2
tff(fact_1011_mult__cancel__right1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C3: A,B2: A] :
( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
<=> ( ( C3 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_right1
tff(fact_1012_mult__cancel__left2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C3: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = C3 )
<=> ( ( C3 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_left2
tff(fact_1013_mult__cancel__left1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C3: A,B2: A] :
( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
<=> ( ( C3 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_left1
tff(fact_1014_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_1015_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_1016_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_1017_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_1018_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_1019_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_1020_zero__le,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).
% zero_le
tff(fact_1021_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
<=> ( N != zero_zero(A) ) ) ) ).
% zero_less_iff_neq_zero
tff(fact_1022_gr__implies__not__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [M2: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N))
=> ( N != zero_zero(A) ) ) ) ).
% gr_implies_not_zero
tff(fact_1023_not__less__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).
% not_less_zero
tff(fact_1024_gr__zeroI,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ( N != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).
% gr_zeroI
tff(fact_1025_mult__not__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) )
=> ( ( A3 != zero_zero(A) )
& ( B2 != zero_zero(A) ) ) ) ) ).
% mult_not_zero
tff(fact_1026_divisors__zero,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% divisors_zero
tff(fact_1027_no__zero__divisors,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) ) ) ) ) ).
% no_zero_divisors
tff(fact_1028_mult__left__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
tff(fact_1029_mult__right__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
tff(fact_1030_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_1031_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( minus_minus(A,A3,B2) = zero_zero(A) ) ) ) ).
% eq_iff_diff_eq_0
tff(fact_1032_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_mono
tff(fact_1033_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_mono'
tff(fact_1034_zero__le__square,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3))) ) ).
% zero_le_square
tff(fact_1035_split__mult__pos__le,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,B2: A] :
( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).
% split_mult_pos_le
tff(fact_1036_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% mult_left_mono_neg
tff(fact_1037_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_nonpos_nonpos
tff(fact_1038_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% mult_left_mono
tff(fact_1039_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).
% mult_right_mono_neg
tff(fact_1040_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).
% mult_right_mono
tff(fact_1041_mult__le__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).
% mult_le_0_iff
tff(fact_1042_split__mult__neg__le,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ).
% split_mult_neg_le
tff(fact_1043_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_nonneg_nonneg
tff(fact_1044_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_nonneg_nonpos
tff(fact_1045_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_nonpos_nonneg
tff(fact_1046_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).
% mult_nonneg_nonpos2
tff(fact_1047_zero__le__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_le_mult_iff
tff(fact_1048_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ordere2520102378445227354miring(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1049_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).
% zero_less_one_class.zero_le_one
tff(fact_1050_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1051_not__one__le__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).
% not_one_le_zero
tff(fact_1052_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,B2)),zero_zero(A))) ) ) ).
% le_iff_diff_le_0
tff(fact_1053_mult__neg__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_neg_neg
tff(fact_1054_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A))) ) ).
% not_square_less_zero
tff(fact_1055_mult__less__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).
% mult_less_0_iff
tff(fact_1056_mult__neg__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_neg_pos
tff(fact_1057_mult__pos__neg,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_pos_neg
tff(fact_1058_mult__pos__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_pos_pos
tff(fact_1059_mult__pos__neg2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).
% mult_pos_neg2
tff(fact_1060_zero__less__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_less_mult_iff
tff(fact_1061_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).
% zero_less_mult_pos
tff(fact_1062_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).
% zero_less_mult_pos2
tff(fact_1063_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% mult_less_cancel_left_neg
tff(fact_1064_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_less_cancel_left_pos
tff(fact_1065_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% mult_strict_left_mono_neg
tff(fact_1066_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% mult_strict_left_mono
tff(fact_1067_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_1068_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).
% mult_strict_right_mono_neg
tff(fact_1069_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).
% mult_strict_right_mono
tff(fact_1070_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_1071_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord2810124833399127020strict(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1072_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).
% less_numeral_extra(1)
tff(fact_1073_zero__less__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).
% zero_less_one
tff(fact_1074_not__one__less__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).
% not_one_less_zero
tff(fact_1075_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,B2)),zero_zero(A))) ) ) ).
% less_iff_diff_less_0
tff(fact_1076_Inf__fin__def,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ( lattic7752659483105999362nf_fin(A) = lattic1715443433743089157tice_F(A,inf_inf(A)) ) ) ).
% Inf_fin_def
tff(fact_1077_mult__le__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% mult_le_cancel_left
tff(fact_1078_mult__le__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% mult_le_cancel_right
tff(fact_1079_mult__left__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_left_less_imp_less
tff(fact_1080_mult__strict__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_strict_mono
tff(fact_1081_mult__less__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_left
tff(fact_1082_mult__right__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_right_less_imp_less
tff(fact_1083_mult__strict__mono_H,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_strict_mono'
tff(fact_1084_mult__less__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_right
tff(fact_1085_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% mult_le_cancel_left_neg
tff(fact_1086_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_le_cancel_left_pos
tff(fact_1087_mult__left__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_left_le_imp_le
tff(fact_1088_mult__right__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_right_le_imp_le
tff(fact_1089_mult__le__less__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_1090_mult__less__le__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_1091_mult__left__le,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),A3)) ) ) ) ).
% mult_left_le
tff(fact_1092_mult__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ) ).
% mult_le_one
tff(fact_1093_mult__right__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).
% mult_right_le_one_le
tff(fact_1094_mult__left__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).
% mult_left_le_one_le
tff(fact_1095_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A4: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)))
=> ! [A11: A] :
( pp(member2(A,A11,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A11)) ) ) ) ) ) ).
% semilattice_order_set.boundedE
tff(fact_1096_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A4: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A5)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4))) ) ) ) ) ).
% semilattice_order_set.boundedI
tff(fact_1097_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A4: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),X4)) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
tff(fact_1098_mult__less__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).
% mult_less_cancel_right2
tff(fact_1099_mult__less__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).
% mult_less_cancel_right1
tff(fact_1100_max__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P4: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P4),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P4),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P4),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P4),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P4),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P4),Y)) ) ) ) ) ).
% max_mult_distrib_left
tff(fact_1101_min__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P4: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P4),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P4),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P4),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P4),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P4),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P4),Y)) ) ) ) ) ).
% min_mult_distrib_left
tff(fact_1102_max__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P4: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P4) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P4)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P4) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P4)) ) ) ) ) ).
% max_mult_distrib_right
tff(fact_1103_min__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P4: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P4) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P4)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P4))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P4) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P4)) ) ) ) ) ).
% min_mult_distrib_right
tff(fact_1104_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( ! [Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),X)),Y)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% field_le_mult_one_interval
tff(fact_1105_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mult_le_cancel_iff1
tff(fact_1106_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mult_le_cancel_iff2
tff(fact_1107_mult__less__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).
% mult_less_iff1
tff(fact_1108_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N = zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
& ( ( N != zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).
% n_lists_Nil
tff(fact_1109_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),X))
<=> ( X != zero_zero(nat) ) ) ).
% nat_geq_1_eq_neqz
tff(fact_1110_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).
% n_lists.simps(1)
tff(fact_1111_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),minus_minus(set(A),A4,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_1112_semilattice__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% semilattice_set.remove
tff(fact_1113_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_1114_semilattice__set_Ounion,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),B3: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)) ) ) ) ) ) ) ).
% semilattice_set.union
tff(fact_1115_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
tff(fact_1116_semilattice__set_Oinsert,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) ) ) ) ) ).
% semilattice_set.insert
tff(fact_1117_semilattice__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),F,X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4),A4)) ) ) ) ) ).
% semilattice_set.closed
tff(fact_1118_semilattice__set_Osubset,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A),B3: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B3)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),A4) ) ) ) ) ) ).
% semilattice_set.subset
tff(fact_1119_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_1120_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
<=> ( B2 = C3 ) ) ) ).
% add_right_cancel
tff(fact_1121_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
<=> ( B2 = C3 ) ) ) ).
% add_left_cancel
tff(fact_1122_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_right
tff(fact_1123_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_left
tff(fact_1124_add__0,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% add_0
tff(fact_1125_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_1126_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_1127_add__cancel__right__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_right_right
tff(fact_1128_add__cancel__right__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_right_left
tff(fact_1129_add__cancel__left__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_left_right
tff(fact_1130_add__cancel__left__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_left_left
tff(fact_1131_double__zero__sym,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% double_zero_sym
tff(fact_1132_add_Oright__neutral,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).
% add.right_neutral
tff(fact_1133_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_right
tff(fact_1134_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_left
tff(fact_1135_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = A3 ) ).
% add_diff_cancel_right'
tff(fact_1136_add__diff__cancel__right,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = minus_minus(A,A3,B2) ) ).
% add_diff_cancel_right
tff(fact_1137_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),A3) = B2 ) ).
% add_diff_cancel_left'
tff(fact_1138_add__diff__cancel__left,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [C3: A,A3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = minus_minus(A,A3,B2) ) ).
% add_diff_cancel_left
tff(fact_1139_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A3,B2)),B2) = A3 ) ).
% diff_add_cancel
tff(fact_1140_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = A3 ) ).
% add_diff_cancel
tff(fact_1141_add__le__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% add_le_same_cancel1
tff(fact_1142_add__le__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% add_le_same_cancel2
tff(fact_1143_le__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).
% le_add_same_cancel1
tff(fact_1144_le__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).
% le_add_same_cancel2
tff(fact_1145_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_1146_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_1147_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_1148_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_1149_less__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).
% less_add_same_cancel2
tff(fact_1150_less__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).
% less_add_same_cancel1
tff(fact_1151_add__less__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% add_less_same_cancel2
tff(fact_1152_add__less__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% add_less_same_cancel1
tff(fact_1153_diff__add__zero,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).
% diff_add_zero
tff(fact_1154_exists__leI,axiom,
! [N: nat,P: fun(nat,bool)] :
( ( ! [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),N))
=> ~ pp(aa(nat,bool,P,N5)) )
=> pp(aa(nat,bool,P,N)) )
=> ? [N6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N))
& pp(aa(nat,bool,P,N6)) ) ) ).
% exists_leI
tff(fact_1155_add_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).
% add.right_commute
tff(fact_1156_add_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).
% add.right_assoc
tff(fact_1157_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
tff(fact_1158_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
tff(fact_1159_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_1160_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).
% ab_semigroup_add_class.add.commute
tff(fact_1161_add_Oright__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A3: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
<=> ( B2 = C3 ) ) ) ).
% add.right_cancel
tff(fact_1162_add_Oleft__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
<=> ( B2 = C3 ) ) ) ).
% add.left_cancel
tff(fact_1163_add_Oassoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).
% add.assoc
tff(fact_1164_group__cancel_Oadd2,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
=> ( 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),A3),B2)) ) ) ) ).
% group_cancel.add2
tff(fact_1165_group__cancel_Oadd1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% group_cancel.add1
tff(fact_1166_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_1167_mlex__leI,axiom,
! [A3: nat,A10: nat,B2: nat,B8: nat,N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),A10))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),B8))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A10),N4)),B8))) ) ) ).
% mlex_leI
tff(fact_1168_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_1169_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_right
tff(fact_1170_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_left
tff(fact_1171_le__iff__add,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ? [C6: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C6) ) ) ).
% le_iff_add
tff(fact_1172_add__right__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).
% add_right_mono
tff(fact_1173_less__eqE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ) ).
% less_eqE
tff(fact_1174_add__left__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).
% add_left_mono
tff(fact_1175_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_mono
tff(fact_1176_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_1177_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_1178_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& ( K = L ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_1179_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% add.group_left_neutral
tff(fact_1180_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).
% add.comm_neutral
tff(fact_1181_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% comm_monoid_add_class.add_0
tff(fact_1182_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_right
tff(fact_1183_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_left
tff(fact_1184_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).
% add_strict_right_mono
tff(fact_1185_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).
% add_strict_left_mono
tff(fact_1186_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_strict_mono
tff(fact_1187_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& ( K = L ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_1188_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_1189_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_1190_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% ring_class.ring_distribs(2)
tff(fact_1191_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).
% ring_class.ring_distribs(1)
tff(fact_1192_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% comm_semiring_class.distrib
tff(fact_1193_distrib__left,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).
% distrib_left
tff(fact_1194_distrib__right,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% distrib_right
tff(fact_1195_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,E3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E3)),C3) ) ).
% combine_common_factor
tff(fact_1196_diff__diff__eq,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A,C3: A] : minus_minus(A,minus_minus(A,A3,B2),C3) = minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).
% diff_diff_eq
tff(fact_1197_add__implies__diff,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [C3: A,B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A3 )
=> ( C3 = minus_minus(A,A3,B2) ) ) ) ).
% add_implies_diff
tff(fact_1198_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A] : minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = minus_minus(A,minus_minus(A,A3,C3),B2) ) ).
% diff_add_eq_diff_diff_swap
tff(fact_1199_diff__add__eq,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A3,B2)),C3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3),B2) ) ).
% diff_add_eq
tff(fact_1200_diff__diff__eq2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A] : minus_minus(A,A3,minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3),B2) ) ).
% diff_diff_eq2
tff(fact_1201_add__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) ) ).
% add_diff_eq
tff(fact_1202_eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,C3: A,B2: A] :
( ( A3 = minus_minus(A,C3,B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C3 ) ) ) ).
% eq_diff_eq
tff(fact_1203_diff__eq__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus(A,A3,B2) = C3 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).
% diff_eq_eq
tff(fact_1204_group__cancel_Osub1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( minus_minus(A,A4,B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),minus_minus(A,A3,B2)) ) ) ) ).
% group_cancel.sub1
tff(fact_1205_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_1206_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_1207_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_1208_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_1209_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> lattic149705377957585745ce_set(A,inf_inf(A)) ) ).
% Inf_fin.semilattice_set_axioms
tff(fact_1210_add_Osafe__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [X: A,Y: A,A3: A,B2: A] :
( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ) ) ).
% add.safe_commute
tff(fact_1211_add__decreasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).
% add_decreasing
tff(fact_1212_add__increasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).
% add_increasing
tff(fact_1213_add__decreasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).
% add_decreasing2
tff(fact_1214_add__increasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).
% add_increasing2
tff(fact_1215_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_nonneg_nonneg
tff(fact_1216_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_nonpos_nonpos
tff(fact_1217_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_1218_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_1219_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_less_le_mono
tff(fact_1220_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_le_less_mono
tff(fact_1221_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1222_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1223_pos__add__strict,axiom,
! [A: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).
% pos_add_strict
tff(fact_1224_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ ! [C2: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
=> ( C2 = zero_zero(A) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_1225_add__pos__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_pos_pos
tff(fact_1226_add__neg__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_neg_neg
tff(fact_1227_diff__le__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,B2)),C3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).
% diff_le_eq
tff(fact_1228_le__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),minus_minus(A,C3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).
% le_diff_eq
tff(fact_1229_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,A3)),A3) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1230_le__add__diff,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3),A3))) ) ) ).
% le_add_diff
tff(fact_1231_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),minus_minus(A,B2,A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B2)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1232_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),minus_minus(A,B2,A3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1233_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),minus_minus(A,B2,A3)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1234_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,A3)),C3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1235_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,A3)),C3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1236_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( minus_minus(A,C3,minus_minus(A,B2,A3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3),B2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1237_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),minus_minus(A,B2,A3)) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1238_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( minus_minus(A,B2,A3) = C3 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1239_less__add__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).
% less_add_one
tff(fact_1240_add__mono1,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).
% add_mono1
tff(fact_1241_diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,B2)),C3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).
% diff_less_eq
tff(fact_1242_less__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),minus_minus(A,C3,B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).
% less_diff_eq
tff(fact_1243_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_1244_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
<=> ( C3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E3)),D3) ) ) ) ).
% eq_add_iff2
tff(fact_1245_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),E3)),C3) = D3 ) ) ) ).
% eq_add_iff1
tff(fact_1246_add__strict__increasing2,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).
% add_strict_increasing2
tff(fact_1247_add__strict__increasing,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).
% add_strict_increasing
tff(fact_1248_add__pos__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_pos_nonneg
tff(fact_1249_add__nonpos__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_nonpos_neg
tff(fact_1250_add__nonneg__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_nonneg_pos
tff(fact_1251_add__neg__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_neg_nonpos
tff(fact_1252_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).
% sum_squares_ge_zero
tff(fact_1253_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).
% not_sum_squares_lt_zero
tff(fact_1254_zero__less__two,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).
% zero_less_two
tff(fact_1255_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E3)),D3))) ) ) ).
% le_add_iff2
tff(fact_1256_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),E3)),C3)),D3)) ) ) ).
% le_add_iff1
tff(fact_1257_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),E3)),C3)),D3)) ) ) ).
% less_add_iff1
tff(fact_1258_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E3)),D3))) ) ) ).
% less_add_iff2
tff(fact_1259_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_1260_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_1261_semilattice__set_Osingleton,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),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_1262_discrete,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).
% discrete
tff(fact_1263_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_1264_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_1265_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_1266_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [R3: A,A3: A,B2: A,C3: A,D3: A] :
( ( R3 != zero_zero(A) )
=> ( ( ( A3 = B2 )
& ( C3 != D3 ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D3)) ) ) ) ) ).
% add_scale_eq_noteq
tff(fact_1267_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( A3 != B2 )
& ( C3 != D3 ) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).
% crossproduct_noteq
tff(fact_1268_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_1269_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_1270_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
( ( ( ( Begin = zero_zero(nat) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
=> ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),slice(A,Begin,minus_minus(nat,End,one_one(nat)),Xs)) ) )
& ( ~ ( ( Begin = zero_zero(nat) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
=> ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = slice(A,minus_minus(nat,Begin,one_one(nat)),minus_minus(nat,End,one_one(nat)),Xs) ) ) ) ).
% slice_Cons
tff(fact_1271_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).
% slice_Nil
tff(fact_1272_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list(A)] : slice(A,I,I,Xs) = nil(A) ).
% slice_eq_bounds_empty
tff(fact_1273_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_1274_mlex__snd__decrI,axiom,
! [A3: nat,A10: nat,B2: nat,B8: nat,N4: nat] :
( ( A3 = A10 )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),B8))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A10),N4)),B8))) ) ) ).
% mlex_snd_decrI
tff(fact_1275_mlex__fst__decrI,axiom,
! [A3: nat,A10: nat,B2: nat,N4: nat,B8: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A10))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B8),N4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A10),N4)),B8))) ) ) ) ).
% mlex_fst_decrI
tff(fact_1276_mlex__bound,axiom,
! [A3: nat,A4: nat,B2: nat,N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N4)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N4))) ) ) ).
% mlex_bound
tff(fact_1277_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A)] :
( ( ( X = Y )
=> ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)) ) )
& ( ( X != Y )
=> ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(A,nat,count_list(A,Xs),Y) ) ) ) ).
% count_list.simps(2)
tff(fact_1278_power__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M2: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ) ) ) ).
% power_decreasing_iff
tff(fact_1279_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),minus_minus(nat,N,one_one(nat))) ) ) ).
% nth_Cons_pos
tff(fact_1280_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),minus_minus(nat,K,one_one(nat)))),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),K)) ) ) ) ).
% gbinomial_reduce_nat
tff(fact_1281_power__minus__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),minus_minus(nat,N,one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).
% power_minus_mult
tff(fact_1282_power__eq__if,axiom,
! [A: $tType] :
( power(A)
=> ! [M2: nat,P4: A] :
( ( ( M2 = zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P4),M2) = one_one(A) ) )
& ( ( M2 != zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P4),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),P4),aa(nat,A,aa(A,fun(nat,A),power_power(A),P4),minus_minus(nat,M2,one_one(nat)))) ) ) ) ) ).
% power_eq_if
tff(fact_1283_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set(A)] :
( pp(member2(A,X,A4))
=> ( finite_card(A,minus_minus(set(A),A4,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,A4),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_1284_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set(A)] :
( ( pp(member2(A,X,A4))
=> ( finite_card(A,minus_minus(set(A),A4,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,A4),one_one(nat)) ) )
& ( ~ pp(member2(A,X,A4))
=> ( finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = finite_card(A,A4) ) ) ) ).
% card_Diff_singleton_if
tff(fact_1285_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).
% power_one
tff(fact_1286_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
<=> ( M2 = N ) ) ) ) ).
% power_inject_exp
tff(fact_1287_card_Oempty,axiom,
! [A: $tType] : finite_card(A,bot_bot(set(A))) = zero_zero(nat) ).
% card.empty
tff(fact_1288_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).
% nth_Cons_0
tff(fact_1289_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).
% gbinomial_0(1)
tff(fact_1290_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: nat,Y: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ) ) ).
% power_strict_increasing_iff
tff(fact_1291_card__0__eq,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( finite_card(A,A4) = zero_zero(nat) )
<=> ( A4 = bot_bot(set(A)) ) ) ) ).
% card_0_eq
tff(fact_1292_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M2: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_1293_power__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: nat,Y: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).
% power_increasing_iff
tff(fact_1294_power__commuting__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N: 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),N)),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),N)) ) ) ) ).
% power_commuting_commutes
tff(fact_1295_power__mult__distrib,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).
% power_mult_distrib
tff(fact_1296_power__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_commutes
tff(fact_1297_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B),L3: list(B),I: nat] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L3) )
=> ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L3),I)) ) ) ).
% map_eq_nth_eq
tff(fact_1298_one__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% one_le_power
tff(fact_1299_left__right__inverse__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N: 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),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).
% left_right_inverse_power
tff(fact_1300_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).
% power_0
tff(fact_1301_power__add,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M2: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_add
tff(fact_1302_card__eq__0__iff,axiom,
! [A: $tType,A4: set(A)] :
( ( finite_card(A,A4) = zero_zero(nat) )
<=> ( ( A4 = bot_bot(set(A)) )
| ~ pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ).
% card_eq_0_iff
tff(fact_1303_power__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))) ) ) ) ).
% power_le_one
tff(fact_1304_power__less__power__Suc,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).
% power_less_power_Suc
tff(fact_1305_power__gt1__lemma,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).
% power_gt1_lemma
tff(fact_1306_power__0__left,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).
% power_0_left
tff(fact_1307_card__1__singletonE,axiom,
! [A: $tType,A4: set(A)] :
( ( finite_card(A,A4) = one_one(nat) )
=> ~ ! [X2: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_1308_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ).
% power_less_imp_less_exp
tff(fact_1309_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N4: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4))) ) ) ) ).
% power_strict_increasing
tff(fact_1310_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N4: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4))) ) ) ) ).
% power_increasing
tff(fact_1311_card__gt__0__iff,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),finite_card(A,A4)))
<=> ( ( A4 != bot_bot(set(A)) )
& pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ).
% card_gt_0_iff
tff(fact_1312_power__Suc__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% power_Suc_less
tff(fact_1313_card__1__singletonI,axiom,
! [A: $tType,S: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( finite_card(A,S) = one_one(nat) )
=> ( pp(member2(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_1314_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N4: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_strict_decreasing
tff(fact_1315_card__Diff1__le,axiom,
! [A: $tType,A4: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A4))) ).
% card_Diff1_le
tff(fact_1316_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N4: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_decreasing
tff(fact_1317_power__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).
% power_le_imp_le_exp
tff(fact_1318_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),minus_minus(nat,N,one_one(nat))) ) ) ) ).
% nth_Cons'
tff(fact_1319_self__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% self_le_power
tff(fact_1320_card__Un__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).
% card_Un_Int
tff(fact_1321_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% one_less_power
tff(fact_1322_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> ( finite_card(A,minus_minus(set(A),A4,B3)) = minus_minus(nat,finite_card(A,A4),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).
% card_Diff_subset_Int
tff(fact_1323_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).
% count_list.simps(1)
tff(fact_1324_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A4)))
<=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
& pp(member2(A,X,A4)) ) ) ).
% card_Diff1_less_iff
tff(fact_1325_card__Diff2__less,axiom,
! [A: $tType,A4: set(A),X: A,Y: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( pp(member2(A,Y,A4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),finite_card(A,minus_minus(set(A),minus_minus(set(A),A4,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,A4))) ) ) ) ).
% card_Diff2_less
tff(fact_1326_card__Diff1__less,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),finite_card(A,A4))) ) ) ).
% card_Diff1_less
tff(fact_1327_card__Un__disjoint,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B3)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_1328_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
( ( X != Y )
=> ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
<=> ( ( aa(nat,A,nth(A,Xs),minus_minus(nat,N,one_one(nat))) = Y )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).
% nth_non_equal_first_eq
tff(fact_1329_foldr__max__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Y: A] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
=> ( ( ( Xs = nil(A) )
=> ( aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),ord_max(A)),Xs),Y) = Y ) )
& ( ( Xs != nil(A) )
=> ( aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),ord_max(A)),Xs),Y) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y) ) ) ) ) ) ).
% foldr_max_sorted
tff(fact_1330_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),minus_minus(nat,To,From)))
=> ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).
% slice_nth
tff(fact_1331_last__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( last(A,Xs) = aa(nat,A,nth(A,Xs),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat))) ) ) ).
% last_conv_nth
tff(fact_1332_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),minus_minus(nat,aa(num,nat,numeral_numeral(nat),V),one_one(nat))) ).
% nth_Cons_numeral
tff(fact_1333_card__insert__disjoint_H,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( minus_minus(nat,finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,A4) ) ) ) ).
% card_insert_disjoint'
tff(fact_1334_card_Oremove,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( finite_card(A,A4) = aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% card.remove
tff(fact_1335_card_Oinsert__remove,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).
% card.insert_remove
tff(fact_1336_card__Suc__Diff1,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( aa(nat,nat,suc,finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) = finite_card(A,A4) ) ) ) ).
% card_Suc_Diff1
tff(fact_1337_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_1338_numeral__times__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).
% numeral_times_numeral
tff(fact_1339_min__Suc__gt_I1_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A3)),B2) = aa(nat,nat,suc,A3) ) ) ).
% min_Suc_gt(1)
tff(fact_1340_min__Suc__gt_I2_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).
% min_Suc_gt(2)
tff(fact_1341_length__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).
% length_map
tff(fact_1342_append__eq__append__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Us2: list(A),Vs: list(A)] :
( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
| ( aa(list(A),nat,size_size(list(A)),Us2) = aa(list(A),nat,size_size(list(A)),Vs) ) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
<=> ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
tff(fact_1343_length__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_rev
tff(fact_1344_length__rotate1,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_rotate1
tff(fact_1345_distrib__left__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [V: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C3)) ) ).
% distrib_left_numeral
tff(fact_1346_distrib__right__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).
% distrib_right_numeral
tff(fact_1347_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [V: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C3)) ) ).
% right_diff_distrib_numeral
tff(fact_1348_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),aa(num,A,numeral_numeral(A),V)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).
% left_diff_distrib_numeral
tff(fact_1349_length__0__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
<=> ( Xs = nil(A) ) ) ).
% length_0_conv
tff(fact_1350_power__add__numeral2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M2: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B2) ) ).
% power_add_numeral2
tff(fact_1351_power__add__numeral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).
% power_add_numeral
tff(fact_1352_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_1353_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_1354_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_1355_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_1356_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).
% nth_Cons_Suc
tff(fact_1357_length__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).
% length_append
tff(fact_1358_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) = finite_card(A,A4) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_1359_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_1360_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))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),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_1361_length__greater__0__conv,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
<=> ( Xs != nil(A) ) ) ).
% length_greater_0_conv
tff(fact_1362_length__concat__rev,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),Xs)) ).
% length_concat_rev
tff(fact_1363_nth__append__length,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).
% nth_append_length
tff(fact_1364_Suc__diff,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
=> ( aa(nat,nat,suc,minus_minus(nat,N,M2)) = minus_minus(nat,N,minus_minus(nat,M2,one_one(nat))) ) ) ) ).
% Suc_diff
tff(fact_1365_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list(A),F: fun(A,B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,B,nth(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)),N) = aa(A,B,F,aa(nat,A,nth(A,Xs),N)) ) ) ).
% nth_map
tff(fact_1366_nth__append__first,axiom,
! [A: $tType,I: nat,L: list(A),L3: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3)),I) = aa(nat,A,nth(A,L),I) ) ) ).
% nth_append_first
tff(fact_1367_nth__append__length__plus,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).
% nth_append_length_plus
tff(fact_1368_length__butlast,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)) = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) ).
% length_butlast
tff(fact_1369_rotate1__length01,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> ( aa(list(A),list(A),rotate1(A),Xs) = Xs ) ) ).
% rotate1_length01
tff(fact_1370_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = minus_minus(nat,To,From) ) ) ) ).
% slice_len
tff(fact_1371_length__ge__1__conv,axiom,
! [A: $tType,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L)))
<=> ( L != nil(A) ) ) ).
% length_ge_1_conv
tff(fact_1372_length__Cons,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_Cons
tff(fact_1373_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
<=> ? [Y3: A,Ys3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% Suc_length_conv
tff(fact_1374_length__Suc__conv,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Y3: A,Ys3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% length_Suc_conv
tff(fact_1375_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).
% Ex_list_of_length
tff(fact_1376_neq__if__length__neq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1377_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2)))) ) ) ) ).
% sorted_iff_nth_Suc
tff(fact_1378_sorted__wrt__less__idx,axiom,
! [Ns: list(nat),I: nat] :
( sorted_wrt(nat,ord_less(nat),Ns)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I))) ) ) ).
% sorted_wrt_less_idx
tff(fact_1379_sorted__wrt__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,P,Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).
% sorted_wrt_nth_less
tff(fact_1380_sorted__wrt__iff__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( sorted_wrt(A,P,Xs)
<=> ! [I2: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ).
% sorted_wrt_iff_nth_less
tff(fact_1381_sorted__wrt01,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> sorted_wrt(A,P,Xs) ) ).
% sorted_wrt01
tff(fact_1382_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
<=> ? [X4: A,Ys3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys3))) ) ) ).
% Suc_le_length_iff
tff(fact_1383_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : sorted_wrt(A,P,nil(A)) ).
% sorted_wrt.simps(1)
tff(fact_1384_length__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( ! [Xs2: list(A)] :
( ! [Ys6: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys6)),aa(list(A),nat,size_size(list(A)),Xs2)))
=> pp(aa(list(A),bool,P,Ys6)) )
=> pp(aa(list(A),bool,P,Xs2)) )
=> pp(aa(list(A),bool,P,Xs)) ) ).
% length_induct
tff(fact_1385_finite__maxlen,axiom,
! [A: $tType,M: set(list(A))] :
( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M))
=> ? [N7: nat] :
! [X3: list(A)] :
( pp(member2(list(A),X3,M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N7)) ) ) ).
% finite_maxlen
tff(fact_1386_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),G),Ys) )
=> ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).
% map_eq_imp_length_eq
tff(fact_1387_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2))),aa(nat,A,nth(A,Xs),I2))) ) ) ) ).
% sorted_rev_iff_nth_Suc
tff(fact_1388_sorted__wrt__mergesort__by__rel,axiom,
! [X6: $tType,R: fun(X6,fun(X6,bool)),Xs: list(X6)] :
( ! [X2: X6,Y2: X6] :
( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Y2))
| pp(aa(X6,bool,aa(X6,fun(X6,bool),R,Y2),X2)) )
=> ( ! [X2: X6,Y2: X6,Z4: X6] :
( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Y2))
=> ( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,Y2),Z4))
=> pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Z4)) ) )
=> sorted_wrt(X6,R,aa(list(X6),list(X6),mergesort_by_rel(X6,R),Xs)) ) ) ).
% sorted_wrt_mergesort_by_rel
tff(fact_1389_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I2: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ) ).
% sorted_iff_nth_mono_less
tff(fact_1390_sorted01,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% sorted01
tff(fact_1391_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% list.size(4)
tff(fact_1392_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [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) = N ) ) ) ).
% length_Suc_rev_conv
tff(fact_1393_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Y3: A,Ys3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% length_Suc_conv_rev
tff(fact_1394_length__append__singleton,axiom,
! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_append_singleton
tff(fact_1395_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% strict_sorted_imp_sorted
tff(fact_1396_sorted__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).
% sorted_nth_mono
tff(fact_1397_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I2: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ) ).
% sorted_iff_nth_mono
tff(fact_1398_sorted2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Zs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).
% sorted2
tff(fact_1399_sorted0,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).
% sorted0
tff(fact_1400_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less(A),nil(A)) ) ).
% strict_sorted_simps(1)
tff(fact_1401_sorted__wrt1,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A] : sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
% sorted_wrt1
tff(fact_1402_sorted__remove1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),A3: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xs)) ) ) ).
% sorted_remove1
tff(fact_1403_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_1404_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_1405_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),N) = aa(nat,A,nth(A,Xs),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(nat,nat,suc,N))) ) ) ).
% rev_nth
tff(fact_1406_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_1407_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws2: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),bool))))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
=> ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws2) )
=> ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,nil(A)),nil(B)),nil(C)),nil(D)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B),Z4: C,Zs3: list(C),W3: D,Ws3: list(D)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys2) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(C),nat,size_size(list(C)),Zs3) )
=> ( ( aa(list(C),nat,size_size(list(C)),Zs3) = aa(list(D),nat,size_size(list(D)),Ws3) )
=> ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs2),Ys2),Zs3),Ws3))
=> pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z4),Zs3)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W3),Ws3))) ) ) ) )
=> pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs),Ys),Zs),Ws2)) ) ) ) ) ) ).
% list_induct4
tff(fact_1408_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),bool)))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
=> ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,nil(A)),nil(B)),nil(C)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B),Z4: C,Zs3: list(C)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys2) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(C),nat,size_size(list(C)),Zs3) )
=> ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs2),Ys2),Zs3))
=> pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z4),Zs3))) ) ) )
=> pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs),Ys),Zs)) ) ) ) ) ).
% list_induct3
tff(fact_1409_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),bool))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys2) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ).
% list_induct2
tff(fact_1410_one__le__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).
% one_le_numeral
tff(fact_1411_list_Osize_I3_J,axiom,
! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).
% list.size(3)
tff(fact_1412_not__numeral__less__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).
% not_numeral_less_one
tff(fact_1413_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),L)))
=> ( L != nil(A) ) ) ).
% len_greater_imp_nonempty
tff(fact_1414_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_1415_impossible__Cons,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)))
=> ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) ) ) ).
% impossible_Cons
tff(fact_1416_power__Suc,axiom,
! [A: $tType] :
( power(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_Suc
tff(fact_1417_power__Suc2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) ) ).
% power_Suc2
tff(fact_1418_list__rest__coinc,axiom,
! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1)))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R2) )
=> ? [R1p: list(A)] : R2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R1) ) ) ).
% list_rest_coinc
tff(fact_1419_nat__compl__induct_H,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N7: nat] :
( ! [Nn: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N7))
=> pp(aa(nat,bool,P,Nn)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N7))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% nat_compl_induct'
tff(fact_1420_nat__compl__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N7: nat] :
( ! [Nn: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N7))
=> pp(aa(nat,bool,P,Nn)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N7))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% nat_compl_induct
tff(fact_1421_nth__equalityI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1422_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
=> ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I2),X_12)) )
<=> ? [Xs3: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
=> pp(aa(A,bool,aa(nat,fun(A,bool),P,I2),aa(nat,A,nth(A,Xs3),I2))) ) ) ) ).
% Skolem_list_nth
tff(fact_1423_list__eq__iff__nth__eq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = Ys )
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1424_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: fun(A,fun(nat,bool))] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
=> ? [Li: A] : pp(aa(nat,bool,aa(A,fun(nat,bool),P,Li),I3)) )
=> ~ ! [L2: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L2) = N )
=> ~ ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
=> pp(aa(nat,bool,aa(A,fun(nat,bool),P,aa(nat,A,nth(A,L2),I4)),I4)) ) ) ) ).
% obtain_list_from_elements
tff(fact_1425_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(nat,nat,suc,A3))) )
<=> ( B2 = aa(nat,nat,suc,A3) ) ) ).
% nat_in_between_eq(1)
tff(fact_1426_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),aa(nat,nat,suc,A3))) )
<=> ( B2 = A3 ) ) ).
% nat_in_between_eq(2)
tff(fact_1427_Suc__to__right,axiom,
! [N: nat,M2: nat] :
( ( aa(nat,nat,suc,N) = M2 )
=> ( N = minus_minus(nat,M2,aa(nat,nat,suc,zero_zero(nat))) ) ) ).
% Suc_to_right
tff(fact_1428_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).
% sorted_rev_nth_mono
tff(fact_1429_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
<=> ! [I2: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J2)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).
% sorted_rev_iff_nth_mono
tff(fact_1430_count__le__length,axiom,
! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).
% count_le_length
tff(fact_1431_sorted1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).
% sorted1
tff(fact_1432_length__code,axiom,
! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).
% length_code
tff(fact_1433_gen__length__def,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,gen_length(A,N),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).
% gen_length_def
tff(fact_1434_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list(B)] : aa(list(B),nat,gen_length(B,N),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),nat,gen_length(B,aa(nat,nat,suc,N)),Xs) ).
% gen_length_code(2)
tff(fact_1435_sorted__butlast,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).
% sorted_butlast
tff(fact_1436_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),X: B] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),X),Xs))) ) ) ).
% sorted_map_remove1
tff(fact_1437_length__compl__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [E: A,L2: list(A)] :
( ! [Ll2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll2)),aa(list(A),nat,size_size(list(A)),L2)))
=> pp(aa(list(A),bool,P,Ll2)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),L2))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_induct
tff(fact_1438_same__length__different,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != Ys )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> ? [Pre: list(A),X2: A,Xs4: list(A),Y2: A,Ys4: list(A)] :
( ( X2 != Y2 )
& ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),Xs4)) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),Ys4)) ) ) ) ) ).
% same_length_different
tff(fact_1439_power__gt1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)))) ) ) ).
% power_gt1
tff(fact_1440_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_1441_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).
% gbinomial_Suc_Suc
tff(fact_1442_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).
% nth_butlast
tff(fact_1443_sorted__in__between,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I: nat,J: nat,L: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),J)))
=> ~ ! [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),J))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),K2)),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))))) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_1444_power__Suc__le__self,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).
% power_Suc_le_self
tff(fact_1445_power__Suc__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).
% power_Suc_less_one
tff(fact_1446_length__compl__rev__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [L2: list(A),E: A] :
( ! [Ll2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll2)),aa(list(A),nat,size_size(list(A)),L2)))
=> pp(aa(list(A),bool,P,Ll2)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),nil(A))))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_rev_induct
tff(fact_1447_card__Suc__eq,axiom,
! [A: $tType,A4: set(A),K: nat] :
( ( finite_card(A,A4) = aa(nat,nat,suc,K) )
<=> ? [B5: A,B9: set(A)] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),B9) )
& ~ pp(member2(A,B5,B9))
& ( finite_card(A,B9) = K )
& ( ( K = zero_zero(nat) )
=> ( B9 = bot_bot(set(A)) ) ) ) ) ).
% card_Suc_eq
tff(fact_1448_card__eq__SucD,axiom,
! [A: $tType,A4: set(A),K: nat] :
( ( finite_card(A,A4) = aa(nat,nat,suc,K) )
=> ? [B4: A,B6: set(A)] :
( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),B6) )
& ~ pp(member2(A,B4,B6))
& ( finite_card(A,B6) = K )
& ( ( K = zero_zero(nat) )
=> ( B6 = bot_bot(set(A)) ) ) ) ) ).
% card_eq_SucD
tff(fact_1449_card__1__singleton__iff,axiom,
! [A: $tType,A4: set(A)] :
( ( finite_card(A,A4) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ? [X4: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).
% card_1_singleton_iff
tff(fact_1450_nth__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Xs),N) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Ys),minus_minus(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).
% nth_append
tff(fact_1451_nz__le__conv__less,axiom,
! [K: nat,M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,K,aa(nat,nat,suc,zero_zero(nat)))),M2)) ) ) ).
% nz_le_conv_less
tff(fact_1452_gbinomial__addition__formula,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),K)) ) ).
% gbinomial_addition_formula
tff(fact_1453_Suc__n__minus__m__eq,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),M2))
=> ( aa(nat,nat,suc,minus_minus(nat,N,M2)) = minus_minus(nat,N,minus_minus(nat,M2,one_one(nat))) ) ) ) ).
% Suc_n_minus_m_eq
tff(fact_1454_length__product__lists,axiom,
! [B: $tType,Xss2: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),aa(list(list(B)),list(list(B)),product_lists(B),Xss2)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),foldr(nat,nat),times_times(nat)),aa(list(list(B)),list(nat),aa(fun(list(B),nat),fun(list(list(B)),list(nat)),map(list(B),nat),size_size(list(B))),Xss2)),one_one(nat)) ).
% length_product_lists
tff(fact_1455_drop__last__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(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_1456_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
=> ( ( N != zero_zero(nat) )
=> ( last(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L)) = aa(nat,A,nth(A,L),minus_minus(nat,N,one_one(nat))) ) ) ) ).
% last_take_nth_conv
tff(fact_1457_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,I)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).
% take_Suc_conv_app_nth
tff(fact_1458_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),L: list(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( sorted_wrt(A,ord_less(A),L)
& ( aa(list(A),set(A),set2(A),L) = A4 )
& ( aa(list(A),nat,size_size(list(A)),L) = finite_card(A,A4) ) )
<=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_1459_enumerate__Suc_H,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = 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)))),N) ) ).
% enumerate_Suc'
tff(fact_1460_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),L))
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J)) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
tff(fact_1461_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,N,aa(nat,nat,suc,zero_zero(nat)))),L) = aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L)) ) ) ).
% take_minus_one_conv_butlast
tff(fact_1462_List_Ofinite__set,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(set(A),bool,finite_finite2(A),aa(list(A),set(A),set2(A),Xs))) ).
% List.finite_set
tff(fact_1463_map__eq__conv,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),G: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs) )
<=> ! [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),Xs)))
=> ( aa(B,A,F,X4) = aa(B,A,G,X4) ) ) ) ).
% map_eq_conv
tff(fact_1464_drop0,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),X3) = X3 ).
% drop0
tff(fact_1465_set__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_rev
tff(fact_1466_drop__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),Xs) ).
% drop_drop
tff(fact_1467_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss2: list(list(A))] :
( ( nil(A) = aa(list(list(A)),list(A),concat(A),Xss2) )
<=> ! [X4: list(A)] :
( pp(member2(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2)))
=> ( X4 = nil(A) ) ) ) ).
% Nil_eq_concat_conv
tff(fact_1468_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss2: list(list(A))] :
( ( aa(list(list(A)),list(A),concat(A),Xss2) = nil(A) )
<=> ! [X4: list(A)] :
( pp(member2(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2)))
=> ( X4 = nil(A) ) ) ) ).
% concat_eq_Nil_conv
tff(fact_1469_distinct__rev,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),rev(A),Xs)))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% distinct_rev
tff(fact_1470_in__set__remove1,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( ( A3 != B2 )
=> ( pp(member2(A,A3,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),B2),Xs))))
<=> pp(member2(A,A3,aa(list(A),set(A),set2(A),Xs))) ) ) ).
% in_set_remove1
tff(fact_1471_set__mergesort__by__rel,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_mergesort_by_rel
tff(fact_1472_take__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),M2)),Xs) ).
% take_take
tff(fact_1473_set__rotate1,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_rotate1
tff(fact_1474_distinct1__rotate,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),rotate1(A),Xs)))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% distinct1_rotate
tff(fact_1475_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = Xs ) ) ).
% in_set_insert
tff(fact_1476_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% distinct_insert
tff(fact_1477_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_1478_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_1479_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X22)) ).
% list.simps(15)
tff(fact_1480_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ).
% take_Suc_Cons
tff(fact_1481_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) )
<=> ( ( N = zero_zero(nat) )
| ( Xs = nil(A) ) ) ) ).
% take_eq_Nil2
tff(fact_1482_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = nil(A) )
<=> ( ( N = zero_zero(nat) )
| ( Xs = nil(A) ) ) ) ).
% take_eq_Nil
tff(fact_1483_take0,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),X3) = nil(A) ).
% take0
tff(fact_1484_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) ).
% drop_Suc_Cons
tff(fact_1485_take__all,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = Xs ) ) ).
% take_all
tff(fact_1486_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = Xs )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% take_all_iff
tff(fact_1487_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).
% nth_take
tff(fact_1488_length__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),N) ).
% length_drop
tff(fact_1489_List_Olast__in__set,axiom,
! [A: $tType,As2: list(A)] :
( ( As2 != nil(A) )
=> pp(member2(A,last(A,As2),aa(list(A),set(A),set2(A),As2))) ) ).
% List.last_in_set
tff(fact_1490_Misc_Olast__in__set,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> pp(member2(A,last(A,L),aa(list(A),set(A),set2(A),L))) ) ).
% Misc.last_in_set
tff(fact_1491_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) = A4 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_1492_append__take__drop__id,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = Xs ).
% append_take_drop_id
tff(fact_1493_length__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).
% length_take
tff(fact_1494_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ).
% not_in_set_insert
tff(fact_1495_List_Oset__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(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_1496_count__notin,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).
% count_notin
tff(fact_1497_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% drop_eq_Nil2
tff(fact_1498_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = nil(A) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% drop_eq_Nil
tff(fact_1499_drop__all,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = nil(A) ) ) ).
% drop_all
tff(fact_1500_take__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Ys)) ).
% take_append
tff(fact_1501_drop__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),minus_minus(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Ys)) ).
% drop_append
tff(fact_1502_last__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( last(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = last(A,Xs) ) ) ).
% last_drop
tff(fact_1503_distinct__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,distinct(A),Xs))
& pp(aa(list(A),bool,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_1504_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(num,nat,numeral_numeral(nat),V)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,aa(num,nat,numeral_numeral(nat),V),one_one(nat))),Xs)) ).
% take_Cons_numeral
tff(fact_1505_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(num,nat,numeral_numeral(nat),V)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),minus_minus(nat,aa(num,nat,numeral_numeral(nat),V),one_one(nat))),Xs) ).
% drop_Cons_numeral
tff(fact_1506_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).
% nth_drop
tff(fact_1507_take__butlast__conv,axiom,
! [A: $tType,L: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(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),butlast(A),L) ).
% take_butlast_conv
tff(fact_1508_set__remove1__eq,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(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_1509_take__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),Xs)) ).
% take_drop
tff(fact_1510_drop__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,M2,N)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ).
% drop_take
tff(fact_1511_subseqs__distinctD,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( pp(member2(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),Ys)) ) ) ).
% subseqs_distinctD
tff(fact_1512_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(list(B),bool,distinct(B),Ys))
=> pp(aa(list(product_prod(A,B)),bool,distinct(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xs),Ys))) ) ) ).
% distinct_product
tff(fact_1513_distinct__concat,axiom,
! [A: $tType,Xs: list(list(A))] :
( pp(aa(list(list(A)),bool,distinct(list(A)),Xs))
=> ( ! [Ys2: list(A)] :
( pp(member2(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> pp(aa(list(A),bool,distinct(A),Ys2)) )
=> ( ! [Ys2: list(A),Zs3: list(A)] :
( pp(member2(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( pp(member2(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( ( Ys2 != 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),Ys2)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) )
=> pp(aa(list(A),bool,distinct(A),aa(list(list(A)),list(A),concat(A),Xs))) ) ) ) ).
% distinct_concat
tff(fact_1514_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B),X: B,Y: B] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L)))
=> ( ( aa(B,A,F,X) = aa(B,A,F,Y) )
=> ( pp(member2(B,X,aa(list(B),set(B),set2(B),L)))
=> ( pp(member2(B,Y,aa(list(B),set(B),set2(B),L)))
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
tff(fact_1515_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(list(A),bool,distinct(A),Xs)) ) ) ).
% distinct.simps(2)
tff(fact_1516_finite__distinct__list,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ? [Xs2: list(A)] :
( ( aa(list(A),set(A),set2(A),Xs2) = A4 )
& pp(aa(list(A),bool,distinct(A),Xs2)) ) ) ).
% finite_distinct_list
tff(fact_1517_take__equalityI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ! [I3: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Ys)
=> ( Xs = Ys ) ) ).
% take_equalityI
tff(fact_1518_distinct__take,axiom,
! [A: $tType,Xs: list(A),I: nat] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs))) ) ).
% distinct_take
tff(fact_1519_distinct__drop,axiom,
! [A: $tType,Xs: list(A),I: nat] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs))) ) ).
% distinct_drop
tff(fact_1520_in__set__takeD,axiom,
! [A: $tType,X: A,N: nat,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_takeD
tff(fact_1521_in__set__dropD,axiom,
! [A: $tType,X: A,N: nat,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_dropD
tff(fact_1522_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),Xs)) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
tff(fact_1523_distinct__product__lists,axiom,
! [A: $tType,Xss2: list(list(A))] :
( ! [X2: list(A)] :
( pp(member2(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2)))
=> pp(aa(list(A),bool,distinct(A),X2)) )
=> pp(aa(list(list(A)),bool,distinct(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2))) ) ).
% distinct_product_lists
tff(fact_1524_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(set(A)),bool,distinct(set(A)),aa(list(list(A)),list(set(A)),aa(fun(list(A),set(A)),fun(list(list(A)),list(set(A))),map(list(A),set(A)),set2(A)),aa(list(A),list(list(A)),subseqs(A),Xs)))) ) ).
% distinct_set_subseqs
tff(fact_1525_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))),aa(list(A),set(A),set2(A),Xs))) ).
% set_drop_subset
tff(fact_1526_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs))),aa(list(A),set(A),set2(A),Xs))) ).
% set_take_subset
tff(fact_1527_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list(A),I: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),Vs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Vs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),Vs))) = bot_bot(set(A)) ) ) ) ).
% set_take_disj_set_drop_if_distinct
tff(fact_1528_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Zs )
<=> ( ( Xs = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xs)),Zs) )
& ( Ys = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xs)),Zs) ) ) ) ).
% append_eq_conv_conj
tff(fact_1529_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),L))),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),L)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),L) ) ) ).
% drop_take_drop_unsplit
tff(fact_1530_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs))) ).
% take_add
tff(fact_1531_set__take__subset__set__take,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)))) ) ).
% set_take_subset_set_take
tff(fact_1532_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)))) ) ).
% set_drop_subset_set_drop
tff(fact_1533_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),As2))
<=> ? [Xs3: list(A),Y3: A,Ys3: list(A)] :
( pp(member2(A,Y3,aa(list(A),set(A),set2(A),Xs3)))
& pp(aa(list(A),bool,distinct(A),Xs3))
& ( As2 = 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),Y3),Ys3)) ) ) ) ).
% not_distinct_conv_prefix
tff(fact_1534_distinct__length__le,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Ys))
=> ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% distinct_length_le
tff(fact_1535_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> ( sorted_wrt(A,ord_less_eq(A),Ys)
=> ( pp(aa(list(A),bool,distinct(A),Ys))
=> ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
tff(fact_1536_card__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ( finite_card(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% card_distinct
tff(fact_1537_distinct__card,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( finite_card(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).
% distinct_card
tff(fact_1538_take__Nil,axiom,
! [A: $tType,N: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),nil(A)) = nil(A) ).
% take_Nil
tff(fact_1539_drop__Nil,axiom,
! [A: $tType,N: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),nil(A)) = nil(A) ).
% drop_Nil
tff(fact_1540_drop__0,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),Xs) = Xs ).
% drop_0
tff(fact_1541_slice__def,axiom,
! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,To,From)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),From),List)) ).
% slice_def
tff(fact_1542_distinct__length__2__or__more,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs))))
<=> ( ( A3 != B2 )
& pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Xs)))
& pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs))) ) ) ).
% distinct_length_2_or_more
tff(fact_1543_distinct_Osimps_I1_J,axiom,
! [A: $tType] : pp(aa(list(A),bool,distinct(A),nil(A))) ).
% distinct.simps(1)
tff(fact_1544_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Xs)) ).
% take_map
tff(fact_1545_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Xs)) ).
% drop_map
tff(fact_1546_distinct__mapI,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B)] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L)))
=> pp(aa(list(B),bool,distinct(B),L)) ) ).
% distinct_mapI
tff(fact_1547_sorted__wrt__take,axiom,
! [A: $tType,F: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
( sorted_wrt(A,F,Xs)
=> sorted_wrt(A,F,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ) ).
% sorted_wrt_take
tff(fact_1548_sorted__wrt__drop,axiom,
! [A: $tType,F: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
( sorted_wrt(A,F,Xs)
=> sorted_wrt(A,F,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ) ).
% sorted_wrt_drop
tff(fact_1549_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X22: list(A),X21: A] :
( pp(member2(A,Y,aa(list(A),set(A),set2(A),X22)))
=> pp(member2(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)))) ) ).
% list.set_intros(2)
tff(fact_1550_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : pp(member2(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)))) ).
% list.set_intros(1)
tff(fact_1551_list_Oset__cases,axiom,
! [A: $tType,E3: A,A3: list(A)] :
( pp(member2(A,E3,aa(list(A),set(A),set2(A),A3)))
=> ( ! [Z22: list(A)] : A3 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Z22)
=> ~ ! [Z1: A,Z22: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z1),Z22) )
=> ~ pp(member2(A,E3,aa(list(A),set(A),set2(A),Z22))) ) ) ) ).
% list.set_cases
tff(fact_1552_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list(A)] :
( pp(member2(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
=> ( ( Y = X )
| pp(member2(A,Y,aa(list(A),set(A),set2(A),Xs))) ) ) ).
% set_ConsD
tff(fact_1553_finite__list,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A4 ) ).
% finite_list
tff(fact_1554_subset__code_I1_J,axiom,
! [A: $tType,Xs: list(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B3))
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(A,X4,B3)) ) ) ).
% subset_code(1)
tff(fact_1555_ex__map__conv,axiom,
! [B: $tType,A: $tType,Ys: list(B),F: fun(A,B)] :
( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs3)
<=> ! [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),Ys)))
=> ? [Xa2: A] : X4 = aa(A,B,F,Xa2) ) ) ).
% ex_map_conv
tff(fact_1556_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F: fun(A,B),G: fun(A,B)] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Ys)))
=> ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys) ) ) ) ).
% map_cong
tff(fact_1557_map__idI,axiom,
! [A: $tType,Xs: list(A),F: fun(A,A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,A,F,X2) = X2 ) )
=> ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F),Xs) = Xs ) ) ).
% map_idI
tff(fact_1558_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs) ) ) ).
% map_ext
tff(fact_1559_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list(A),Xa: list(A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z4: A,Za: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),X)))
=> ( pp(member2(A,Za,aa(list(A),set(A),set2(A),Xa)))
=> ( ( aa(A,B,F,Z4) = aa(A,B,Fa,Za) )
=> ( Z4 = Za ) ) ) )
=> ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Fa),Xa) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
tff(fact_1560_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list(A),F: fun(A,B),G: fun(A,B)] :
( ! [Z4: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),X)))
=> ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),X) ) ) ).
% list.map_cong0
tff(fact_1561_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list(A),Ya: list(A),F: fun(A,B),G: fun(A,B)] :
( ( X = Ya )
=> ( ! [Z4: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),Ya)))
=> ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ya) ) ) ) ).
% list.map_cong
tff(fact_1562_drop__butlast,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ).
% drop_butlast
tff(fact_1563_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool))] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Q,X2),Y2)) ) ) )
=> ( sorted_wrt(A,P,Xs)
=> sorted_wrt(A,Q,Xs) ) ) ).
% sorted_wrt_mono_rel
tff(fact_1564_distinct__butlast,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),butlast(A),Xs))) ) ).
% distinct_butlast
tff(fact_1565_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : pp(aa(list(A),bool,distinct(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A4))) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_1566_distinct__remove1,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs))) ) ).
% distinct_remove1
tff(fact_1567_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_butlastD
tff(fact_1568_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,L: list(B),K: list(B),F: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A5: A,X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),L)))
=> ( aa(A,A,aa(B,fun(A,A),F,X2),A5) = aa(A,A,aa(B,fun(A,A),G,X2),A5) ) )
=> ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),L),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),K),B2) ) ) ) ) ).
% foldr_cong
tff(fact_1569_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),Y),Xs)))) ) ).
% notin_set_remove1
tff(fact_1570_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs) = Xs ) ) ).
% remove1_idem
tff(fact_1571_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
=> ( ( Xs_1 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1) )
& ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1)),Ys_2) ) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
=> ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Ys_1)),Xs_1) = Ys_1 )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Ys_1)),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
tff(fact_1572_drop__rev,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),N)),Xs)) ).
% drop_rev
tff(fact_1573_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),I)),aa(list(A),list(A),rev(A),Xs)) ).
% rev_drop
tff(fact_1574_rev__take,axiom,
! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),I)),aa(list(A),list(A),rev(A),Xs)) ).
% rev_take
tff(fact_1575_take__rev,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),N)),Xs)) ).
% take_rev
tff(fact_1576_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),Xs))
=> ~ ! [Y2: A,Ys2: list(A)] :
( pp(aa(list(A),bool,distinct(A),Ys2))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Ys2)))
=> ! [Zs3: 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))),Zs3)) ) ) ) ).
% not_distinct_split_distinct
tff(fact_1577_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ? [X2: list(A)] :
( ( aa(list(A),set(A),set2(A),X2) = A4 )
& sorted_wrt(A,ord_less_eq(A),X2)
& pp(aa(list(A),bool,distinct(A),X2))
& ! [Y4: list(A)] :
( ( ( aa(list(A),set(A),set2(A),Y4) = A4 )
& sorted_wrt(A,ord_less_eq(A),Y4)
& pp(aa(list(A),bool,distinct(A),Y4)) )
=> ( Y4 = X2 ) ) ) ) ) ).
% finite_sorted_distinct_unique
tff(fact_1578_distinct__Ex1,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ? [X2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),X2) = X )
& ! [Y4: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),Y4) = X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
tff(fact_1579_subseqs__refl,axiom,
! [A: $tType,Xs: list(A)] : pp(member2(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs)))) ).
% subseqs_refl
tff(fact_1580_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_1581_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F: fun(A,list(B)),G: fun(A,list(B))] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,list(B),F,X2) = aa(A,list(B),G,X2) ) )
=> ( bind(A,B,Xs,F) = bind(A,B,Ys,G) ) ) ) ).
% list_bind_cong
tff(fact_1582_list__ex1__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( list_ex1(A,P,Xs)
<=> ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4))
& ! [Y3: A] :
( ( pp(member2(A,Y3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,Y3)) )
=> ( Y3 = X4 ) ) ) ) ).
% list_ex1_iff
tff(fact_1583_merge__list__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ls: list(list(A)),As2: list(list(A))] :
( ! [L2: list(A)] :
( pp(member2(list(A),L2,aa(list(list(A)),set(list(A)),set2(list(A)),Ls)))
=> ( pp(aa(list(A),bool,distinct(A),L2))
& sorted_wrt(A,ord_less_eq(A),L2) ) )
=> ( ! [L2: list(A)] :
( pp(member2(list(A),L2,aa(list(list(A)),set(list(A)),set2(list(A)),As2)))
=> ( pp(aa(list(A),bool,distinct(A),L2))
& sorted_wrt(A,ord_less_eq(A),L2) ) )
=> ( pp(aa(list(A),bool,distinct(A),merge_list(A,As2,Ls)))
& sorted_wrt(A,ord_less_eq(A),merge_list(A,As2,Ls))
& ( aa(list(A),set(A),set2(A),merge_list(A,As2,Ls)) = aa(list(A),set(A),set2(A),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As2),Ls))) ) ) ) ) ) ).
% merge_list_correct
tff(fact_1584_in__set__member,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> member(A,Xs,X) ) ).
% in_set_member
tff(fact_1585_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N: nat,L: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L))))
<=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
& ( X = aa(nat,A,nth(A,L),I2) ) ) ) ).
% in_set_drop_conv_nth
tff(fact_1586_take__0,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),Xs) = nil(A) ).
% take_0
tff(fact_1587_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list(A),X: A,Xs6: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs6) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xs) = Xs6 ) ) ).
% drop_eq_ConsD
tff(fact_1588_distinct__singleton,axiom,
! [A: $tType,X: A] : pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ).
% distinct_singleton
tff(fact_1589_sorted__take,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),N: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ) ) ).
% sorted_take
tff(fact_1590_sorted__drop,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),N: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ) ) ).
% sorted_drop
tff(fact_1591_nth__via__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),Y: A,Ys: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
=> ( aa(nat,A,nth(A,Xs),N) = Y ) ) ).
% nth_via_drop
tff(fact_1592_distinct__match,axiom,
! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl))))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl2)) )
<=> ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
tff(fact_1593_merge__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L12: list(A),L23: list(A)] :
( ( pp(aa(list(A),bool,distinct(A),L12))
& sorted_wrt(A,ord_less_eq(A),L12) )
=> ( ( pp(aa(list(A),bool,distinct(A),L23))
& sorted_wrt(A,ord_less_eq(A),L23) )
=> ( pp(aa(list(A),bool,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_1594_empty__set,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).
% empty_set
tff(fact_1595_set__subset__Cons,axiom,
! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ).
% set_subset_Cons
tff(fact_1596_split__list,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ? [Ys2: list(A),Zs3: 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(A,fun(list(A),list(A)),cons(A),X),Zs3)) ) ).
% split_list
tff(fact_1597_split__list__last,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ? [Ys2: list(A),Zs3: 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(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Zs3))) ) ) ).
% split_list_last
tff(fact_1598_split__list__prop,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ? [Ys2: list(A),X2: A] :
( ? [Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3))
& pp(aa(A,bool,P,X2)) ) ) ).
% split_list_prop
tff(fact_1599_split__list__first,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ? [Ys2: list(A),Zs3: 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(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Ys2))) ) ) ).
% split_list_first
tff(fact_1600_split__list__propE,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ~ ! [Ys2: list(A),X2: A] :
( ? [Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3))
=> ~ pp(aa(A,bool,P,X2)) ) ) ).
% split_list_propE
tff(fact_1601_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),Xs6: list(A),Ys7: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Ys)))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys7)) )
<=> ( ( Xs = Xs6 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
tff(fact_1602_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [Ys3: list(A),Zs2: list(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),X),Zs2)) ) ).
% in_set_conv_decomp
tff(fact_1603_split__list__last__prop,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ? [Ys2: list(A),X2: A,Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3)) )
& pp(aa(A,bool,P,X2))
& ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Zs3)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).
% split_list_last_prop
tff(fact_1604_split__list__first__prop,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ? [Ys2: list(A),X2: A] :
( ? [Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3))
& pp(aa(A,bool,P,X2))
& ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys2)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).
% split_list_first_prop
tff(fact_1605_split__list__last__propE,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ~ ! [Ys2: list(A),X2: A,Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3)) )
=> ( pp(aa(A,bool,P,X2))
=> ~ ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Zs3)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ) ).
% split_list_last_propE
tff(fact_1606_split__list__first__propE,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ~ ! [Ys2: list(A),X2: A] :
( ? [Zs3: 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(A,fun(list(A),list(A)),cons(A),X2),Zs3))
=> ( pp(aa(A,bool,P,X2))
=> ~ ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys2)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ) ).
% split_list_first_propE
tff(fact_1607_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [Ys3: list(A),Zs2: list(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),X),Zs2)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Zs2))) ) ) ).
% in_set_conv_decomp_last
tff(fact_1608_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [Ys3: list(A),Zs2: list(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),X),Zs2)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Ys3))) ) ) ).
% in_set_conv_decomp_first
tff(fact_1609_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) )
<=> ? [Ys3: list(A),X4: A,Zs2: list(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),X4),Zs2)) )
& pp(aa(A,bool,P,X4))
& ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Zs2)))
=> ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).
% split_list_last_prop_iff
tff(fact_1610_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) )
<=> ? [Ys3: list(A),X4: A] :
( ? [Zs2: list(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),X4),Zs2))
& pp(aa(A,bool,P,X4))
& ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys3)))
=> ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).
% split_list_first_prop_iff
tff(fact_1611_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list(A)] :
( pp(member2(A,E3,aa(list(A),set(A),set2(A),L)))
=> ~ ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L22)) ) ).
% in_set_list_format
tff(fact_1612_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list(A),Y: A] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),L)))
=> ( pp(member2(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_1613_strict__sorted__equal,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( sorted_wrt(A,ord_less(A),Xs)
=> ( sorted_wrt(A,ord_less(A),Ys)
=> ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
tff(fact_1614_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_1615_sorted__wrt__append,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( sorted_wrt(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( sorted_wrt(A,P,Xs)
& sorted_wrt(A,P,Ys)
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Xa2)) ) ) ) ) ).
% sorted_wrt_append
tff(fact_1616_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
( ( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
| pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys)))) )
=> pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))))) ) ).
% in_set_butlast_appendI
tff(fact_1617_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs))),aa(list(A),set(A),set2(A),Xs))) ).
% set_remove1_subset
tff(fact_1618_remove1__append,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
( ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)),Ys) ) )
& ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),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),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Ys)) ) ) ) ).
% remove1_append
tff(fact_1619_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
( pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))
=> pp(member2(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs)))) ) ).
% Cons_in_subseqsD
tff(fact_1620_subset__code_I2_J,axiom,
! [B: $tType,A4: set(B),Ys: list(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),coset(B,Ys)))
<=> ! [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),Ys)))
=> ~ pp(member2(B,X4,A4)) ) ) ).
% subset_code(2)
tff(fact_1621_List_Oinsert__def,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = Xs ) )
& ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ) ).
% List.insert_def
tff(fact_1622_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs: list(A),F: fun(A,B)] :
( ( Xs != nil(A) )
=> pp(member2(A,arg_min_list(A,B,F,Xs),aa(list(A),set(A),set2(A),Xs))) ) ) ).
% arg_min_list_in
tff(fact_1623_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list(A),Xss2: list(list(A))] :
( pp(member2(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2))))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss2) ) ) ).
% in_set_product_lists_length
tff(fact_1624_length__n__lists__elem,axiom,
! [A: $tType,Ys: list(A),N: nat,Xs: list(A)] :
( pp(member2(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs))))
=> ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).
% length_n_lists_elem
tff(fact_1625_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs))) ) ) ).
% id_take_nth_drop
tff(fact_1626_strict__sorted__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,ord_less(A),L)
<=> ( sorted_wrt(A,ord_less_eq(A),L)
& pp(aa(list(A),bool,distinct(A),L)) ) ) ) ).
% strict_sorted_iff
tff(fact_1627_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),Ws2))
=> ? [Xs2: list(A),Ys2: list(A),Zs3: list(A),Y2: A] : Ws2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),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))),Zs3)))) ) ).
% not_distinct_decomp
tff(fact_1628_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list(A),I: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
<=> ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
tff(fact_1629_distinct__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( I2 != J2 )
=> ( aa(nat,A,nth(A,Xs),I2) != aa(nat,A,nth(A,Xs),J2) ) ) ) ) ) ).
% distinct_conv_nth
tff(fact_1630_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ).
% take_butlast
tff(fact_1631_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_pos_if_in_set
tff(fact_1632_sorted__simps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Ys: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
<=> ( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Ys)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) )
& sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).
% sorted_simps(2)
tff(fact_1633_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Ys: list(A)] :
( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
<=> ( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Ys)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
& sorted_wrt(A,ord_less(A),Ys) ) ) ) ).
% strict_sorted_simps(2)
tff(fact_1634_sorted__append,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( sorted_wrt(A,ord_less_eq(A),Xs)
& sorted_wrt(A,ord_less_eq(A),Ys)
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa2)) ) ) ) ) ) ).
% sorted_append
tff(fact_1635_all__set__conv__nth,axiom,
! [A: $tType,L: list(A),P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),L)))
=> pp(aa(A,bool,P,X4)) )
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),I2))) ) ) ).
% all_set_conv_nth
tff(fact_1636_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X4)) )
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2))) ) ) ).
% all_set_conv_all_nth
tff(fact_1637_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1638_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1639_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).
% list_ball_nth
tff(fact_1640_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member2(A,aa(nat,A,nth(A,Xs),N),aa(list(A),set(A),set2(A),Xs))) ) ).
% nth_mem
tff(fact_1641_card__length,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% card_length
tff(fact_1642_butlast__subset,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( ( Xs != nil(A) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))),A4)) ) ) ).
% butlast_subset
tff(fact_1643_remove1__split,axiom,
! [A: $tType,A3: A,Xs: list(A),Ys: list(A)] :
( pp(member2(A,A3,aa(list(A),set(A),set2(A),Xs)))
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xs) = Ys )
<=> ? [Ls3: list(A),Rs: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Rs)) )
& ~ pp(member2(A,A3,aa(list(A),set(A),set2(A),Ls3)))
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls3),Rs) ) ) ) ) ).
% remove1_split
tff(fact_1644_subset__code_I3_J,axiom,
! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).
% subset_code(3)
tff(fact_1645_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_1646_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),minus_minus(nat,N,one_one(nat))),Xs) ) ) ) ).
% drop_Cons'
tff(fact_1647_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys)))
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K))
=> ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K),Ys) ) ) ) ) ).
% nth_take_lemma
tff(fact_1648_distinct__idx,axiom,
! [B: $tType,A: $tType,F: fun(B,A),L: list(B),I: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L)))
=> ( ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L),J)) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
tff(fact_1649_butlast__conv__take,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat))),Xs) ).
% butlast_conv_take
tff(fact_1650_length__remove1,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)) = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) ) )
& ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).
% length_remove1
tff(fact_1651_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,N,one_one(nat))),Xs)) ) ) ) ).
% take_Cons'
tff(fact_1652_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs) ) ) ).
% Cons_nth_drop_Suc
tff(fact_1653_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,N,one_one(nat))),Xs) ) ) ).
% butlast_take
tff(fact_1654_sorted__append__bigger,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Y: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),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_1655_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
<=> ( N = zero_zero(nat) ) ) ) ) ).
% nth_equal_first_eq
tff(fact_1656_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ~ ! [L2: list(A)] :
( sorted_wrt(A,ord_less(A),L2)
=> ( ( aa(list(A),set(A),set2(A),L2) = A4 )
=> ( aa(list(A),nat,size_size(list(A)),L2) != finite_card(A,A4) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
tff(fact_1657_distinct__sorted__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(list(A),bool,distinct(A),L))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))) ) ) ) ) ) ).
% distinct_sorted_mono
tff(fact_1658_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),L))
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J)) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
tff(fact_1659_set__union,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),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_union
tff(fact_1660_can__select__set__list__ex1,axiom,
! [A: $tType,P: fun(A,bool),A4: list(A)] :
( can_select(A,P,aa(list(A),set(A),set2(A),A4))
<=> list_ex1(A,P,A4) ) ).
% can_select_set_list_ex1
tff(fact_1661_mergesort__remdups__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( pp(aa(list(A),bool,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_1662_length__transpose__sorted,axiom,
! [A: $tType,Xs: list(list(A))] :
( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
=> ( ( ( Xs = nil(list(A)) )
=> ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
& ( ( Xs != nil(list(A)) )
=> ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).
% length_transpose_sorted
tff(fact_1663_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ~ ! [L2: list(B)] :
( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L2))
=> ( ( aa(list(B),set(B),set2(B),L2) = A4 )
=> ( aa(list(B),nat,size_size(list(B)),L2) != finite_card(B,A4) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
tff(fact_1664_distinct__union,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xs),Ys)))
<=> pp(aa(list(A),bool,distinct(A),Ys)) ) ).
% distinct_union
tff(fact_1665_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: 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),F),A3),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),F),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Ys))) ) ).
% horner_sum_append
tff(fact_1666_distinct__concat__iff,axiom,
! [A: $tType,Xs: list(list(A))] :
( pp(aa(list(A),bool,distinct(A),aa(list(list(A)),list(A),concat(A),Xs)))
<=> ( pp(aa(list(list(A)),bool,distinct(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),removeAll(list(A)),nil(A)),Xs)))
& ! [Ys3: list(A)] :
( pp(member2(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> pp(aa(list(A),bool,distinct(A),Ys3)) )
& ! [Ys3: list(A),Zs2: list(A)] :
( ( pp(member2(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
& pp(member2(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
& ( Ys3 != Zs2 ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) ) ).
% distinct_concat_iff
tff(fact_1667_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs) = Xs ) ) ).
% removeAll_id
tff(fact_1668_removeAll__append,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Ys)) ).
% removeAll_append
tff(fact_1669_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: 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),F),A3),nil(B)) = zero_zero(A) ) ).
% horner_sum_simps(1)
tff(fact_1670_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs))) ) ).
% horner_sum_simps(2)
tff(fact_1671_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(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_1672_can__select__def,axiom,
! [A: $tType,P: fun(A,bool),A4: set(A)] :
( can_select(A,P,A4)
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& pp(aa(A,bool,P,X4))
& ! [Y3: A] :
( ( pp(member2(A,Y3,A4))
& pp(aa(A,bool,P,Y3)) )
=> ( Y3 = X4 ) ) ) ) ).
% can_select_def
tff(fact_1673_transpose_Osimps_I1_J,axiom,
! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).
% transpose.simps(1)
tff(fact_1674_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A)] :
( ( ( X = Y )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs) ) )
& ( ( X != Y )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)) ) ) ) ).
% removeAll.simps(2)
tff(fact_1675_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),nil(A)) = nil(A) ).
% removeAll.simps(1)
tff(fact_1676_distinct__removeAll,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs))) ) ).
% distinct_removeAll
tff(fact_1677_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss2: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) = transpose(A,Xss2) ).
% transpose.simps(2)
tff(fact_1678_transpose__empty,axiom,
! [A: $tType,Xs: list(list(A))] :
( ( transpose(A,Xs) = nil(list(A)) )
<=> ! [X4: list(A)] :
( pp(member2(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( X4 = nil(A) ) ) ) ).
% transpose_empty
tff(fact_1679_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_removeAll_less_eq
tff(fact_1680_transpose__map__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),Xs)) = aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),transpose(B,Xs)) ).
% transpose_map_map
tff(fact_1681_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),Xs: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
=> pp(aa(list(B),bool,distinct(B),Xs)) ) ) ).
% folding_insort_key.distinct_if_distinct_map
tff(fact_1682_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs) ) ) ).
% distinct_remove1_removeAll
tff(fact_1683_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,aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)) ).
% insert_code(2)
tff(fact_1684_remove__code_I1_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)) ).
% remove_code(1)
tff(fact_1685_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_removeAll_less
tff(fact_1686_sorted__transpose,axiom,
! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),transpose(A,Xs)))) ).
% sorted_transpose
tff(fact_1687_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B),L: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),L))
& ( aa(list(B),set(B),set2(B),L) = A4 )
& ( aa(list(B),nat,size_size(list(B)),L) = finite_card(B,A4) ) )
<=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4) = L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_1688_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4)) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_1689_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),Xs: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S))
=> ( sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> ( pp(aa(list(B),bool,distinct(B),Xs))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
tff(fact_1690_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))),nil(A))) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs) ) ) ).
% take_hd_drop
tff(fact_1691_set__update__distinct,axiom,
! [A: $tType,Xs: list(A),N: nat,X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),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),N)),bot_bot(set(A))))) ) ) ) ).
% set_update_distinct
tff(fact_1692_is__empty__set,axiom,
! [A: $tType,Xs: list(A)] :
( is_empty(A,aa(list(A),set(A),set2(A),Xs))
<=> pp(aa(list(A),bool,null(A),Xs)) ) ).
% is_empty_set
tff(fact_1693_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4) = nil(B) )
<=> ( A4 = bot_bot(set(B)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_1694_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list(A),I5: fun(set(A),fun(B,bool)),Sigma_0: B,F: fun(B,fun(A,B))] :
( pp(aa(list(A),bool,distinct(A),S))
=> ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,aa(list(A),set(A),set2(A),S)),Sigma_0))
=> ( ! [X2: A,It: set(A),Sigma: B] :
( pp(member2(A,X2,It))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S)))
=> ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,It),Sigma))
=> pp(aa(B,bool,aa(set(A),fun(B,bool),I5,minus_minus(set(A),It,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F,Sigma),X2))) ) ) )
=> pp(aa(B,bool,aa(set(A),fun(B,bool),I5,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),Sigma_0),S))) ) ) ) ).
% distinct_foldl_invar
tff(fact_1695_list__update__overwrite,axiom,
! [A: $tType,Xs: list(A),I: nat,X: A,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I),Y) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),Y) ).
% list_update_overwrite
tff(fact_1696_list__update__nonempty,axiom,
! [A: $tType,Xs: list(A),K: nat,X: A] :
( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
% list_update_nonempty
tff(fact_1697_length__list__update,axiom,
! [A: $tType,Xs: list(A),I: nat,X: A] : aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_list_update
tff(fact_1698_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list(A),X: A] :
( ( I != J )
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).
% nth_list_update_neq
tff(fact_1699_list__update__id,axiom,
! [A: $tType,Xs: list(A),I: nat] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(nat,A,nth(A,Xs),I)) = Xs ).
% list_update_id
tff(fact_1700_foldl__append,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),A3),Xs)),Ys) ).
% foldl_append
tff(fact_1701_take__update,axiom,
! [A: $tType,N: nat,L: list(A),I: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L)),I),X) ).
% take_update
tff(fact_1702_take__update__swap,axiom,
! [A: $tType,M2: nat,Xs: list(A),N: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M2),Xs)),N),X) ).
% take_update_swap
tff(fact_1703_list__update__beyond,axiom,
! [A: $tType,Xs: list(A),I: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X) = Xs ) ) ).
% list_update_beyond
tff(fact_1704_hd__append2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ).
% hd_append2
tff(fact_1705_take__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A),Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),M2),Y)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ).
% take_update_cancel
tff(fact_1706_drop__upd__irrelevant,axiom,
! [A: $tType,M2: nat,N: nat,L: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),M2),X)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L) ) ) ).
% drop_upd_irrelevant
tff(fact_1707_drop__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xs) ) ) ).
% drop_update_cancel
tff(fact_1708_list__update__length,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).
% list_update_length
tff(fact_1709_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list(A),J: nat,X: A] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),J),X)),I) = aa(nat,A,nth(A,L),I) ) ) ).
% nth_update_invalid
tff(fact_1710_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I) = X ) ) ).
% nth_list_update_eq
tff(fact_1711_hd__take,axiom,
! [A: $tType,J: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).
% hd_take
tff(fact_1712_set__swap,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(nat,A,nth(A,Xs),J))),J),aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).
% set_swap
tff(fact_1713_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(list(A),bool,distinct(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(nat,A,nth(A,Xs),J))),J),aa(nat,A,nth(A,Xs),I))))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ) ) ).
% distinct_swap
tff(fact_1714_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),aa(fun(A,fun(list(B),A)),fun(A,fun(list(list(B)),A)),foldl(A,list(B)),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F)),A3),Xs) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),A3),aa(list(list(B)),list(B),concat(B),Xs)) ).
% foldl_foldl_conv_concat
tff(fact_1715_foldl__A1__eq,axiom,
! [A: $tType,F: fun(A,fun(A,A)),N: A,I: A,Ww: list(A)] :
( ! [E: A] : aa(A,A,aa(A,fun(A,A),F,N),E) = E
=> ( ! [E: A] : aa(A,A,aa(A,fun(A,A),F,E),N) = E
=> ( ! [A5: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B4),C2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B4)),C2)
=> ( aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),F),I),Ww) = aa(A,A,aa(A,fun(A,A),F,I),aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),F),N),Ww)) ) ) ) ) ).
% foldl_A1_eq
tff(fact_1716_list__update__swap,axiom,
! [A: $tType,I: nat,I6: nat,Xs: list(A),X: A,X7: A] :
( ( I != I6 )
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I6),X7) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I6),X7)),I),X) ) ) ).
% list_update_swap
tff(fact_1717_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : sorted8670434370408473282of_set(A,B,Less_eq) = sorted8670434370408473282of_set(A,B,Less_eq) ).
% linorder.sorted_key_list_of_set.cong
tff(fact_1718_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),I),V) = nil(A) ).
% list_update.simps(1)
tff(fact_1719_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),I),Y) = nil(A) ).
% list_update_code(1)
tff(fact_1720_map__update,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),K: nat,Y: B] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Xs),K),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)),K),aa(B,A,F,Y)) ).
% map_update
tff(fact_1721_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = X21 ).
% list.sel(1)
tff(fact_1722_butlast__update_H,axiom,
! [A: $tType,L: list(A),I: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),L)),I),X) = aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),X)) ).
% butlast_update'
tff(fact_1723_foldl__Cons,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,B)),A3: B,X: A,Xs: list(A)] : aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),aa(A,B,aa(B,fun(A,B),F,A3),X)),Xs) ).
% foldl_Cons
tff(fact_1724_foldl__Nil,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,B)),A3: B] : aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),A3),nil(A)) = A3 ).
% foldl_Nil
tff(fact_1725_foldl__cong,axiom,
! [A: $tType,B: $tType,A3: A,B2: A,L: list(B),K: list(B),F: fun(A,fun(B,A)),G: fun(A,fun(B,A))] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A5: A,X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),L)))
=> ( aa(B,A,aa(A,fun(B,A),F,A5),X2) = aa(B,A,aa(A,fun(B,A),G,A5),X2) ) )
=> ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),A3),L) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),B2),K) ) ) ) ) ).
% foldl_cong
tff(fact_1726_hd__concat,axiom,
! [A: $tType,Xs: list(list(A))] :
( ( Xs != nil(list(A)) )
=> ( ( aa(list(list(A)),list(A),hd(list(A)),Xs) != nil(A) )
=> ( aa(list(A),A,hd(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).
% hd_concat
tff(fact_1727_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).
% list_update_code(2)
tff(fact_1728_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),I: nat,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,I)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),Y)) ).
% list_update_code(3)
tff(fact_1729_set__update__subsetI,axiom,
! [A: $tType,Xs: list(A),A4: set(A),X: A,I: nat] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4))
=> ( pp(member2(A,X,A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X))),A4)) ) ) ).
% set_update_subsetI
tff(fact_1730_hd__in__set,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> pp(member2(A,aa(list(A),A,hd(A),Xs),aa(list(A),set(A),set2(A),Xs))) ) ).
% hd_in_set
tff(fact_1731_list_Oset__sel_I1_J,axiom,
! [A: $tType,A3: list(A)] :
( ( A3 != nil(A) )
=> pp(member2(A,aa(list(A),A,hd(A),A3),aa(list(A),set(A),set2(A),A3))) ) ).
% list.set_sel(1)
tff(fact_1732_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list(A),F: fun(A,B)] :
( ( A3 != nil(A) )
=> ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),A3)) = aa(A,B,F,aa(list(A),A,hd(A),A3)) ) ) ).
% list.map_sel(1)
tff(fact_1733_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,B)] :
( ( Xs != nil(A) )
=> ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)) = aa(A,B,F,aa(list(A),A,hd(A),Xs)) ) ) ).
% hd_map
tff(fact_1734_hd__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Ys) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ) ).
% hd_append
tff(fact_1735_longest__common__prefix,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
? [Ps2: list(A),Xs4: list(A),Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Xs4) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Ys4) )
& ( ( Xs4 = nil(A) )
| ( Ys4 = nil(A) )
| ( aa(list(A),A,hd(A),Xs4) != aa(list(A),A,hd(A),Ys4) ) ) ) ).
% longest_common_prefix
tff(fact_1736_hd__Nil__eq__last,axiom,
! [A: $tType] : aa(list(A),A,hd(A),nil(A)) = last(A,nil(A)) ).
% hd_Nil_eq_last
tff(fact_1737_last__rev,axiom,
! [A: $tType,Xs: list(A)] : last(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),A,hd(A),Xs) ).
% last_rev
tff(fact_1738_hd__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),rev(A),Xs)) = last(A,Xs) ).
% hd_rev
tff(fact_1739_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(A),fun(list(list(A)),list(A))),foldl(list(A),list(A)),append(A)),I),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(A),fun(list(list(A)),list(A))),foldl(list(A),list(A)),append(A)),nil(A)),Ww)) ).
% foldl_conc_empty_eq
tff(fact_1740_foldl__absorb1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),times_times(A)),one_one(A)),Zs)) = aa(list(A),A,aa(A,fun(list(A),A),aa(fun(A,fun(A,A)),fun(A,fun(list(A),A)),foldl(A,A),times_times(A)),X),Zs) ) ).
% foldl_absorb1
tff(fact_1741_foldl__un__empty__eq,axiom,
! [A: $tType,I: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(set(A),fun(list(set(A)),set(A))),foldl(set(A),set(A)),sup_sup(set(A))),I),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(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_1742_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list(A),F: fun(B,fun(A,B)),A3: B,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),A3),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),aa(A,B,aa(B,fun(A,B),F,aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F),A3),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs))),X)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xs)) ) ) ).
% foldl_list_update
tff(fact_1743_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : ~ pp(aa(list(A),bool,null(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) ).
% null_rec(1)
tff(fact_1744_eq__Nil__null,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs = nil(A) )
<=> pp(aa(list(A),bool,null(A),Xs)) ) ).
% eq_Nil_null
tff(fact_1745_null__rec_I2_J,axiom,
! [B: $tType] : pp(aa(list(B),bool,null(B),nil(B))) ).
% null_rec(2)
tff(fact_1746_set__update__subset__insert,axiom,
! [A: $tType,Xs: list(A),I: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),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_1747_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list(A),X: A,Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),Y))))
<=> ( ( X = Y )
| ! [Y3: A] : pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),Y3)))) ) ) ) ).
% in_set_upd_eq_aux
tff(fact_1748_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list(A),I: nat,Y: A] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),Y))))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( X != Y ) )
=> pp(member2(A,X,aa(list(A),set(A),set2(A),L))) ) ) ).
% in_set_upd_cases
tff(fact_1749_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list(A),X: A,Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),Y))))
<=> ( ( X = Y )
| ( pp(member2(A,X,aa(list(A),set(A),set2(A),L)))
& ! [Y3: A] : pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),Y3)))) ) ) ) ) ).
% in_set_upd_eq
tff(fact_1750_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)))) ) ).
% set_update_memI
tff(fact_1751_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list(A),Ys: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),I),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),Ys) ) ) ).
% list_update_append1
tff(fact_1752_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list(A),X: A] :
( ( ( ( I = J )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))) )
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),X)),J) = X ) )
& ( ~ ( ( I = J )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))) )
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),X)),J) = aa(nat,A,nth(A,L),J) ) ) ) ).
% nth_list_update'
tff(fact_1753_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X) = Xs )
<=> ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).
% list_update_same_conv
tff(fact_1754_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( ( I = J )
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = X ) )
& ( ( I != J )
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).
% nth_list_update
tff(fact_1755_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M2),Xs)),minus_minus(nat,N,M2)),X) ) ) ).
% drop_update_swap
tff(fact_1756_hd__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).
% hd_conv_nth
tff(fact_1757_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list(A),I: nat] :
( pp(aa(list(A),bool,distinct(A),Pq))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Pq),I),last(A,Pq))))) ) ).
% distinct_butlast_swap
tff(fact_1758_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2))] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
=> ( ! [L1: list(A),L22: list(A),X2: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X2)),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),X2),nil(A)))),L22)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),aa(fun(Sigma2,fun(A,Sigma2)),fun(Sigma2,fun(list(A),Sigma2)),foldl(Sigma2,A),F),Sigma_0),L0)),L0),nil(A))) ) ) ).
% foldl_rule
tff(fact_1759_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
=> ( ! [L1: list(A),L22: list(A),X2: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X2)),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),X2),nil(A)))),L22)) ) )
=> ( ! [Sigma: Sigma2] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L0),nil(A)))
=> pp(aa(Sigma2,bool,P,Sigma)) )
=> pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),aa(fun(Sigma2,fun(A,Sigma2)),fun(Sigma2,fun(list(A),Sigma2)),foldl(Sigma2,A),F),Sigma_0),L0))) ) ) ) ).
% foldl_rule_P
tff(fact_1760_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2))] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
=> ( ! [L1: list(A),L22: list(A),X2: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) )
=> ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)))
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X2)),L22)) ) )
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),aa(fun(Sigma2,fun(A,Sigma2)),fun(Sigma2,fun(list(A),Sigma2)),foldl(Sigma2,A),F),Sigma_0),L0)),nil(A))) ) ) ).
% foldl_rule_aux
tff(fact_1761_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
=> ( ! [L1: list(A),L22: list(A),X2: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)) )
=> ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L22)))
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X2)),L22)) ) )
=> ( ! [Sigma: Sigma2] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),nil(A)))
=> pp(aa(Sigma2,bool,P,Sigma)) )
=> pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),aa(fun(Sigma2,fun(A,Sigma2)),fun(Sigma2,fun(list(A),Sigma2)),foldl(Sigma2,A),F),Sigma_0),L0))) ) ) ) ).
% foldl_rule_aux_P
tff(fact_1762_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A),X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)),Ys) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Ys),minus_minus(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),X)) ) ) ) ).
% list_update_append
tff(fact_1763_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(A,B,F,aa(nat,A,nth(A,L),I)) = aa(A,B,F,X) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),X)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),L) ) ) ).
% map_upd_eq
tff(fact_1764_sorted__hd__min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),Xs)),X3)) ) ) ) ) ).
% sorted_hd_min
tff(fact_1765_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list(A),X: A] :
( ( ( K = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) )
=> ( aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = aa(list(A),list(A),butlast(A),Xs) ) )
& ( ( K != minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) )
=> ( aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),Xs)),K),X) ) ) ) ).
% butlast_list_update
tff(fact_1766_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).
% hd_drop_conv_nth
tff(fact_1767_sorted__hd__last,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( ( L != nil(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),L)),last(A,L))) ) ) ) ).
% sorted_hd_last
tff(fact_1768_hd__last__singletonI,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( aa(list(A),A,hd(A),Xs) = last(A,Xs) )
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A)) ) ) ) ) ).
% hd_last_singletonI
tff(fact_1769_hd__butlast,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).
% hd_butlast
tff(fact_1770_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),bot_bot(set(B))) = nil(B) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_1771_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B),B3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),S))
=> ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),B3) )
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( A4 = B3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_1772_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),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),I)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),L),I),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_1773_rev__update,axiom,
! [A: $tType,K: nat,Xs: list(A),Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),rev(A),Xs)),minus_minus(nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),K),one_one(nat))),Y) ) ) ).
% rev_update
tff(fact_1774_last__list__update,axiom,
! [A: $tType,Xs: list(A),K: nat,X: A] :
( ( Xs != nil(A) )
=> ( ( ( K = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) )
=> ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = X ) )
& ( ( K != minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) )
=> ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = last(A,Xs) ) ) ) ) ).
% last_list_update
tff(fact_1775_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),A,hd(A),slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).
% slice_head
tff(fact_1776_distinct__list__update,axiom,
! [A: $tType,Xs: list(A),A3: A,I: nat] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ~ pp(member2(A,A3,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),I)),bot_bot(set(A))))))
=> pp(aa(list(A),bool,distinct(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),A3))) ) ) ).
% distinct_list_update
tff(fact_1777_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4)) = A4 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
tff(fact_1778_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4)) = finite_card(B,A4) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
tff(fact_1779_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4)))) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_1780_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4))) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_1781_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),S))
=> sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4))) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_1782_take__update__last,axiom,
! [A: $tType,N: nat,List: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),List)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),List)),N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).
% take_update_last
tff(fact_1783_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list(A),A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs))) ) ) ).
% upd_conv_take_nth_drop
tff(fact_1784_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),minus_minus(set(B),A4,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_1785_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ~ pp(member2(B,X,A4))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F),A4)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_1786_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),X: B] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(list(A),bool,distinct(A),L))
=> ( aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),L)),I),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),fun_upd(A,B,F,aa(nat,A,nth(A,L),I),X)),L) ) ) ) ).
% map_distinct_upd_conv
tff(fact_1787_min__list__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( min_list(A,Xs) = lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),Xs)) ) ) ) ).
% min_list_Min
tff(fact_1788_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),minus_minus(nat,J,aa(nat,nat,suc,I))))
=> ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_1789_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),aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),L))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),L)),nil(A))) = aa(list(A),list(A),rev(A),L) ) ) ).
% rev_split_conv
tff(fact_1790_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list(set(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L)))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),minus_minus(nat,J,aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(set(A))),drop(set(A)),J),L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(set(A))),drop(set(A)),minus_minus(nat,J,aa(nat,nat,suc,zero_zero(nat)))),L))) ) ) ) ).
% Un_set_drop_extend
tff(fact_1791_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( ( aa(set(A),A,complete_Sup_Sup(A),A4) = bot_bot(A) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> ( X4 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(1)
tff(fact_1792_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A4) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> ( X4 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(2)
tff(fact_1793_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_1794_tl__append2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),Ys) ) ) ).
% tl_append2
tff(fact_1795_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list(A),F: fun(A,B),V: B] :
( ~ pp(member2(A,Y,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),fun_upd(A,B,F,Y,V)),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) ) ) ).
% map_fun_upd
tff(fact_1796_butlast__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xs)) ).
% butlast_rev
tff(fact_1797_length__tl,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat)) ).
% length_tl
tff(fact_1798_hd__Cons__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),tl(A),Xs)) = Xs ) ) ).
% hd_Cons_tl
tff(fact_1799_list_Ocollapse,axiom,
! [A: $tType,List: list(A)] :
( ( List != nil(A) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).
% list.collapse
tff(fact_1800_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list(A),X: A] :
( ( L != nil(A) )
=> ( ( ( X = aa(list(A),A,hd(A),L) )
| pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),L)))) )
<=> pp(member2(A,X,aa(list(A),set(A),set2(A),L))) ) ) ).
% in_hd_or_tl_conv
tff(fact_1801_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : insort_key(A,B,Less_eq) = insort_key(A,B,Less_eq) ).
% linorder.insort_key.cong
tff(fact_1802_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = X22 ).
% list.sel(3)
tff(fact_1803_list_Osel_I2_J,axiom,
! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).
% list.sel(2)
tff(fact_1804_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_tlD
tff(fact_1805_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_1806_Union__empty__conv,axiom,
! [A: $tType,A4: set(set(A))] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) = bot_bot(set(A)) )
<=> ! [X4: set(A)] :
( pp(member2(set(A),X4,A4))
=> ( X4 = bot_bot(set(A)) ) ) ) ).
% Union_empty_conv
tff(fact_1807_empty__Union__conv,axiom,
! [A: $tType,A4: set(set(A))] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) )
<=> ! [X4: set(A)] :
( pp(member2(set(A),X4,A4))
=> ( X4 = bot_bot(set(A)) ) ) ) ).
% empty_Union_conv
tff(fact_1808_map__tl,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ).
% map_tl
tff(fact_1809_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(set(A))),drop(set(A)),N),L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(set(A)),list(set(A)),aa(nat,fun(list(set(A)),list(set(A))),take(set(A)),N),L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).
% Union_take_drop_id
tff(fact_1810_distinct__tl,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),tl(A),Xs))) ) ).
% distinct_tl
tff(fact_1811_tl__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xs)) ).
% tl_drop
tff(fact_1812_butlast__tl,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),butlast(A),Xs)) ).
% butlast_tl
tff(fact_1813_less__eq__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),U: A] :
( ! [V2: A] :
( pp(member2(A,V2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2)) )
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ).
% less_eq_Sup
tff(fact_1814_tl__obtain__elem,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
=> ~ ! [E: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),nil(A)) ) ) ).
% tl_obtain_elem
tff(fact_1815_tl__Nil,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
<=> ( ( Xs = nil(A) )
| ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).
% tl_Nil
tff(fact_1816_Nil__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
<=> ( ( Xs = nil(A) )
| ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).
% Nil_tl
tff(fact_1817_list_Oset__sel_I2_J,axiom,
! [A: $tType,A3: list(A),X: A] :
( ( A3 != nil(A) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A3))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),A3))) ) ) ).
% list.set_sel(2)
tff(fact_1818_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list(A),F: fun(A,B)] :
( ( A3 != nil(A) )
=> ( aa(list(B),list(B),tl(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),A3)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(list(A),list(A),tl(A),A3)) ) ) ).
% list.map_sel(2)
tff(fact_1819_sorted__tl,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).
% sorted_tl
tff(fact_1820_insert__partition,axiom,
! [A: $tType,X: set(A),F2: set(set(A))] :
( ~ pp(member2(set(A),X,F2))
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F2)))
=> ! [Xa4: set(A)] :
( pp(member2(set(A),Xa4,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F2)))
=> ( ( X2 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),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)),F2)) = bot_bot(set(A)) ) ) ) ).
% insert_partition
tff(fact_1821_Union__disjoint,axiom,
! [A: $tType,C4: set(set(A)),A4: 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)),C4)),A4) = bot_bot(set(A)) )
<=> ! [X4: set(A)] :
( pp(member2(set(A),X4,C4))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),A4) = bot_bot(set(A)) ) ) ) ).
% Union_disjoint
tff(fact_1822_Union__Int__subset,axiom,
! [A: $tType,A4: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A4),B3))),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)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)))) ).
% Union_Int_subset
tff(fact_1823_take__tl,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs)) ).
% take_tl
tff(fact_1824_drop__Suc,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xs)) ).
% drop_Suc
tff(fact_1825_list_Oexpand,axiom,
! [A: $tType,List: list(A),List2: list(A)] :
( ( ( List = nil(A) )
<=> ( List2 = nil(A) ) )
=> ( ( ( List != nil(A) )
=> ( ( List2 != nil(A) )
=> ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
& ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
tff(fact_1826_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( X != aa(list(A),A,hd(A),Xs) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).
% not_hd_in_tl
tff(fact_1827_tl__last,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),tl(A),Xs) != nil(A) )
=> ( last(A,Xs) = last(A,aa(list(A),list(A),tl(A),Xs)) ) ) ).
% tl_last
tff(fact_1828_last__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( ( Xs = nil(A) )
| ( aa(list(A),list(A),tl(A),Xs) != nil(A) ) )
=> ( last(A,aa(list(A),list(A),tl(A),Xs)) = last(A,Xs) ) ) ).
% last_tl
tff(fact_1829_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),J))
=> ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).
% sorted_list_of_set_greaterThanLessThan
tff(fact_1830_rev__butlast__is__tl__rev,axiom,
! [A: $tType,L: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),butlast(A),L)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),rev(A),L)) ).
% rev_butlast_is_tl_rev
tff(fact_1831_finite__Sup__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] :
( pp(member2(A,X2,A4))
=> ( pp(member2(A,Y2,A4))
=> pp(member2(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2),A4)) ) )
=> pp(member2(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4)) ) ) ) ) ).
% finite_Sup_in
tff(fact_1832_Sup__inter__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ).
% Sup_inter_less_eq
tff(fact_1833_tl__subset,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( ( Xs != nil(A) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))),A4)) ) ) ).
% tl_subset
tff(fact_1834_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( Xs != nil(A) )
=> ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).
% Misc.nth_tl
tff(fact_1835_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),P: fun(B,fun(A,bool))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ! [N7: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,Ys),N7)),aa(nat,A,nth(A,Xs),N7))) )
=> ! [N8: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N8),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
=> pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Ys)),N8)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N8))) ) ) ) ).
% list_take_induct_tl2
tff(fact_1836_Max__Sup,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).
% Max_Sup
tff(fact_1837_list_Oexhaust__sel,axiom,
! [A: $tType,List: list(A)] :
( ( List != nil(A) )
=> ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).
% list.exhaust_sel
tff(fact_1838_Sup__fin__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic5882676163264333800up_fin(A,A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).
% Sup_fin_Sup
tff(fact_1839_tl__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,N,one_one(nat))),aa(list(A),list(A),tl(A),Xs)) ).
% tl_take
tff(fact_1840_distinct__hd__tl,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ( X = aa(list(A),A,hd(A),Xs) )
=> ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).
% distinct_hd_tl
tff(fact_1841_butlast__rev__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).
% butlast_rev_tl
tff(fact_1842_remove1__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),aa(list(A),A,hd(A),Xs)),Xs) = aa(list(A),list(A),tl(A),Xs) ) ) ).
% remove1_tl
tff(fact_1843_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,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)),aa(fun(B,set(A)),fun(list(B),list(set(A))),map(B,set(A)),F),L))) ).
% list_collect_set_as_map
tff(fact_1844_List_Onth__tl,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).
% List.nth_tl
tff(fact_1845_card__partition,axiom,
! [A: $tType,C4: set(set(A)),K: nat] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),C4))
=> ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))
=> ( ! [C2: set(A)] :
( pp(member2(set(A),C2,C4))
=> ( finite_card(A,C2) = K ) )
=> ( ! [C13: set(A),C24: set(A)] :
( pp(member2(set(A),C13,C4))
=> ( pp(member2(set(A),C24,C4))
=> ( ( C13 != C24 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C13),C24) = bot_bot(set(A)) ) ) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(set(A),C4)) = finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) ) ) ) ) ) ).
% card_partition
tff(fact_1846_take__Suc,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xs))) ) ) ).
% take_Suc
tff(fact_1847_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),rotate1(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A))) ) ) ).
% rotate1_hd_tl
tff(fact_1848_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C3,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_greaterThanLessThan
tff(fact_1849_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% greaterThanLessThan_empty_iff2
tff(fact_1850_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% greaterThanLessThan_empty_iff
tff(fact_1851_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
=> ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanLessThan_empty
tff(fact_1852_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_1853_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_1854_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))) ) ) ) ).
% Nitpick.size_list_simp(2)
tff(fact_1855_cSup__least,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z2)) ) ) ) ).
% cSup_least
tff(fact_1856_cSup__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3)) )
=> ( ! [Y2: A] :
( ! [X3: A] :
( pp(member2(A,X3,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y2)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = A3 ) ) ) ) ) ).
% cSup_eq_non_empty
tff(fact_1857_less__cSupE,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [Y: A,X5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5)))
=> ( ( X5 != bot_bot(set(A)) )
=> ~ ! [X2: A] :
( pp(member2(A,X2,X5))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X2)) ) ) ) ) ).
% less_cSupE
tff(fact_1858_less__cSupD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X5)))
=> ? [X2: A] :
( pp(member2(A,X2,X5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2)) ) ) ) ) ).
% less_cSupD
tff(fact_1859_finite__Sup__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X5))
=> ( ( X5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A3))
<=> ! [X4: A] :
( pp(member2(A,X4,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ) ) ).
% finite_Sup_less_iff
tff(fact_1860_cSup__eq__Max,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( pp(aa(set(A),bool,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_1861_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( pp(aa(set(A),bool,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_1862_Sup__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( ( S = bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = X ) )
& ( ( S != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ) ) ).
% Sup_insert_finite
tff(fact_1863_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B3: set(A),A3: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A3) = bot_bot(A) )
<=> ! [X4: A] :
( pp(member2(A,X4,B3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).
% Sup_inf_eq_bot_iff
tff(fact_1864_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType,Xs: list(A),F: fun(A,nat)] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),nat,size_list(A,F),Xs) = zero_zero(nat) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),nat,size_list(A,F),Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F),aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).
% Nitpick.size_list_simp(1)
tff(fact_1865_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),minus_minus(nat,J,I)))
=> ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_1866_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] :
( pp(aa(B,bool,P,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)))
<=> ~ ( ( ( List = nil(A) )
& ~ pp(aa(B,bool,P,F1)) )
| ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
& ~ pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).
% list.split_sel_asm
tff(fact_1867_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] :
( pp(aa(B,bool,P,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)))
<=> ( ( ( List = nil(A) )
=> pp(aa(B,bool,P,F1)) )
& ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
=> pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).
% list.split_sel
tff(fact_1868_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat,S4: A] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),set_ord_lessThan(A,S4)))))
=> ( infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),set_ord_lessThan(A,S4)),N) = infini527867602293511546merate(A,S,N) ) ) ) ) ).
% finite_enumerate_initial_segment
tff(fact_1869_Pow__set_I1_J,axiom,
! [A: $tType] : pow(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),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_1870_PowI,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(member2(set(A),A4,pow(A,B3))) ) ).
% PowI
tff(fact_1871_Pow__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(member2(set(A),A4,pow(A,B3)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% Pow_iff
tff(fact_1872_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
=> ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanAtMost_empty
tff(fact_1873_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).
% greaterThanAtMost_empty_iff
tff(fact_1874_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_1875_size__list__append,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F),Xs)),aa(list(A),nat,size_list(A,F),Ys)) ).
% size_list_append
tff(fact_1876_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).
% sorted_list_of_set_lessThan_Suc
tff(fact_1877_Pow__Int__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A4)),pow(A,B3)) ).
% Pow_Int_eq
tff(fact_1878_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))),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_1879_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_greaterThanAtMost
tff(fact_1880_Pow__empty,axiom,
! [A: $tType] : pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_empty
tff(fact_1881_Pow__singleton__iff,axiom,
! [A: $tType,X5: set(A),Y5: set(A)] :
( ( pow(A,X5) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Y5),bot_bot(set(set(A)))) )
<=> ( ( X5 = bot_bot(set(A)) )
& ( Y5 = bot_bot(set(A)) ) ) ) ).
% Pow_singleton_iff
tff(fact_1882_Pow__top,axiom,
! [A: $tType,A4: set(A)] : pp(member2(set(A),A4,pow(A,A4))) ).
% Pow_top
tff(fact_1883_Pow__bottom,axiom,
! [A: $tType,B3: set(A)] : pp(member2(set(A),bot_bot(set(A)),pow(A,B3))) ).
% Pow_bottom
tff(fact_1884_PowD,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(member2(set(A),A4,pow(A,B3)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3)) ) ).
% PowD
tff(fact_1885_Pow__mono,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow(A,A4)),pow(A,B3))) ) ).
% Pow_mono
tff(fact_1886_Un__Pow__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A4)),pow(A,B3))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))) ).
% Un_Pow_subset
tff(fact_1887_lessThan__non__empty,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] : set_ord_lessThan(A,X) != bot_bot(set(A)) ) ).
% lessThan_non_empty
tff(fact_1888_Pow__not__empty,axiom,
! [A: $tType,A4: set(A)] : pow(A,A4) != bot_bot(set(set(A))) ).
% Pow_not_empty
tff(fact_1889_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: fun(A,fun(list(A),B)),X21: A,X22: list(A)] : aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(list(A),B,aa(A,fun(list(A),B),F22,X21),X22) ).
% list.simps(5)
tff(fact_1890_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: fun(A,fun(list(A),B))] : aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),nil(A)) = F1 ).
% list.simps(4)
tff(fact_1891_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(6)
tff(fact_1892_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( linorder(A)
& order_bot(A) )
=> ! [N: A] :
( ( set_ord_lessThan(A,N) = bot_bot(set(A)) )
<=> ( N = bot_bot(A) ) ) ) ).
% Iio_eq_empty_iff
tff(fact_1893_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A3)),set_ord_lessThan(A,B2)) = set_ord_lessThan(A,aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).
% greaterThan_Int_greaterThan
tff(fact_1894_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: fun(A,nat)] : aa(list(A),nat,size_list(A,X),nil(A)) = zero_zero(nat) ).
% list.size_gen(1)
tff(fact_1895_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(list(A),nat,size_list(A,F),Xs))) ) ) ).
% size_list_estimation
tff(fact_1896_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F),Xs))) ) ) ).
% size_list_estimation'
tff(fact_1897_size__list__pointwise,axiom,
! [A: $tType,Xs: list(A),F: fun(A,nat),G: fun(A,nat)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,G,X2))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_list(A,F),Xs)),aa(list(A),nat,size_list(A,G),Xs))) ) ).
% size_list_pointwise
tff(fact_1898_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = bot_bot(set(A)) )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).
% Ioc_disjoint
tff(fact_1899_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(2)
tff(fact_1900_Iio__Int__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,K: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(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)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(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)))) = bot_bot(set(A)) ) ) ) ) ).
% Iio_Int_singleton
tff(fact_1901_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I)),J))
=> ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).
% sorted_list_of_set_greaterThanAtMost
tff(fact_1902_list_Ocase__eq__if,axiom,
! [B: $tType,A: $tType,List: list(A),F1: B,F22: fun(A,fun(list(A),B))] :
( ( ( List = nil(A) )
=> ( aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List) = F1 ) )
& ( ( List != nil(A) )
=> ( aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List) = aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ) ).
% list.case_eq_if
tff(fact_1903_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_1904_butlast__upd__last__eq,axiom,
! [A: $tType,L: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(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),aa(list(A),list(A),aa(nat,fun(list(A),list(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_1905_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),Ys: list(B)] :
( inj_on(B,A,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
=> ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> ( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
=> ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys))
=> ( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys)))
=> ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
tff(fact_1906_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),X: B] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),linord329482645794927042rt_key(B,A,F,X,Xs))) ) ) ).
% sorted_insort_insert_key
tff(fact_1907_distinct__adj__append__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,distinct_adj(A),Xs))
& pp(aa(list(A),bool,distinct_adj(A),Ys))
& ( ( Xs = nil(A) )
| ( Ys = nil(A) )
| ( last(A,Xs) != aa(list(A),A,hd(A),Ys) ) ) ) ) ).
% distinct_adj_append_iff
tff(fact_1908_numeral__plus__one,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).
% numeral_plus_one
tff(fact_1909_one__plus__numeral,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).
% one_plus_numeral
tff(fact_1910_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),rotate1(A),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% nth_rotate1
tff(fact_1911_inj__on__rev,axiom,
! [A: $tType,A4: set(list(A))] : inj_on(list(A),list(A),rev(A),A4) ).
% inj_on_rev
tff(fact_1912_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X: B,Y: B,Xs: list(B)] :
( pp(aa(list(B),bool,distinct_adj(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Xs))))
<=> ( ( X != Y )
& pp(aa(list(B),bool,distinct_adj(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Xs))) ) ) ).
% distinct_adj_Cons_Cons
tff(fact_1913_distinct__adj__rev,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),rev(A),Xs)))
<=> pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ).
% distinct_adj_rev
tff(fact_1914_numeral__eq__one__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: num] :
( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
<=> ( N = one2 ) ) ) ).
% numeral_eq_one_iff
tff(fact_1915_one__eq__numeral__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: num] :
( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
<=> ( one2 = N ) ) ) ).
% one_eq_numeral_iff
tff(fact_1916_mod__by__1,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).
% mod_by_1
tff(fact_1917_numeral__le__one__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).
% numeral_le_one_iff
tff(fact_1918_one__less__numeral__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).
% one_less_numeral_iff
tff(fact_1919_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_1920_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
<=> ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
tff(fact_1921_inj__on__Cons1,axiom,
! [A: $tType,X: A,A4: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A4) ).
% inj_on_Cons1
tff(fact_1922_distinct__adj__map__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A)] :
( inj_on(A,B,F,aa(list(A),set(A),set2(A),Xs))
=> ( pp(aa(list(B),bool,distinct_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)))
<=> pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ) ).
% distinct_adj_map_iff
tff(fact_1923_distinct__adj__mapI,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,B)] :
( pp(aa(list(A),bool,distinct_adj(A),Xs))
=> ( inj_on(A,B,F,aa(list(A),set(A),set2(A),Xs))
=> pp(aa(list(B),bool,distinct_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs))) ) ) ).
% distinct_adj_mapI
tff(fact_1924_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
=> ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).
% divmod_digit_0(2)
tff(fact_1925_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> inj_on(B,A,F,S) ) ).
% folding_insort_key.inj_on
tff(fact_1926_distinct__adj__ConsD,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
=> pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ).
% distinct_adj_ConsD
tff(fact_1927_distinct__adj__Nil,axiom,
! [A: $tType] : pp(aa(list(A),bool,distinct_adj(A),nil(A))) ).
% distinct_adj_Nil
tff(fact_1928_left__add__twice,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),B2) ) ).
% left_add_twice
tff(fact_1929_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_1930_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_1931_distinct__adj__mapD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
=> pp(aa(list(B),bool,distinct_adj(B),Xs)) ) ).
% distinct_adj_mapD
tff(fact_1932_distinct__adj__appendD2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
=> pp(aa(list(A),bool,distinct_adj(A),Ys)) ) ).
% distinct_adj_appendD2
tff(fact_1933_distinct__adj__appendD1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
=> pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ).
% distinct_adj_appendD1
tff(fact_1934_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_1935_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% power2_eq_square
tff(fact_1936_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_1937_mod__double__modulus,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) = modulo_modulo(A,X,M2) )
| ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M2)),M2) ) ) ) ) ) ).
% mod_double_modulus
tff(fact_1938_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
=> ( minus_minus(A,modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_1939_inj__on__mult,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,A4: set(A)] :
( ( A3 != zero_zero(A) )
=> inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A4) ) ) ).
% inj_on_mult
tff(fact_1940_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,B4: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),nil(A))) ) ).
% list_decomp_2
tff(fact_1941_card__2__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
<=> ? [X4: A,Y3: A] :
( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))) )
& ( X4 != Y3 ) ) ) ).
% card_2_iff
tff(fact_1942_mult__numeral__1,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).
% mult_numeral_1
tff(fact_1943_mult__numeral__1__right,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).
% mult_numeral_1_right
tff(fact_1944_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_1945_distinct__insort__insert,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),F: fun(B,A),X: B] :
( pp(aa(list(B),bool,distinct(B),Xs))
=> pp(aa(list(B),bool,distinct(B),linord329482645794927042rt_key(B,A,F,X,Xs))) ) ) ).
% distinct_insort_insert
tff(fact_1946_length__subseqs,axiom,
! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),aa(list(A),list(list(A)),subseqs(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_subseqs
tff(fact_1947_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_1948_power__odd__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).
% power_odd_eq
tff(fact_1949_distinct__adj__singleton,axiom,
! [B: $tType,X: B] : pp(aa(list(B),bool,distinct_adj(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))) ).
% distinct_adj_singleton
tff(fact_1950_map__inj__on,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(B)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
=> ( inj_on(B,A,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
=> ( Xs = Ys ) ) ) ).
% map_inj_on
tff(fact_1951_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A),Ys: list(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
=> ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Ys) )
<=> ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
tff(fact_1952_distinct__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
<=> ( pp(aa(list(B),bool,distinct(B),Xs))
& inj_on(B,A,F,aa(list(B),set(B),set2(B),Xs)) ) ) ).
% distinct_map
tff(fact_1953_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_1954_inj__on__nth,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ! [X2: nat] :
( pp(member2(nat,X2,I5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs))) )
=> inj_on(nat,A,nth(A,Xs),I5) ) ) ).
% inj_on_nth
tff(fact_1955_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X: A,Xs: list(A)] :
( inj_on(A,B,F,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),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),removeAll(B),aa(A,B,F,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)) ) ) ).
% map_removeAll_inj_on
tff(fact_1956_distinct__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( ( Xs = nil(A) )
| ( ( X != aa(list(A),A,hd(A),Xs) )
& pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ) ) ).
% distinct_adj_Cons
tff(fact_1957_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_1958_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_1959_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_1960_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_1961_mask__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat,M2: 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))),N),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) = 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),M2),N)),one_one(A)) ) ).
% mask_mod_exp
tff(fact_1962_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N,M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) ) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_1963_mod__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self1
tff(fact_1964_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = zero_zero(A) ) ).
% mod_mult_self2_is_0
tff(fact_1965_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),B2) = zero_zero(A) ) ).
% mod_mult_self1_is_0
tff(fact_1966_bits__mod__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).
% bits_mod_by_1
tff(fact_1967_mod__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C3: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A3),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self4
tff(fact_1968_mod__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C3: A,B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self3
tff(fact_1969_mod__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self2
tff(fact_1970_mod__mult__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).
% mod_mult_eq
tff(fact_1971_mod__mult__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C3: A,A10: A,B2: A,B8: A] :
( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,A10,C3) )
=> ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B8,C3) )
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A10),B8),C3) ) ) ) ) ).
% mod_mult_cong
tff(fact_1972_mod__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C3) ) ).
% mod_mult_mult2
tff(fact_1973_mult__mod__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).
% mult_mod_right
tff(fact_1974_mod__mult__left__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).
% mod_mult_left_eq
tff(fact_1975_mod__mult__right__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).
% mod_mult_right_eq
tff(fact_1976_mod__eqE,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,B2,C3) )
=> ~ ! [D4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D4)) ) ) ).
% mod_eqE
tff(fact_1977_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).
% inj_on_empty
tff(fact_1978_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list(A),A4: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A4))
=> ( inj_on(A,B,F,A4)
=> ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),inv_on(A,B,F,A4)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),L)) = L ) ) ) ).
% inj_on_map_inv_f
tff(fact_1979_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_1980_even__succ__mod__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ) ) ).
% even_succ_mod_exp
tff(fact_1981_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = divide_divide(A,A3,B2) ) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_1982_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: 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))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% one_mod_2_pow_eq
tff(fact_1983_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] :
( pp(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))),N),one_one(A))))
<=> ( N = zero_zero(nat) ) ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_1984_times__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C3) ) ).
% times_divide_eq_left
tff(fact_1985_divide__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A3,B2),C3) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).
% divide_divide_eq_left
tff(fact_1986_divide__divide__eq__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A,C3: A] : divide_divide(A,A3,divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2) ) ).
% divide_divide_eq_right
tff(fact_1987_times__divide__eq__right,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).
% times_divide_eq_right
tff(fact_1988_div__by__1,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : divide_divide(A,A3,one_one(A)) = A3 ) ).
% div_by_1
tff(fact_1989_bits__div__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : divide_divide(A,A3,one_one(A)) = A3 ) ).
% bits_div_by_1
tff(fact_1990_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).
% of_bool_eq(2)
tff(fact_1991_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: bool] :
( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
<=> pp(P) ) ) ).
% of_bool_eq_1_iff
tff(fact_1992_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X: A] :
( inj_on(A,B,F,A4)
=> ( pp(member2(A,X,A4))
=> ( aa(B,A,inv_on(A,B,F,A4),aa(A,B,F,X)) = X ) ) ) ).
% inv_on_f_f
tff(fact_1993_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> ( ( C3 = zero_zero(A) )
| pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% dvd_mult_cancel_left
tff(fact_1994_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> ( ( C3 = zero_zero(A) )
| pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% dvd_mult_cancel_right
tff(fact_1995_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)))
<=> pp(dvd_dvd(A,B2,C3)) ) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_1996_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)))
<=> pp(dvd_dvd(A,B2,C3)) ) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_1997_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),A3) = B2 ) ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_1998_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = A3 ) ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1999_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( field(A)
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3 = zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = zero_zero(A) ) )
& ( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2000_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2001_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( field(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2002_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2003_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( field(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_2004_div__mult__mult1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% div_mult_mult1
tff(fact_2005_div__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).
% div_mult_mult2
tff(fact_2006_div__mult__mult1__if,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3 = zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = zero_zero(A) ) )
& ( ( C3 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).
% div_mult_mult1_if
tff(fact_2007_div__self,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).
% div_self
tff(fact_2008_divide__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( divide_divide(A,A3,B2) = one_one(A) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
tff(fact_2009_one__eq__divide__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( one_one(A) = divide_divide(A,A3,B2) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
tff(fact_2010_divide__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).
% divide_self
tff(fact_2011_divide__self__if,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( ( A3 = zero_zero(A) )
=> ( divide_divide(A,A3,A3) = zero_zero(A) ) )
& ( ( A3 != zero_zero(A) )
=> ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ) ).
% divide_self_if
tff(fact_2012_divide__eq__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( ( divide_divide(A,B2,A3) = one_one(A) )
<=> ( ( A3 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
tff(fact_2013_eq__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( ( one_one(A) = divide_divide(A,B2,A3) )
<=> ( ( A3 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
tff(fact_2014_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( ( divide_divide(A,one_one(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% one_divide_eq_0_iff
tff(fact_2015_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( ( zero_zero(A) = divide_divide(A,one_one(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% zero_eq_1_divide_iff
tff(fact_2016_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3))))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_2017_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_2018_unit__prod,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A))) ) ) ) ).
% unit_prod
tff(fact_2019_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,A3)) = B2 ) ) ) ).
% dvd_mult_div_cancel
tff(fact_2020_dvd__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A3)),A3) = B2 ) ) ) ).
% dvd_div_mult_self
tff(fact_2021_unit__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,divide_divide(A,A3,B2),one_one(A))) ) ) ) ).
% unit_div
tff(fact_2022_unit__div__1__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> pp(dvd_dvd(A,divide_divide(A,one_one(A),A3),one_one(A))) ) ) ).
% unit_div_1_unit
tff(fact_2023_unit__div__1__div__1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A3)) = A3 ) ) ) ).
% unit_div_1_div_1
tff(fact_2024_of__bool__less__one__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
<=> ~ pp(P) ) ) ).
% of_bool_less_one_iff
tff(fact_2025_of__bool__not__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = minus_minus(A,one_one(A),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).
% of_bool_not_iff
tff(fact_2026_divide__le__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% divide_le_0_1_iff
tff(fact_2027_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% zero_le_divide_1_iff
tff(fact_2028_divide__less__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% divide_less_0_1_iff
tff(fact_2029_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% divide_less_eq_1_neg
tff(fact_2030_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% divide_less_eq_1_pos
tff(fact_2031_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% less_divide_eq_1_neg
tff(fact_2032_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% less_divide_eq_1_pos
tff(fact_2033_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% zero_less_divide_1_iff
tff(fact_2034_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).
% le_divide_eq_numeral1(1)
tff(fact_2035_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W: num,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)))) ) ) ).
% divide_le_eq_numeral1(1)
tff(fact_2036_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W: num] :
( ( A3 = divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)) )
<=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) = B2 ) )
& ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2037_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W: num,A3: A] :
( ( divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)) = A3 )
<=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) ) )
& ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2038_div__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self1
tff(fact_2039_div__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self2
tff(fact_2040_div__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self3
tff(fact_2041_div__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self4
tff(fact_2042_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),A3) ) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2043_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2044_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W: num,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)))) ) ) ).
% divide_less_eq_numeral1(1)
tff(fact_2045_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B2)) ) ) ).
% less_divide_eq_numeral1(1)
tff(fact_2046_unit__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A3)),A3) = B2 ) ) ) ).
% unit_div_mult_self
tff(fact_2047_unit__mult__div__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,one_one(A),A3)) = divide_divide(A,B2,A3) ) ) ) ).
% unit_mult_div_div
tff(fact_2048_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% le_divide_eq_1_pos
tff(fact_2049_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% le_divide_eq_1_neg
tff(fact_2050_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% divide_le_eq_1_pos
tff(fact_2051_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% divide_le_eq_1_neg
tff(fact_2052_even__mult__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
<=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
| pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),B2)) ) ) ) ).
% even_mult_iff
tff(fact_2053_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_2054_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_2055_even__plus__one__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))))
<=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3)) ) ) ).
% even_plus_one_iff
tff(fact_2056_even__succ__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_2
tff(fact_2057_odd__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).
% odd_succ_div_two
tff(fact_2058_even__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_two
tff(fact_2059_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).
% odd_two_times_div_two_succ
tff(fact_2060_one__div__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: 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))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% one_div_2_pow_eq
tff(fact_2061_bits__1__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: 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))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% bits_1_div_exp
tff(fact_2062_even__succ__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ) ).
% even_succ_div_exp
tff(fact_2063_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( divide_divide(A,B2,A3) = divide_divide(A,C3,A3) )
<=> ( B2 = C3 ) ) ) ) ).
% unit_div_cancel
tff(fact_2064_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,divide_divide(A,A3,B2),C3))
<=> pp(dvd_dvd(A,A3,C3)) ) ) ) ).
% div_unit_dvd_iff
tff(fact_2065_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,divide_divide(A,C3,B2)))
<=> pp(dvd_dvd(A,A3,C3)) ) ) ) ).
% dvd_div_unit_iff
tff(fact_2066_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,D3: A,C3: A] :
( pp(dvd_dvd(A,B2,A3))
=> ( pp(dvd_dvd(A,D3,C3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),divide_divide(A,C3,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_2067_dvd__mult__imp__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2))
=> pp(dvd_dvd(A,A3,divide_divide(A,B2,C3))) ) ) ).
% dvd_mult_imp_div
tff(fact_2068_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3),A3))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ).
% dvd_div_mult2_eq
tff(fact_2069_div__div__eq__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C3,B2))
=> ( pp(dvd_dvd(A,B2,A3))
=> ( divide_divide(A,A3,divide_divide(A,B2,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),C3) ) ) ) ) ).
% div_div_eq_right
tff(fact_2070_div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C3,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ) ) ).
% div_mult_swap
tff(fact_2071_dvd__div__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C3,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C3) ) ) ) ).
% dvd_div_mult
tff(fact_2072_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( A3 != zero_zero(A) )
=> ( ( C3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,C3,D3))
=> ( ( divide_divide(A,B2,A3) = divide_divide(A,D3,C3) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_2073_dvd__div__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,B2: A,A3: A] :
( ( C3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,C3,B2))
=> ( pp(dvd_dvd(A,A3,divide_divide(A,B2,C3)))
<=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2)) ) ) ) ) ).
% dvd_div_iff_mult
tff(fact_2074_div__dvd__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( ( B2 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,A3))
=> ( pp(dvd_dvd(A,divide_divide(A,A3,B2),C3))
<=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ) ).
% div_dvd_iff_mult
tff(fact_2075_dvd__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,A3,B2))
=> ( ( divide_divide(A,B2,A3) = C3 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) ) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_2076_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( divide_divide(A,A3,B2) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ) ).
% unit_div_eq_0_iff
tff(fact_2077_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,C3,one_one(A)))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_2078_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ) ) ).
% unit_div_mult_swap
tff(fact_2079_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),C3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2) ) ) ) ).
% unit_div_commute
tff(fact_2080_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C3,one_one(A)))
=> ( pp(dvd_dvd(A,B2,A3))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ) ).
% div_mult_unit2
tff(fact_2081_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( A3 = divide_divide(A,C3,B2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C3 ) ) ) ) ).
% unit_eq_div2
tff(fact_2082_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( divide_divide(A,A3,B2) = C3 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).
% unit_eq_div1
tff(fact_2083_of__bool__conj,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).
% of_bool_conj
tff(fact_2084_dvdE,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,A3))
=> ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).
% dvdE
tff(fact_2085_dvdI,axiom,
! [A: $tType] :
( dvd(A)
=> ! [A3: A,B2: A,K: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
=> pp(dvd_dvd(A,B2,A3)) ) ) ).
% dvdI
tff(fact_2086_dvd__def,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,A3))
<=> ? [K4: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K4) ) ) ).
% dvd_def
tff(fact_2087_dvd__mult,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(dvd_dvd(A,A3,C3))
=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ).
% dvd_mult
tff(fact_2088_dvd__mult2,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,B2))
=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ).
% dvd_mult2
tff(fact_2089_dvd__mult__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3))
=> pp(dvd_dvd(A,A3,C3)) ) ) ).
% dvd_mult_left
tff(fact_2090_dvd__triv__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ).
% dvd_triv_left
tff(fact_2091_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C3: A,D3: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,C3,D3))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ).
% mult_dvd_mono
tff(fact_2092_dvd__mult__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3))
=> pp(dvd_dvd(A,B2,C3)) ) ) ).
% dvd_mult_right
tff(fact_2093_dvd__triv__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ).
% dvd_triv_right
tff(fact_2094_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A3,B2),C3) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).
% divide_divide_eq_left'
tff(fact_2095_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_2096_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_2097_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : pp(dvd_dvd(A,one_one(A),A3)) ) ).
% one_dvd
tff(fact_2098_unit__imp__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,B2,A3)) ) ) ).
% unit_imp_dvd
tff(fact_2099_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ) ).
% dvd_unit_imp_unit
tff(fact_2100_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [B4: A] :
( ( B4 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B4,one_one(A)))
=> ( ( divide_divide(A,one_one(A),A3) = B4 )
=> ( ( divide_divide(A,one_one(A),B4) = A3 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B4) = one_one(A) )
=> ( divide_divide(A,C3,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B4) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_2101_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_2102_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_2103_even__two__times__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) = A3 ) ) ) ).
% even_two_times_div_two
tff(fact_2104_of__bool__less__eq__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).
% of_bool_less_eq_one
tff(fact_2105_split__of__bool__asm,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P4: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P4)))
<=> ~ ( ( pp(P4)
& ~ pp(aa(A,bool,P,one_one(A))) )
| ( ~ pp(P4)
& ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool_asm
tff(fact_2106_split__of__bool,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P4: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P4)))
<=> ( ( pp(P4)
=> pp(aa(A,bool,P,one_one(A))) )
& ( ~ pp(P4)
=> pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool
tff(fact_2107_of__bool__def,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P4: bool] :
( ( pp(P4)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P4) = one_one(A) ) )
& ( ~ pp(P4)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P4) = zero_zero(A) ) ) ) ) ).
% of_bool_def
tff(fact_2108_not__is__unit__0,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ~ pp(dvd_dvd(A,zero_zero(A),one_one(A))) ) ).
% not_is_unit_0
tff(fact_2109_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_2110_divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C3: A,A3: A] :
( ( divide_divide(A,B2,C3) = A3 )
<=> ( ( ( C3 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) )
& ( ( C3 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq
tff(fact_2111_eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = divide_divide(A,B2,C3) )
<=> ( ( ( C3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 ) )
& ( ( C3 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq
tff(fact_2112_divide__eq__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C3: A,B2: A,A3: A] :
( ( C3 != zero_zero(A) )
=> ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) )
=> ( divide_divide(A,B2,C3) = A3 ) ) ) ) ).
% divide_eq_imp
tff(fact_2113_eq__divide__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 )
=> ( A3 = divide_divide(A,B2,C3) ) ) ) ) ).
% eq_divide_imp
tff(fact_2114_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C3: A,B2: A,A3: A] :
( ( C3 != zero_zero(A) )
=> ( ( divide_divide(A,B2,C3) = A3 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2115_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C3: A,A3: A,B2: A] :
( ( C3 != zero_zero(A) )
=> ( ( A3 = divide_divide(A,B2,C3) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 ) ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2116_right__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( divide_divide(A,A3,B2) = one_one(A) )
<=> ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
tff(fact_2117_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) )
<=> ( B2 = C3 ) ) ) ) ).
% unit_mult_right_cancel
tff(fact_2118_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) )
<=> ( B2 = C3 ) ) ) ) ).
% unit_mult_left_cancel
tff(fact_2119_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3))
<=> pp(dvd_dvd(A,B2,C3)) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_2120_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> pp(dvd_dvd(A,A3,C3)) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_2121_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3))
<=> pp(dvd_dvd(A,A3,C3)) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_2122_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(dvd_dvd(A,A3,C3)) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_2123_is__unit__mult__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A)))
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
& pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% is_unit_mult_iff
tff(fact_2124_power__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,one_one(A),A3)),N) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_one_over
tff(fact_2125_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [A5: A] :
( ( divide_divide(A,A5,aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
=> pp(aa(A,bool,P,A5)) )
=> ( ! [A5: A,B4: bool] :
( pp(aa(A,bool,P,A5))
=> ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5)),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A5)))) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% bits_induct
tff(fact_2126_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2127_unity__coeff__ex,axiom,
! [A: $tType] :
( ( dvd(A)
& semiring_0(A) )
=> ! [P: fun(A,bool),L: A] :
( ? [X4: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
<=> ? [X4: A] :
( pp(dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
& pp(aa(A,bool,P,X4)) ) ) ) ).
% unity_coeff_ex
tff(fact_2128_divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% divide_less_eq
tff(fact_2129_less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq
tff(fact_2130_neg__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).
% neg_divide_less_eq
tff(fact_2131_neg__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% neg_less_divide_eq
tff(fact_2132_pos__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% pos_divide_less_eq
tff(fact_2133_pos__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).
% pos_less_divide_eq
tff(fact_2134_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),Z2)) ) ) ) ).
% mult_imp_div_pos_less
tff(fact_2135_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),divide_divide(A,X,Y))) ) ) ) ).
% mult_imp_less_div_pos
tff(fact_2136_divide__strict__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).
% divide_strict_left_mono
tff(fact_2137_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_2138_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ).
% unit_dvdE
tff(fact_2139_divide__less__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_less_eq_1
tff(fact_2140_less__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% less_divide_eq_1
tff(fact_2141_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B2: A,C3: A] :
( ( aa(num,A,numeral_numeral(A),W) = divide_divide(A,B2,C3) )
<=> ( ( ( C3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3) = B2 ) )
& ( ( C3 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2142_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C3: A,W: num] :
( ( divide_divide(A,B2,C3) = aa(num,A,numeral_numeral(A),W) )
<=> ( ( ( C3 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3) ) )
& ( ( C3 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2143_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B2) = B2 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_2144_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B2,Z2)) = A3 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B2,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),A3),Z2)),B2),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_2145_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_2146_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_2147_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_2148_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_2149_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_2150_div__add__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ).
% div_add_self1
tff(fact_2151_div__add__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ).
% div_add_self2
tff(fact_2152_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( minus_minus(A,A3,divide_divide(A,B2,Z2)) = A3 ) )
& ( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,A3,divide_divide(A,B2,Z2)) = divide_divide(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2),B2),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_2153_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_2154_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_2155_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_2156_gt__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).
% gt_half_sum
tff(fact_2157_less__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).
% less_half_sum
tff(fact_2158_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D2: A,T4: A] :
( pp(dvd_dvd(A,D3,D2))
=> ! [X3: A,K3: A] :
( ~ pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T4)))
<=> ~ pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2))),T4))) ) ) ) ).
% inf_period(4)
tff(fact_2159_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D2: A,T4: A] :
( pp(dvd_dvd(A,D3,D2))
=> ! [X3: A,K3: A] :
( pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T4)))
<=> pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,X3,aa(A,A,aa(A,fun(A,A),times_times(A),K3),D2))),T4))) ) ) ) ).
% inf_period(3)
tff(fact_2160_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).
% unit_imp_mod_eq_0
tff(fact_2161_mult__div__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))),modulo_modulo(A,A3,B2)) = A3 ) ).
% mult_div_mod_eq
tff(fact_2162_mod__mult__div__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))) = A3 ) ).
% mod_mult_div_eq
tff(fact_2163_mod__div__mult__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)) = A3 ) ).
% mod_div_mult_eq
tff(fact_2164_div__mult__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).
% div_mult_mod_eq
tff(fact_2165_mod__div__decomp,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2)) ) ).
% mod_div_decomp
tff(fact_2166_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).
% cancel_div_mod_rules(1)
tff(fact_2167_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).
% cancel_div_mod_rules(2)
tff(fact_2168_div__mult1__eq,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C3)),C3)) ) ).
% div_mult1_eq
tff(fact_2169_is__unit__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),one_one(A)))
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
| ( N = zero_zero(nat) ) ) ) ) ).
% is_unit_power_iff
tff(fact_2170_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))) = modulo_modulo(A,A3,B2) ) ).
% minus_mult_div_eq_mod
tff(fact_2171_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2)) ) ).
% minus_mod_eq_mult_div
tff(fact_2172_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2) ) ).
% minus_mod_eq_div_mult
tff(fact_2173_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)) = modulo_modulo(A,A3,B2) ) ).
% minus_div_mult_eq_mod
tff(fact_2174_even__mask__div__iff_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M2: nat,N: nat] :
( pp(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))),M2),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).
% even_mask_div_iff'
tff(fact_2175_exp__div__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] : divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,M2,N))) ) ).
% exp_div_exp_eq
tff(fact_2176_even__mask__div__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] :
( pp(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))),M2),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).
% even_mask_div_iff
tff(fact_2177_divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% divide_le_eq
tff(fact_2178_le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq
tff(fact_2179_divide__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).
% divide_left_mono
tff(fact_2180_neg__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).
% neg_divide_le_eq
tff(fact_2181_neg__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% neg_le_divide_eq
tff(fact_2182_pos__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% pos_divide_le_eq
tff(fact_2183_pos__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).
% pos_le_divide_eq
tff(fact_2184_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),Z2)) ) ) ) ).
% mult_imp_div_pos_le
tff(fact_2185_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),divide_divide(A,X,Y))) ) ) ) ).
% mult_imp_le_div_pos
tff(fact_2186_divide__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).
% divide_left_mono_neg
tff(fact_2187_divide__le__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_le_eq_1
tff(fact_2188_le__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% le_divide_eq_1
tff(fact_2189_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) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W,Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),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_2190_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_2191_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_2192_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) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W,Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),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_2193_evenE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ~ ! [B4: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4) ) ) ).
% evenE
tff(fact_2194_odd__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),one_one(A))) ) ).
% odd_one
tff(fact_2195_dvd__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,M2: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( pp(dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
<=> ( pp(dvd_dvd(A,X,one_one(A)))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ) ).
% dvd_power_iff
tff(fact_2196_dvd__power,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
| ( X = one_one(A) ) )
=> pp(dvd_dvd(A,X,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).
% dvd_power
tff(fact_2197_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,M2: nat,N: nat] :
( pp(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),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
| ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
& pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N,M2))))) ) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_2198_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_2199_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_2200_scaling__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [U: A,V: A,R3: A,S4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),S4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R3),minus_minus(A,V,U)),S4))),V)) ) ) ) ) ).
% scaling_mono
tff(fact_2201_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
=> ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,divide_divide(A,A3,B2),C3))),modulo_modulo(A,A3,B2)) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2202_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_2203_power__diff__power__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,N: nat,M2: nat] :
( ( A3 != zero_zero(A) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
=> ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),minus_minus(nat,M2,N)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
=> ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),minus_minus(nat,N,M2))) ) ) ) ) ) ).
% power_diff_power_eq
tff(fact_2204_odd__card__imp__not__empty,axiom,
! [A: $tType,A4: set(A)] :
( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),finite_card(A,A4)))
=> ( A4 != bot_bot(set(A)) ) ) ).
% odd_card_imp_not_empty
tff(fact_2205_dvd__partition,axiom,
! [A: $tType,C4: set(set(A)),K: nat] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,C4))
=> pp(dvd_dvd(nat,K,finite_card(A,X2))) )
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,C4))
=> ! [Xa4: set(A)] :
( pp(member2(set(A),Xa4,C4))
=> ( ( X2 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),Xa4) = bot_bot(set(A)) ) ) ) )
=> pp(dvd_dvd(nat,K,finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)))) ) ) ) ).
% dvd_partition
tff(fact_2206_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ~ ! [B4: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4)),one_one(A)) ) ) ).
% oddE
tff(fact_2207_parity__cases,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
=> ~ ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).
% parity_cases
tff(fact_2208_mod2__eq__if,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) )
& ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ) ).
% mod2_eq_if
tff(fact_2209_exp__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M2: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) ) ).
% exp_mod_exp
tff(fact_2210_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = divide_divide(A,A3,B2) ) ) ) ) ).
% divmod_digit_0(1)
tff(fact_2211_inj__on__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
( ( inj_on(A,B,F,A4)
| inj_on(A,B,F,B3) )
=> inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ).
% inj_on_Int
tff(fact_2212_flip__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% flip_bit_0
tff(fact_2213_set__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% set_bit_0
tff(fact_2214_unset__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% unset_bit_0
tff(fact_2215_set__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),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,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% set_bit_Suc
tff(fact_2216_flip__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),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,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% flip_bit_Suc
tff(fact_2217_unset__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),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,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% unset_bit_Suc
tff(fact_2218_signed__take__bit__Suc,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),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,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% signed_take_bit_Suc
tff(fact_2219_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).
% signed_take_bit_Suc_1
tff(fact_2220_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_2221_signed__take__bit__rec,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) )
& ( ( N != zero_zero(nat) )
=> ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),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,N,one_one(nat)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ) ) ).
% signed_take_bit_rec
tff(fact_2222_take__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
& ( ( N != zero_zero(nat) )
=> ( bit_se2584673776208193580ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,minus_minus(nat,N,one_one(nat)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).
% take_bit_rec
tff(fact_2223_gbinomial__absorption_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),minus_minus(nat,K,one_one(nat)))) ) ) ) ).
% gbinomial_absorption'
tff(fact_2224_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( ( divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
=> ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
& ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( bit_se2584673776208193580ke_bit(A,N,A3) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N),one_one(A)) ) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_2225_xor__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3)))) ) ).
% xor_one_eq
tff(fact_2226_one__xor__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3)))) ) ).
% one_xor_eq
tff(fact_2227_neg__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) )
<=> ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
tff(fact_2228_add_Oinverse__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).
% add.inverse_inverse
tff(fact_2229_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( uminus(B)
=> ! [A4: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X)) ) ).
% uminus_apply
tff(fact_2230_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
<=> ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_2231_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_2232_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4)) ) ).
% Compl_subset_Compl_iff
tff(fact_2233_Compl__anti__mono,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B3)),aa(set(A),set(A),uminus_uminus(set(A)),A4))) ) ).
% Compl_anti_mono
tff(fact_2234_neg__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% neg_le_iff_le
tff(fact_2235_compl__le__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% compl_le_compl_iff
tff(fact_2236_neg__equal__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( aa(A,A,uminus_uminus(A),A3) = A3 )
<=> ( A3 = zero_zero(A) ) ) ) ).
% neg_equal_zero
tff(fact_2237_equal__neg__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( A3 = aa(A,A,uminus_uminus(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% equal_neg_zero
tff(fact_2238_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] :
( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% neg_equal_0_iff_equal
tff(fact_2239_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
<=> ( zero_zero(A) = A3 ) ) ) ).
% neg_0_equal_iff_equal
tff(fact_2240_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_2241_neg__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% neg_less_iff_less
tff(fact_2242_compl__less__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% compl_less_compl_iff
tff(fact_2243_minus__add__distrib,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).
% minus_add_distrib
tff(fact_2244_minus__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).
% minus_add_cancel
tff(fact_2245_add__minus__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).
% add_minus_cancel
tff(fact_2246_mult__minus__left,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% mult_minus_left
tff(fact_2247_minus__mult__minus,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) ) ).
% minus_mult_minus
tff(fact_2248_mult__minus__right,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% mult_minus_right
tff(fact_2249_minus__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),minus_minus(A,A3,B2)) = minus_minus(A,B2,A3) ) ).
% minus_diff_eq
tff(fact_2250_Compl__disjoint2,axiom,
! [A: $tType,A4: 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)),A4)),A4) = bot_bot(set(A)) ).
% Compl_disjoint2
tff(fact_2251_Compl__disjoint,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = bot_bot(set(A)) ).
% Compl_disjoint
tff(fact_2252_inter__compl__diff__conv,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = minus_minus(set(A),A4,B3) ).
% inter_compl_diff_conv
tff(fact_2253_Diff__Compl,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : minus_minus(set(A),A4,aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).
% Diff_Compl
tff(fact_2254_Compl__Diff__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),minus_minus(set(A),A4,B3)) = 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)),A4)),B3) ).
% Compl_Diff_eq
tff(fact_2255_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% neg_0_le_iff_le
tff(fact_2256_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% neg_le_0_iff_le
tff(fact_2257_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% less_eq_neg_nonpos
tff(fact_2258_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% neg_less_eq_nonneg
tff(fact_2259_less__neg__neg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% less_neg_neg
tff(fact_2260_neg__less__pos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% neg_less_pos
tff(fact_2261_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% neg_0_less_iff_less
tff(fact_2262_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% neg_less_0_iff_less
tff(fact_2263_ab__left__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).
% ab_left_minus
tff(fact_2264_add_Oright__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).
% add.right_inverse
tff(fact_2265_diff__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : minus_minus(A,zero_zero(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ).
% diff_0
tff(fact_2266_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_2267_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_2268_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = minus_minus(A,B2,A3) ) ).
% uminus_add_conv_diff
tff(fact_2269_diff__minus__eq__add,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).
% diff_minus_eq_add
tff(fact_2270_div__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A] : divide_divide(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).
% div_minus1_right
tff(fact_2271_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_2272_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_2273_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_2274_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_2275_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_2276_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_2277_subset__Compl__singleton,axiom,
! [A: $tType,A4: set(A),B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
<=> ~ pp(member2(A,B2,A4)) ) ).
% subset_Compl_singleton
tff(fact_2278_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% boolean_algebra.de_Morgan_disj
tff(fact_2279_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% boolean_algebra.de_Morgan_conj
tff(fact_2280_take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).
% take_bit_Suc_1
tff(fact_2281_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_2282_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,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_2283_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_2284_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_2285_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_2286_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_2287_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] :
( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
<=> ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
tff(fact_2288_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
tff(fact_2289_mod__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).
% mod_minus1_right
tff(fact_2290_left__minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat,A3: 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))),N)),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))),N)),A3)) = A3 ) ).
% left_minus_one_mult_self
tff(fact_2291_minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: 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))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).
% minus_one_mult_self
tff(fact_2292_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit(A,N,one_one(A)) = zero_zero(A) )
<=> ( N = zero_zero(nat) ) ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_2293_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).
% mult_neg_numeral_simps(3)
tff(fact_2294_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).
% mult_neg_numeral_simps(2)
tff(fact_2295_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).
% mult_neg_numeral_simps(1)
tff(fact_2296_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_2297_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))))
<=> ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_2298_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W: num,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).
% divide_le_eq_numeral1(2)
tff(fact_2299_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).
% le_divide_eq_numeral1(2)
tff(fact_2300_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W: num] :
( ( A3 = divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
<=> ( ( ( 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),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2 ) )
& ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_2301_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W: num,A3: A] :
( ( divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A3 )
<=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) )
& ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_2302_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))))
<=> ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_2303_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W: num,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2)) ) ) ).
% divide_less_eq_numeral1(2)
tff(fact_2304_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))) ) ) ).
% less_divide_eq_numeral1(2)
tff(fact_2305_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_2306_take__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% take_bit_of_1
tff(fact_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)),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),M2),one2))) ) ).
% diff_numeral_special(4)
tff(fact_2314_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : minus_minus(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).
% diff_numeral_special(3)
tff(fact_2315_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_2316_power__minus1__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: 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))),N)) = one_one(A) ) ).
% power_minus1_even
tff(fact_2317_neg__one__odd__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% neg_one_odd_power
tff(fact_2318_neg__one__even__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).
% neg_one_even_power
tff(fact_2319_take__bit__of__exp,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M2: nat,N: nat] : bit_se2584673776208193580ke_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).
% take_bit_of_exp
tff(fact_2320_take__bit__of__2,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_of_2
tff(fact_2321_minus__equation__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = B2 )
<=> ( aa(A,A,uminus_uminus(A),B2) = A3 ) ) ) ).
% minus_equation_iff
tff(fact_2322_equation__minus__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),B2) )
<=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% equation_minus_iff
tff(fact_2323_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( uminus(B)
=> ! [A4: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X3) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X3)) ) ).
% fun_Compl_def
tff(fact_2324_le__imp__neg__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% le_imp_neg_le
tff(fact_2325_minus__le__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).
% minus_le_iff
tff(fact_2326_le__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% le_minus_iff
tff(fact_2327_compl__le__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% compl_le_swap2
tff(fact_2328_compl__le__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% compl_le_swap1
tff(fact_2329_compl__mono,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).
% compl_mono
tff(fact_2330_minus__less__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).
% minus_less_iff
tff(fact_2331_less__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% less_minus_iff
tff(fact_2332_compl__less__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% compl_less_swap2
tff(fact_2333_compl__less__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% compl_less_swap1
tff(fact_2334_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).
% add.inverse_distrib_swap
tff(fact_2335_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,K: A,A3: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).
% group_cancel.neg1
tff(fact_2336_square__eq__iff,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
<=> ( ( A3 = B2 )
| ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).
% square_eq_iff
tff(fact_2337_minus__mult__commute,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% minus_mult_commute
tff(fact_2338_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_2339_minus__diff__commute,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B2: A,A3: A] : minus_minus(A,aa(A,A,uminus_uminus(A),B2),A3) = minus_minus(A,aa(A,A,uminus_uminus(A),A3),B2) ) ).
% minus_diff_commute
tff(fact_2340_minus__assn__def,axiom,
! [A3: assn,B2: assn] : minus_minus(assn,A3,B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B2)) ).
% minus_assn_def
tff(fact_2341_div__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,M2: nat,N: nat] : divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = divide_divide(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% div_mult2_eq'
tff(fact_2342_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_2343_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_2344_add_Oinverse__unique,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).
% add.inverse_unique
tff(fact_2345_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).
% ab_group_add_class.ab_left_minus
tff(fact_2346_add__eq__0__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
<=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% add_eq_0_iff
tff(fact_2347_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% le_minus_one_simps(4)
tff(fact_2348_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).
% le_minus_one_simps(2)
tff(fact_2349_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_2350_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% less_minus_one_simps(4)
tff(fact_2351_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).
% less_minus_one_simps(2)
tff(fact_2352_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% numeral_neq_neg_one
tff(fact_2353_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).
% one_neq_neg_numeral
tff(fact_2354_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_2355_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2356_diff__conv__add__uminus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : minus_minus(A,A3,B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% diff_conv_add_uminus
tff(fact_2357_group__cancel_Osub2,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B3: A,K: A,B2: A,A3: A] :
( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
=> ( minus_minus(A,A3,B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),minus_minus(A,A3,B2)) ) ) ) ).
% group_cancel.sub2
tff(fact_2358_inf__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).
% inf_cancel_left1
tff(fact_2359_inf__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).
% inf_cancel_left2
tff(fact_2360_diff__eq,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : minus_minus(A,X,Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).
% diff_eq
tff(fact_2361_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)))
<=> ( A4 = bot_bot(set(A)) ) ) ).
% subset_Compl_self_eq
tff(fact_2362_Compl__Un,axiom,
! [A: $tType,A4: set(A),B3: 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)),A4),B3)) = 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)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).
% Compl_Un
tff(fact_2363_Compl__Int,axiom,
! [A: $tType,A4: set(A),B3: 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)),A4),B3)) = 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)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).
% Compl_Int
tff(fact_2364_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_2365_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_2366_gbinomial__negated__upper,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),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),A3),one_one(A))),K)) ) ).
% gbinomial_negated_upper
tff(fact_2367_gbinomial__index__swap,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: 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),N)),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))),N)),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))),N)) ) ).
% gbinomial_index_swap
tff(fact_2368_coset__def,axiom,
! [A: $tType,Xs: list(A)] : coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xs)) ).
% coset_def
tff(fact_2369_compl__coset,axiom,
! [A: $tType,Xs: list(A)] : aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% compl_coset
tff(fact_2370_gbinomial__minus,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),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),A3),aa(nat,A,semiring_1_of_nat(A),K)),one_one(A))),K)) ) ).
% gbinomial_minus
tff(fact_2371_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).
% le_minus_one_simps(1)
tff(fact_2372_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% le_minus_one_simps(3)
tff(fact_2373_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% less_minus_one_simps(3)
tff(fact_2374_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).
% less_minus_one_simps(1)
tff(fact_2375_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).
% not_one_le_neg_numeral
tff(fact_2376_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% not_numeral_le_neg_one
tff(fact_2377_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% neg_numeral_le_neg_one
tff(fact_2378_neg__one__le__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).
% neg_one_le_numeral
tff(fact_2379_neg__numeral__le__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).
% neg_numeral_le_one
tff(fact_2380_neg__numeral__less__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).
% neg_numeral_less_one
tff(fact_2381_neg__one__less__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).
% neg_one_less_numeral
tff(fact_2382_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% not_numeral_less_neg_one
tff(fact_2383_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).
% not_one_less_neg_numeral
tff(fact_2384_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).
% not_neg_one_less_neg_numeral
tff(fact_2385_eq__minus__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) )
<=> ( ( ( C3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( C3 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_2386_minus__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C3: A,A3: A] :
( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) = A3 )
<=> ( ( ( C3 != zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) )
& ( ( C3 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_2387_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A,C3: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) = C3 )
<=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_2388_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( C3 = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_2389_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).
% mult_1s_ring_1(2)
tff(fact_2390_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).
% mult_1s_ring_1(1)
tff(fact_2391_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( divide_divide(A,A3,B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_2392_inf__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% inf_shunt
tff(fact_2393_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_2394_shunt1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2))) ) ) ).
% shunt1
tff(fact_2395_shunt2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ) ).
% shunt2
tff(fact_2396_sup__neg__inf,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [P4: A,Q2: A,R3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P4),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P4),aa(A,A,uminus_uminus(A),Q2))),R3)) ) ) ).
% sup_neg_inf
tff(fact_2397_power__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = 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))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_minus
tff(fact_2398_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set(A),B3: 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)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B3))) ) ).
% disjoint_eq_subset_Compl
tff(fact_2399_Compl__insert,axiom,
! [A: $tType,X: A,A4: 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),A4)) = minus_minus(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).
% Compl_insert
tff(fact_2400_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),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,N)),one_one(A)) ) ).
% take_bit_Suc_minus_1_eq
tff(fact_2401_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_2402_mod__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,M2: nat,N: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),modulo_modulo(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M2))) ) ).
% mod_mult2_eq'
tff(fact_2403_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% pos_minus_divide_less_eq
tff(fact_2404_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_less_minus_divide_eq
tff(fact_2405_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_less_eq
tff(fact_2406_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% neg_less_minus_divide_eq
tff(fact_2407_minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% minus_divide_less_eq
tff(fact_2408_less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% less_minus_divide_eq
tff(fact_2409_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C3: A,W: num] :
( ( divide_divide(A,B2,C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
<=> ( ( ( C3 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3) ) )
& ( ( C3 = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_2410_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B2: A,C3: A] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = divide_divide(A,B2,C3) )
<=> ( ( ( C3 != 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))),C3) = B2 ) )
& ( ( C3 = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_2411_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: 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,A3,Z2))),B2) = B2 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_2412_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_2413_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2)),B2) = divide_divide(A,minus_minus(A,aa(A,A,uminus_uminus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(6)
tff(fact_2414_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( minus_minus(A,divide_divide(A,A3,Z2),B2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( Z2 != zero_zero(A) )
=> ( minus_minus(A,divide_divide(A,A3,Z2),B2) = divide_divide(A,minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_2415_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_2416_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),K)) ) ).
% gbinomial_absorb_comp
tff(fact_2417_gbinomial__mult__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),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,A3),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,A3),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1
tff(fact_2418_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),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,A3),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1'
tff(fact_2419_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_Suc_bit0
tff(fact_2420_le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% le_minus_divide_eq
tff(fact_2421_minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% minus_divide_le_eq
tff(fact_2422_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% neg_le_minus_divide_eq
tff(fact_2423_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_le_eq
tff(fact_2424_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_le_minus_divide_eq
tff(fact_2425_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).
% pos_minus_divide_le_eq
tff(fact_2426_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_2427_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_2428_Suc__times__gbinomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).
% Suc_times_gbinomial
tff(fact_2429_gbinomial__absorption,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),K)) ) ).
% gbinomial_absorption
tff(fact_2430_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,M2: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M2)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M2)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,minus_minus(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),minus_minus(nat,M2,K))) ) ) ) ).
% gbinomial_trinomial_revision
tff(fact_2431_power2__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A3: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
<=> ( ( A3 = one_one(A) )
| ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% power2_eq_1_iff
tff(fact_2432_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( 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),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),minus_minus(nat,N,K)) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2433_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C3: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_2434_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_2435_gbinomial__factors,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,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),A3),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).
% gbinomial_factors
tff(fact_2436_gbinomial__rec,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).
% gbinomial_rec
tff(fact_2437_square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,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_2438_minus__power__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A3: A,N: 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),A3)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).
% minus_power_mult_self
tff(fact_2439_minus__one__power__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
& ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% minus_one_power_iff
tff(fact_2440_power__minus1__odd,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: 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))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% power_minus1_odd
tff(fact_2441_take__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% take_bit_Suc
tff(fact_2442_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_2443_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_2444_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_2445_of__nat__Suc,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M2)) ) ).
% of_nat_Suc
tff(fact_2446_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_2447_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
<=> ( N = one_one(nat) ) ) ) ).
% of_nat_1_eq_iff
tff(fact_2448_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B3) )
<=> ( A4 = B3 ) ) ).
% Compl_eq_Compl_iff
tff(fact_2449_Compl__iff,axiom,
! [A: $tType,C3: A,A4: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4)))
<=> ~ pp(member2(A,C3,A4)) ) ).
% Compl_iff
tff(fact_2450_ComplI,axiom,
! [A: $tType,C3: A,A4: set(A)] :
( ~ pp(member2(A,C3,A4))
=> pp(member2(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4))) ) ).
% ComplI
tff(fact_2451_of__nat__mult,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_mult
tff(fact_2452_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
<=> ( N = one_one(nat) ) ) ) ).
% of_nat_eq_1_iff
tff(fact_2453_double__complement,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = A4 ).
% double_complement
tff(fact_2454_ComplD,axiom,
! [A: $tType,C3: A,A4: set(A)] :
( pp(member2(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4)))
=> ~ pp(member2(A,C3,A4)) ) ).
% ComplD
tff(fact_2455_inf__nat__def,axiom,
inf_inf(nat) = ord_min(nat) ).
% inf_nat_def
tff(fact_2456_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_2457_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_2458_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
=> ~ ! [N7: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N7)))),E3)) ) ) ).
% nat_approx_posE
tff(fact_2459_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ? [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N7)),X))) ) ) ).
% ex_less_of_nat_mult
tff(fact_2460_pochhammer__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N: 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))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,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))),N)))),comm_s3205402744901411588hammer(A,Z2,N))),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)))),N)) ) ).
% pochhammer_double
tff(fact_2461_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N),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),N))) ) ).
% push_bit_numeral_minus_1
tff(fact_2462_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_2463_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_2464_pochhammer__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).
% pochhammer_0
tff(fact_2465_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_2466_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_2467_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_2468_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_2469_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_2470_push__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),A3) = bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% push_bit_Suc
tff(fact_2471_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_2472_push__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).
% push_bit_of_1
tff(fact_2473_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_2474_numeral__Bit1,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),bit1(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).
% numeral_Bit1
tff(fact_2475_pochhammer__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).
% pochhammer_0_left
tff(fact_2476_flip__bit__eq__xor,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).
% flip_bit_eq_xor
tff(fact_2477_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),A3)),A3) ) ).
% power3_eq_cube
tff(fact_2478_push__bit__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,N,A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% push_bit_double
tff(fact_2479_pochhammer__rec,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).
% pochhammer_rec
tff(fact_2480_pochhammer__rec_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z2,N)) ) ).
% pochhammer_rec'
tff(fact_2481_pochhammer__Suc,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).
% pochhammer_Suc
tff(fact_2482_pochhammer__product_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N: nat,M2: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N)),M2)) ) ).
% pochhammer_product'
tff(fact_2483_card__3__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
<=> ? [X4: A,Y3: A,Z3: A] :
( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),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))))) )
& ( X4 != Y3 )
& ( Y3 != Z3 )
& ( X4 != Z3 ) ) ) ).
% card_3_iff
tff(fact_2484_push__bit__eq__mult,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).
% push_bit_eq_mult
tff(fact_2485_pochhammer__product,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M2: nat,N: nat,Z2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( comm_s3205402744901411588hammer(A,Z2,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M2)),minus_minus(nat,N,M2))) ) ) ) ).
% pochhammer_product
tff(fact_2486_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,R3,aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).
% pochhammer_absorb_comp
tff(fact_2487_pochhammer__minus,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).
% pochhammer_minus
tff(fact_2488_pochhammer__minus_H,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).
% pochhammer_minus'
tff(fact_2489_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),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,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).
% take_bit_Suc_bit1
tff(fact_2490_fact__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: 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))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,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))),N))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).
% fact_double
tff(fact_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_and_Oright__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).
% and.right_neutral
tff(fact_2498_and_Oleft__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).
% and.left_neutral
tff(fact_2499_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_2500_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_2501_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_2502_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_2503_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_2504_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_2505_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_2506_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_2507_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_2508_fact__Suc,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).
% fact_Suc
tff(fact_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
tff(fact_2518_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_2519_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
& ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_2520_fact__ge__1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).
% fact_ge_1
tff(fact_2521_pochhammer__fact,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_semiring_1(A) )
=> ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).
% pochhammer_fact
tff(fact_2522_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] : pp(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,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).
% fact_fact_dvd_fact
tff(fact_2523_set__bit__eq__or,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) ) ).
% set_bit_eq_or
tff(fact_2524_choose__dvd,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> pp(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,N,K))),semiring_char_0_fact(A,N))) ) ) ).
% choose_dvd
tff(fact_2525_and__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% and_one_eq
tff(fact_2526_one__and__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% one_and_eq
tff(fact_2527_fact__num__eq__if,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M2: nat] :
( ( ( M2 = zero_zero(nat) )
=> ( semiring_char_0_fact(A,M2) = one_one(A) ) )
& ( ( M2 != zero_zero(nat) )
=> ( semiring_char_0_fact(A,M2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),semiring_char_0_fact(A,minus_minus(nat,M2,one_one(nat)))) ) ) ) ) ).
% fact_num_eq_if
tff(fact_2528_fact__reduce,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,minus_minus(nat,N,one_one(nat)))) ) ) ) ).
% fact_reduce
tff(fact_2529_pochhammer__same,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_ring_1(A)
& semiri3467727345109120633visors(A) )
=> ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = 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))),N)),semiring_char_0_fact(A,N)) ) ).
% pochhammer_same
tff(fact_2530_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_2531_or__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))) ) ).
% or_one_eq
tff(fact_2532_one__or__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))) ) ).
% one_or_eq
tff(fact_2533_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = divide_divide(A,comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer'
tff(fact_2534_gbinomial__pochhammer,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),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),A3),K)),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer
tff(fact_2535_mask__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).
% mask_numeral
tff(fact_2536_mask__Suc__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = 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,N))) ) ).
% mask_Suc_double
tff(fact_2537_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N),one_one(A)) ) ).
% mask_eq_exp_minus_1
tff(fact_2538_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [J3: nat] : ~ pp(bit_se5641148757651400278ts_bit(A,A3,aa(nat,nat,suc,J3)))
=> ( pp(bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),N))
<=> ( ( ( N = zero_zero(nat) )
=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3)) )
& ( ( N != zero_zero(nat) )
=> pp(bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2),N)) ) ) ) ) ) ).
% bit_sum_mult_2_cases
tff(fact_2539_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M2) ) ).
% diff_numeral_special(8)
tff(fact_2540_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: 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),N))) = neg_numeral_sub(A,N,one2) ) ).
% diff_numeral_special(7)
tff(fact_2541_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_2542_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).
% take_bit_minus_one_eq_mask
tff(fact_2543_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : minus_minus(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).
% diff_numeral_special(1)
tff(fact_2544_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : minus_minus(A,aa(num,A,numeral_numeral(A),M2),one_one(A)) = neg_numeral_sub(A,M2,one2) ) ).
% diff_numeral_special(2)
tff(fact_2545_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: 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),M2))) = neg_numeral_sub(A,one2,M2) ) ).
% add_neg_numeral_special(1)
tff(fact_2546_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A)) = neg_numeral_sub(A,one2,M2) ) ).
% add_neg_numeral_special(2)
tff(fact_2547_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M2,one2) ) ).
% add_neg_numeral_special(3)
tff(fact_2548_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one2) ) ).
% add_neg_numeral_special(4)
tff(fact_2549_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M2: num] : minus_minus(A,aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M2,one2)),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) ) ).
% minus_sub_one_diff_one
tff(fact_2550_bit__1__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,one_one(A),N))
<=> ( N = zero_zero(nat) ) ) ) ).
% bit_1_iff
tff(fact_2551_not__bit__1__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : ~ pp(bit_se5641148757651400278ts_bit(A,one_one(A),aa(nat,nat,suc,N))) ) ).
% not_bit_1_Suc
tff(fact_2552_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: num] : ~ pp(bit_se5641148757651400278ts_bit(A,one_one(A),aa(num,nat,numeral_numeral(nat),N))) ) ).
% bit_numeral_simps(1)
tff(fact_2553_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,A3,N))
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),bit_se4730199178511100633sh_bit(A,N,one_one(A))) != zero_zero(A) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_2554_even__bit__succ__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( pp(bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),N))
<=> ( pp(bit_se5641148757651400278ts_bit(A,A3,N))
| ( N = zero_zero(nat) ) ) ) ) ) ).
% even_bit_succ_iff
tff(fact_2555_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ( pp(bit_se5641148757651400278ts_bit(A,A3,N))
<=> ( pp(bit_se5641148757651400278ts_bit(A,minus_minus(A,A3,one_one(A)),N))
| ( N = zero_zero(nat) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_2556_signed__take__bit__def,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se2584673776208193580ke_bit(A,N,A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),bit_se5641148757651400278ts_bit(A,A3,N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).
% signed_take_bit_def
tff(fact_2557_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).
% sorted_list_of_set_atMost_Suc
tff(fact_2558_power__minus_H,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,N: 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)),N) = 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))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).
% power_minus'
tff(fact_2559_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_2560_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_2561_division__decomp,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
=> ? [B10: A,C7: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B10),C7) )
& pp(dvd_dvd(A,B10,B2))
& pp(dvd_dvd(A,C7,C3)) ) ) ) ).
% division_decomp
tff(fact_2562_Int__atMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,A3)),set_ord_atMost(A,B2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).
% Int_atMost
tff(fact_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).
% push_bit_minus_one_eq_not_mask
tff(fact_2572_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_2573_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [H: A] : bot_bot(set(A)) != set_ord_atMost(A,H) ) ).
% not_empty_eq_Iic_eq_empty
tff(fact_2574_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_2575_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_2576_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).
% minus_eq_not_plus_1
tff(fact_2577_not__eq__complement,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = minus_minus(A,aa(A,A,uminus_uminus(A),A3),one_one(A)) ) ).
% not_eq_complement
tff(fact_2578_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),minus_minus(A,A3,one_one(A))) ) ).
% minus_eq_not_minus_1
tff(fact_2579_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)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(3)
tff(fact_2580_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)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(1)
tff(fact_2581_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se4730199178511100633sh_bit(A,N,one_one(A)))) ) ).
% unset_bit_eq_and_not
tff(fact_2582_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)),set_ord_lessThan(A,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).
% ivl_disj_un_singleton(2)
tff(fact_2583_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_2584_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).
% distrib_left_NO_MATCH
tff(fact_2585_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),C3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).
% distrib_right_NO_MATCH
tff(fact_2586_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),C3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),C3) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_2587_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),minus_minus(A,B2,C3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_2588_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_2589_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_2590_dvd__productE,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [P4: A,A3: A,B2: A] :
( pp(dvd_dvd(A,P4,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> ~ ! [X2: A,Y2: A] :
( ( P4 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y2) )
=> ( pp(dvd_dvd(A,X2,A3))
=> ~ pp(dvd_dvd(A,Y2,B2)) ) ) ) ) ).
% dvd_productE
tff(fact_2591_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X: B,B2: A,A3: A] :
( nO_MATCH(B,A,X,B2)
=> ( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ) ).
% div_add_self2_no_field
tff(fact_2592_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X: B,B2: A,A3: A] :
( nO_MATCH(B,A,X,B2)
=> ( ( B2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ) ).
% div_add_self1_no_field
tff(fact_2593_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_2594_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_2595_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xs),Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),divide_divide(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).
% product_nth
tff(fact_2596_rotate__drop__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs)) ).
% rotate_drop_take
tff(fact_2597_rotate__is__Nil__conv,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
% rotate_is_Nil_conv
tff(fact_2598_set__rotate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_rotate
tff(fact_2599_length__rotate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_rotate
tff(fact_2600_distinct__rotate,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% distinct_rotate
tff(fact_2601_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) ).
% rotate_Suc
tff(fact_2602_rotate__length01,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = Xs ) ) ).
% rotate_length01
tff(fact_2603_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = Xs ) ) ).
% rotate_id
tff(fact_2604_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
( ! [F4: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
=> ~ ! [F4: fun(A,B),A5: A,As: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F4),aa(list(B),product_prod(list(A),list(B)),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),As)),Bs2)) ) ).
% map_tailrec_rev.cases
tff(fact_2605_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
( ! [F4: 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))),F4),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)))
=> ( ! [F4: fun(A,fun(B,C)),A5: A,As: list(A),B4: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F4),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),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))
=> ( ! [A5: fun(A,fun(B,C)),V2: A,Va2: 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),V2),Va2)),nil(B)))
=> ~ ! [A5: fun(A,fun(B,C)),V2: B,Va2: 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),V2),Va2))) ) ) ) ).
% zipf.cases
tff(fact_2606_list__assn_Ocases,axiom,
! [A: $tType,C: $tType,X: product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))] :
( ! [P2: fun(A,fun(C,assn))] : X != aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),P2),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),nil(A)),nil(C)))
=> ( ! [P2: fun(A,fun(C,assn)),A5: A,As: list(A),C2: C,Cs: list(C)] : X != aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),P2),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),C2),Cs)))
=> ( ! [Uu2: fun(A,fun(C,assn)),V2: A,Va2: list(A)] : X != aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),Uu2),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)),nil(C)))
=> ~ ! [Uu2: fun(A,fun(C,assn)),V2: C,Va2: list(C)] : X != aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),Uu2),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),nil(A)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va2))) ) ) ) ).
% list_assn.cases
tff(fact_2607_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
( ! [R4: fun(A,fun(A,bool)),X2: A,Xs2: list(A),Y2: A,Ys2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),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),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
=> ( ! [R4: fun(A,fun(A,bool)),Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
=> ~ ! [R4: fun(A,fun(A,bool)),V2: A,Va2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),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),V2),Va2))) ) ) ).
% mergesort_by_rel_merge.cases
tff(fact_2608_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
( ! [R4: fun(A,fun(A,bool)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),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)))
=> ~ ! [R4: fun(A,fun(A,bool)),Sl: list(A),X2: A,Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),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),X2),Xs2))) ) ).
% quicksort_by_rel.cases
tff(fact_2609_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))] :
( ! [P2: fun(A,fun(B,bool))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),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)))
=> ( ! [P2: fun(A,fun(B,bool)),A5: A,As: list(A),B4: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),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),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))
=> ( ! [P2: fun(A,fun(B,bool)),V2: A,Va2: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),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),V2),Va2)),nil(B)))
=> ~ ! [P2: fun(A,fun(B,bool)),V2: B,Va2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P2),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),V2),Va2))) ) ) ) ).
% list_all_zip.cases
tff(fact_2610_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S: set(product_prod(A,B)),P: fun(A,fun(B,bool))] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S))
=> ( ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) )
=> ? [A5: A,B4: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4),S))
& pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B4)) ) ) ) ).
% bex2I
tff(fact_2611_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
~ ! [F4: fun(A,B),A5: A,B4: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F4),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)) ).
% pairself.cases
tff(fact_2612_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z2: A,X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_2613_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_2614_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A,Z2: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_2615_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_2616_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_2617_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_2618_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),One) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_2619_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,aa(A,fun(A,A),Xor,X),Y)) = Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_2620_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,One),X) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_2621_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),X)),aa(A,A,aa(A,fun(A,A),Disj,Z2),X)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_2622_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_2623_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,Compl,X) = aa(A,A,Compl,Y) )
<=> ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_2624_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),X) = Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_2625_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y))) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_2626_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,X),Z2)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
tff(fact_2627_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
tff(fact_2628_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
tff(fact_2629_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,X)) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
tff(fact_2630_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),X) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),X) = One )
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Y) = One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
tff(fact_2631_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_2632_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
tff(fact_2633_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
tff(fact_2634_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),Zero) = X ) ) ).
% abstract_boolean_algebra.disj_zero_right
tff(fact_2635_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),Zero) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
tff(fact_2636_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),One) = One ) ) ).
% abstract_boolean_algebra.disj_one_right
tff(fact_2637_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,X),Y)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
tff(fact_2638_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,X),Y)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
tff(fact_2639_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,Zero),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
tff(fact_2640_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),One) = X ) ) ).
% abstract_boolean_algebra.conj_one_right
tff(fact_2641_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,One),X) = One ) ) ).
% abstract_boolean_algebra.disj_one_left
tff(fact_2642_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,Compl,X)) = X ) ) ).
% abstract_boolean_algebra.double_compl
tff(fact_2643_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,X),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,X),Y) = One )
=> ( aa(A,A,Compl,X) = Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
tff(fact_2644_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,Zero) = One ) ) ).
% abstract_boolean_algebra.compl_zero
tff(fact_2645_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,One) = Zero ) ) ).
% abstract_boolean_algebra.compl_one
tff(fact_2646_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
tff(fact_2647_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))] :
( ! [P2: fun(A,bool),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P2),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)))
=> ~ ! [P2: fun(A,bool),Yes: list(A),No: list(A),X2: A,Xs2: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P2),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),X2),Xs2))) ) ).
% partition_rev.cases
tff(fact_2648_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),X2: 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),X2),nil(A)))
=> ~ ! [Xs12: list(A),Xs22: list(A),X1: A,X23: A,Xs2: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) ) ) ).
% mergesort_by_rel_split.cases
tff(fact_2649_rotate__rotate,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),Xs) ).
% rotate_rotate
tff(fact_2650_rotate__map,axiom,
! [A: $tType,B: $tType,N: nat,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),rotate(B),N),Xs)) ).
% rotate_map
tff(fact_2651_rotate1__rotate__swap,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rotate1(A),Xs)) ).
% rotate1_rotate_swap
tff(fact_2652_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: product_prod(fun(A,B),list(A))] :
( ! [F4: fun(A,B),X2: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
=> ( ! [F4: fun(A,B),X2: A,Y2: A,Zs3: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)))
=> ~ ! [A5: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A5),nil(A)) ) ) ) ).
% arg_min_list.cases
tff(fact_2653_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P2: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),nil(A))
=> ~ ! [P2: fun(A,fun(A,bool)),X2: A,Ys2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)) ) ).
% sorted_wrt.cases
tff(fact_2654_successively_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P2: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),nil(A))
=> ( ! [P2: fun(A,fun(A,bool)),X2: A] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
=> ~ ! [P2: fun(A,fun(A,bool)),X2: A,Y2: A,Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ).
% successively.cases
tff(fact_2655_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod(list(A),list(A))] :
( ! [Ys2: 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)),Ys2)
=> ( ! [Xs2: 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)),Xs2),nil(A))
=> ~ ! [X2: A,Xs2: list(A),Y2: A,Ys2: 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),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)) ) ) ).
% shuffles.cases
tff(fact_2656_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)
=> ( ! [V2: A,Va2: 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),V2),Va2)),nil(A))
=> ~ ! [X1: 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),X1),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)) ) ) ) ).
% merge.cases
tff(fact_2657_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))) )
=> ( ! [L2: 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)),L2),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)),L2: 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)),L2),nil(list(A))))
=> ~ ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls2: 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),Ls2))) ) ) ) ) ) ).
% merge_list.cases
tff(fact_2658_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,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)),aa(fun(B,product_prod(A,B)),fun(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xs),Ys)) ).
% product.simps(2)
tff(fact_2659_rotate__append,axiom,
! [A: $tType,L: list(A),Q2: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q2),L) ).
% rotate_append
tff(fact_2660_rotate__conv__mod,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).
% rotate_conv_mod
tff(fact_2661_rotate__rev,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).
% rotate_rev
tff(fact_2662_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list(A),M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% nth_rotate
tff(fact_2663_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( Xs != nil(A) )
=> ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% hd_rotate_conv_nth
tff(fact_2664_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,R3: A,Q2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),minus_minus(A,R3,aa(num,A,numeral_numeral(A),L))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R3) ) ) ) ) ).
% divmod_step_eq
tff(fact_2665_lexord__take__index__conv,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R3)))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
& ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),X)),Y) = X ) )
| ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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))))
& ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),X) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Y) )
& pp(member2(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),I2)),aa(nat,A,nth(A,Y),I2)),R3)) ) ) ) ).
% lexord_take_index_conv
tff(fact_2666_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ~ ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys)))
=> ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I3),Ys) )
=> ~ pp(member2(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)),R3)) ) ) ) ) ).
% lex_take_index
tff(fact_2667_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod(list(A),list(A))] :
( ! [Ys2: 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)),Ys2)
=> ~ ! [X2: A,Xs2: list(A),Ys2: 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),X2),Xs2)),Ys2) ) ).
% subset_eq_mset_impl.cases
tff(fact_2668_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).
% diff_numeral_special(5)
tff(fact_2669_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : minus_minus(A,aa(num,A,numeral_numeral(A),M2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M2)) ) ).
% diff_numeral_special(6)
tff(fact_2670_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)),lexord(A,R3)))
<=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3))
| ( ( A3 = B2 )
& pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R3))) ) ) ) ).
% lexord_cons_cons
tff(fact_2671_lexord__Nil__left,axiom,
! [A: $tType,Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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)),nil(A)),Y),lexord(A,R3)))
<=> ? [A6: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),X4) ) ).
% lexord_Nil_left
tff(fact_2672_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M2))) ) ).
% add_neg_numeral_special(6)
tff(fact_2673_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).
% add_neg_numeral_special(5)
tff(fact_2674_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
<=> ( ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3))
& ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
| ( ( X = Y )
& pp(member2(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,R3))) ) ) ) ).
% Cons_in_lex
tff(fact_2675_lexord__lex,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),lex(A,R3)))
<=> ( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R3)))
& ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).
% lexord_lex
tff(fact_2676_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
~ ! [R4: fun(A,fun(A,bool)),Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R4),Xs2) ).
% mergesort_by_rel.cases
tff(fact_2677_lexord__linear,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: list(A),Y: list(A)] :
( ! [A5: A,B4: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),R3))
| ( A5 = B4 )
| pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5),R3)) )
=> ( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R3)))
| ( X = Y )
| pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X),lexord(A,R3))) ) ) ).
% lexord_linear
tff(fact_2678_lexord__irreflexive,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
( ! [X2: A] : ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R3))
=> ~ pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lexord(A,R3))) ) ).
% lexord_irreflexive
tff(fact_2679_lexord__Nil__right,axiom,
! [A: $tType,X: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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),nil(A)),lexord(A,R3))) ).
% lexord_Nil_right
tff(fact_2680_lexord__append__leftI,axiom,
! [A: $tType,U: list(A),V: list(A),R3: set(product_prod(A,A)),X: list(A)] :
( pp(member2(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,R3)))
=> pp(member2(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,R3))) ) ).
% lexord_append_leftI
tff(fact_2681_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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),nil(A)),lex(A,R3))) ).
% Nil2_notin_lex
tff(fact_2682_Nil__notin__lex,axiom,
! [A: $tType,Ys: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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)),nil(A)),Ys),lex(A,R3))) ).
% Nil_notin_lex
tff(fact_2683_lex__append__leftI,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),R3: set(product_prod(A,A)),Xs: list(A)] :
( pp(member2(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),Zs),lex(A,R3)))
=> pp(member2(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),Zs)),lex(A,R3))) ) ).
% lex_append_leftI
tff(fact_2684_lexord__partial__trans,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
( ! [X2: A,Y2: A,Z4: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2),R3))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z4),R3))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z4),R3)) ) ) )
=> ( pp(member2(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,R3)))
=> ( pp(member2(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),Zs),lexord(A,R3)))
=> pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),lexord(A,R3))) ) ) ) ).
% lexord_partial_trans
tff(fact_2685_lexord__append__leftD,axiom,
! [A: $tType,X: list(A),U: list(A),V: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ( ! [A5: A] : ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R3))
=> pp(member2(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,R3))) ) ) ).
% lexord_append_leftD
tff(fact_2686_lexord__append__rightI,axiom,
! [A: $tType,Y: list(A),X: list(A),R3: set(product_prod(A,A))] :
( ? [B11: A,Z5: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B11),Z5)
=> pp(member2(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),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),Y)),lexord(A,R3))) ) ).
% lexord_append_rightI
tff(fact_2687_lexord__sufE,axiom,
! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs)),lexord(A,R3)))
=> ( ( Xs != Ys )
=> ( ( 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)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
=> pp(member2(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,R3))) ) ) ) ) ).
% lexord_sufE
tff(fact_2688_lex__append__left__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( ! [X2: A] : ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R3))
=> ( pp(member2(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),Zs)),lex(A,R3)))
<=> pp(member2(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),Zs),lex(A,R3))) ) ) ).
% lex_append_left_iff
tff(fact_2689_lex__append__leftD,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( ! [X2: A] : ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R3))
=> ( pp(member2(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),Zs)),lex(A,R3)))
=> pp(member2(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),Zs),lex(A,R3))) ) ) ).
% lex_append_leftD
tff(fact_2690_lex__append__rightI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Vs: list(A),Us2: list(A)] :
( pp(member2(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,R3)))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us2) )
=> pp(member2(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),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),lex(A,R3))) ) ) ).
% lex_append_rightI
tff(fact_2691_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3))
=> pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R3))) ) ).
% lexord_append_left_rightI
tff(fact_2692_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),Zs)),lexord(A,R3)))
<=> ( ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R3)) )
| pp(member2(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),Zs),lexord(A,R3))) ) ) ).
% lexord_same_pref_iff
tff(fact_2693_lexord__sufI,axiom,
! [A: $tType,U: list(A),W: list(A),R3: set(product_prod(A,A)),V: list(A),Z2: list(A)] :
( pp(member2(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),W),lexord(A,R3)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U)))
=> pp(member2(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),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z2)),lexord(A,R3))) ) ) ).
% lexord_sufI
tff(fact_2694_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_2695_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),M2),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns)),lenlex(A,R3)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
| ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),N),R3)) )
| ( ( M2 = N )
& pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R3))) ) ) ) ).
% Cons_lenlex_iff
tff(fact_2696_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
<=> ? [Y3: A,N3: nat] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N3)),Y3),R3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
& ( Ys = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N3),Y3) ) ) ) ).
% listrel1_iff_update
tff(fact_2697_nth__step__trancl,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),N: nat,M2: nat] :
( ! [N7: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),one_one(nat))))
=> pp(member2(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,N7))),aa(nat,A,nth(A,Xs),N7)),R)) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),aa(nat,A,nth(A,Xs),M2)),transitive_trancl(A,R))) ) ) ) ).
% nth_step_trancl
tff(fact_2698_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).
% nth_zip
tff(fact_2699_list__assn_Opelims,axiom,
! [A: $tType,C: $tType,X: fun(A,fun(C,assn)),Xa: list(A),Xb: list(C),Y: assn] :
( ( vEBT_List_list_assn(A,C,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),vEBT_L4249061453398456502sn_rel(A,C),aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),X),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(C) )
=> ( ( Y = one_one(assn) )
=> ~ accp(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),vEBT_L4249061453398456502sn_rel(A,C),aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),X),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),nil(A)),nil(C)))) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [C2: C,Cs: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),C2),Cs) )
=> ( ( Y = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(C,assn,aa(A,fun(C,assn),X,A5),C2)),vEBT_List_list_assn(A,C,X,As,Cs)) )
=> ~ accp(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),vEBT_L4249061453398456502sn_rel(A,C),aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),X),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),C2),Cs)))) ) ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xb = nil(C) )
=> ( ( Y = bot_bot(assn) )
=> ~ accp(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),vEBT_L4249061453398456502sn_rel(A,C),aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),X),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)),nil(C)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: C,Va2: list(C)] :
( ( Xb = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va2) )
=> ( ( Y = bot_bot(assn) )
=> ~ accp(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),vEBT_L4249061453398456502sn_rel(A,C),aa(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),aa(fun(A,fun(C,assn)),fun(product_prod(list(A),list(C)),product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C)))),product_Pair(fun(A,fun(C,assn)),product_prod(list(A),list(C))),X),aa(list(C),product_prod(list(A),list(C)),aa(list(A),fun(list(C),product_prod(list(A),list(C))),product_Pair(list(A),list(C)),nil(A)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),V2),Va2)))) ) ) ) ) ) ) ) ) ).
% list_assn.pelims
tff(fact_2700_Gcd__0__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( gcd_Gcd(A,A4) = zero_zero(A) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),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_2701_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A)))) ).
% trancl_single
tff(fact_2702_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A10: list(A),B8: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ( aa(list(A),nat,size_size(list(A)),A10) = aa(list(B),nat,size_size(list(B)),B8) )
=> ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B2) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A10),B8) )
<=> ( ( A3 = A10 )
& ( B2 = B8 ) ) ) ) ) ).
% zip_eq_zip_same_len
tff(fact_2703_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) ).
% zip_Cons_Cons
tff(fact_2704_Gcd__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_empty
tff(fact_2705_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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)),nil(A)),Ns),lenlex(A,R3)))
<=> ( Ns != nil(A) ) ) ).
% Nil_lenlex_iff1
tff(fact_2706_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys) = nil(product_prod(A,B)) )
<=> ( ( Xs = nil(A) )
| ( Ys = nil(B) ) ) ) ).
% zip_eq_Nil_iff
tff(fact_2707_Nil__eq__zip__iff,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( nil(product_prod(A,B)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys) )
<=> ( ( Xs = nil(A) )
| ( Ys = nil(B) ) ) ) ).
% Nil_eq_zip_iff
tff(fact_2708_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),nil(A)),Ys) = nil(product_prod(A,B)) ).
% zip_Nil
tff(fact_2709_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
<=> ( ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3))
& ( Xs = Ys ) )
| ( ( X = Y )
& pp(member2(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,R3))) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_2710_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list(A),Us2: list(B),Ys: list(A),Vs: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us2) )
=> ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs)) = 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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Us2)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Ys),Vs)) ) ) ).
% zip_append
tff(fact_2711_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).
% length_zip
tff(fact_2712_listrel1__mono,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S4))
=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel1(A,S4))) ) ).
% listrel1_mono
tff(fact_2713_trancl__sub,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),transitive_trancl(A,R))) ).
% trancl_sub
tff(fact_2714_trancl__mono__mp,axiom,
! [A: $tType,U2: set(product_prod(A,A)),V3: set(product_prod(A,A)),X: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V3))
=> ( pp(member2(product_prod(A,A),X,transitive_trancl(A,U2)))
=> pp(member2(product_prod(A,A),X,transitive_trancl(A,V3))) ) ) ).
% trancl_mono_mp
tff(fact_2715_trancl__sub__insert__trancl,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R)),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),R)))) ).
% trancl_sub_insert_trancl
tff(fact_2716_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A10: list(A),B8: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ( aa(list(A),nat,size_size(list(A)),A10) = aa(list(B),nat,size_size(list(B)),B8) )
=> ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B2) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A10),B8) )
=> ( ( A3 = A10 )
& ( B2 = B8 ) ) ) ) ) ).
% zip_inj
tff(fact_2717_take__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),take(product_prod(A,B)),N),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ).
% take_zip
tff(fact_2718_drop__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),drop(product_prod(A,B)),N),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys)) ).
% drop_zip
tff(fact_2719_Gcd__1,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( pp(member2(A,one_one(A),A4))
=> ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ).
% Gcd_1
tff(fact_2720_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))] :
( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),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) )
=> ~ ! [X2: A,Xs4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4) )
=> ! [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),X2),Y2) )
=> ( Xys != aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs4),Ys4) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_2721_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xs),Xs))))
<=> ( pp(member2(A,A3,aa(list(A),set(A),set2(A),Xs)))
& ( A3 = B2 ) ) ) ).
% zip_same
tff(fact_2722_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member2(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
=> ~ ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(member2(B,Y,aa(list(B),set(B),set2(B),Ys))) ) ) ).
% in_set_zipE
tff(fact_2723_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member2(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% set_zip_leftD
tff(fact_2724_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member2(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
=> pp(member2(B,Y,aa(list(B),set(B),set2(B),Ys))) ) ).
% set_zip_rightD
tff(fact_2725_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I: nat,X: A,Ys: list(B),Y: B] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)),I),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_update
tff(fact_2726_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),rev(A),Xs)),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) ) ) ).
% zip_rev
tff(fact_2727_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list(A)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),nil(B)) = nil(product_prod(A,B)) ).
% zip.simps(1)
tff(fact_2728_listrel1I2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),X: A] :
( pp(member2(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,R3)))
=> pp(member2(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),X),Ys)),listrel1(A,R3))) ) ).
% listrel1I2
tff(fact_2729_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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)),nil(A)),Xs),listrel1(A,R3))) ).
% not_Nil_listrel1
tff(fact_2730_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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),nil(A)),listrel1(A,R3))) ).
% not_listrel1_Nil
tff(fact_2731_listrel1__eq__len,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).
% listrel1_eq_len
tff(fact_2732_append__listrel1I,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Us2: list(A),Vs: list(A)] :
( ( ( pp(member2(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,R3)))
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& pp(member2(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)),Us2),Vs),listrel1(A,R3))) ) )
=> pp(member2(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),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),listrel1(A,R3))) ) ).
% append_listrel1I
tff(fact_2733_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
~ ! [L1: list(A),L22: list(B)] :
( ( L = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),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_2734_Gcd__eq__1__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,A4: set(A)] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(member2(A,A3,A4))
=> ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ) ).
% Gcd_eq_1_I
tff(fact_2735_distinct__zipI2,axiom,
! [B: $tType,A: $tType,Ys: list(A),Xs: list(B)] :
( pp(aa(list(A),bool,distinct(A),Ys))
=> pp(aa(list(product_prod(B,A)),bool,distinct(product_prod(B,A)),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xs),Ys))) ) ).
% distinct_zipI2
tff(fact_2736_distinct__zipI1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(product_prod(A,B)),bool,distinct(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))) ) ).
% distinct_zipI1
tff(fact_2737_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) )
=> ( pp(member2(B,Y,aa(list(B),set(B),set2(B),Ys)))
=> ~ ! [X2: A] : ~ pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)))) ) ) ).
% in_set_impl_in_set_zip2
tff(fact_2738_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) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ~ ! [Y2: B] : ~ pp(member2(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)))) ) ) ).
% in_set_impl_in_set_zip1
tff(fact_2739_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(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),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).
% hd_zip
tff(fact_2740_lenlex__irreflexive,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
( ! [X2: A] : ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R3))
=> ~ pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R3))) ) ).
% lenlex_irreflexive
tff(fact_2741_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),bool)] :
( ! [Zs3: list(A),Ws3: list(B),N7: nat] :
( ( aa(list(A),nat,size_size(list(A)),Zs3) = aa(list(B),nat,size_size(list(B)),Ws3) )
=> ( ( N7 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
=> ( ( Zs3 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N7),Xs) )
=> ( ( Ws3 = aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N7),Ys) )
=> pp(aa(list(product_prod(A,B)),bool,P,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Zs3),Ws3))) ) ) ) )
=> pp(aa(list(product_prod(A,B)),bool,P,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))) ) ).
% zip_obtain_same_length
tff(fact_2742_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] : ~ pp(member2(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)),Ns),nil(A)),lenlex(A,R3))) ).
% Nil_lenlex_iff2
tff(fact_2743_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ( ! [X2: A] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys) )
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R3)) )
=> ~ ! [Zs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs3) )
=> ~ pp(member2(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)),Zs3),Ys),listrel1(A,R3))) ) ) ) ).
% Cons_listrel1E2
tff(fact_2744_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ( ! [Y2: A] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) )
=> ~ pp(member2(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)) )
=> ~ ! [Zs3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3) )
=> ~ pp(member2(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),Zs3),listrel1(A,R3))) ) ) ) ).
% Cons_listrel1E1
tff(fact_2745_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3))
=> pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R3))) ) ).
% listrel1I1
tff(fact_2746_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),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(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_2747_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Us2: list(A),Vs: list(A),Ys: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3))
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
=> ( ( 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),Vs)) )
=> pp(member2(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,R3))) ) ) ) ).
% listrel1I
tff(fact_2748_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> ~ ! [X2: A,Y2: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2),R3))
=> ! [Us3: list(A),Vs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Vs3)) )
=> ( Ys != 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),Y2),Vs3)) ) ) ) ) ).
% listrel1E
tff(fact_2749_lenlex__length,axiom,
! [A: $tType,Ms: list(A),Ns: list(A),R3: set(product_prod(A,A))] :
( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R3)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).
% lenlex_length
tff(fact_2750_lenlex__append1,axiom,
! [A: $tType,Us2: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
( pp(member2(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)),Us2),Xs),lenlex(A,R)))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> pp(member2(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),Us2),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),lenlex(A,R))) ) ) ).
% lenlex_append1
tff(fact_2751_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs) = 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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),aa(list(A),nat,size_size(list(A)),Xs)),Zs))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Ys),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),aa(list(A),nat,size_size(list(A)),Xs)),Zs))) ).
% zip_append1
tff(fact_2752_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = 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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs)),Ys)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs)),Zs)) ).
% zip_append2
tff(fact_2753_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
<=> ( ( pp(member2(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,R3)))
& ( X = Y ) )
| ( ( Xs = Ys )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_2754_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: fun(A,B),Xa: list(A),Xb: list(B),Y: list(B)] :
( ( map_tailrec_rev(A,B,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),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) )
=> ( ( Y = Xb )
=> ~ accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),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)),Xb))) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( ( Y = map_tailrec_rev(A,B,X,As,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X,A5)),Xb)) )
=> ~ accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),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),As)),Xb))) ) ) ) ) ) ).
% map_tailrec_rev.pelims
tff(fact_2755_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,minus_minus(nat,finite_card(product_prod(A,A),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).
% trancl_set_ntrancl
tff(fact_2756_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
<=> ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
& boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff_def
tff(fact_2757_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
=> boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
tff(fact_2758_lenlex__append2,axiom,
! [A: $tType,R: set(product_prod(A,A)),Us2: list(A),Xs: list(A),Ys: list(A)] :
( irrefl(A,R)
=> ( pp(member2(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),Us2),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys)),lenlex(A,R)))
<=> pp(member2(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),lenlex(A,R))) ) ) ).
% lenlex_append2
tff(fact_2759_minus__Max__eq__Min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),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_2760_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),X: B,A4: set(B)] :
( ( B2 = aa(B,A,F,X) )
=> ( pp(member2(B,X,A4))
=> pp(member2(A,B2,aa(set(B),set(A),image2(B,A,F),A4))) ) ) ).
% image_eqI
tff(fact_2761_image__is__empty,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
( ( aa(set(B),set(A),image2(B,A,F),A4) = bot_bot(set(A)) )
<=> ( A4 = bot_bot(set(B)) ) ) ).
% image_is_empty
tff(fact_2762_empty__is__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F),A4) )
<=> ( A4 = bot_bot(set(B)) ) ) ).
% empty_is_image
tff(fact_2763_image__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(set(B),set(A),image2(B,A,F),bot_bot(set(B))) = bot_bot(set(A)) ).
% image_empty
tff(fact_2764_image__insert,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A3: B,B3: set(B)] : aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F,A3)),aa(set(B),set(A),image2(B,A,F),B3)) ).
% image_insert
tff(fact_2765_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A4: set(A),F: fun(A,B)] :
( pp(member2(A,X,A4))
=> ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F,X)),aa(set(A),set(B),image2(A,B,F),A4)) = aa(set(A),set(B),image2(A,B,F),A4) ) ) ).
% insert_image
tff(fact_2766_op__conc__empty__img__id,axiom,
! [A: $tType,L7: set(list(A))] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L7) = L7 ).
% op_conc_empty_img_id
tff(fact_2767_image__update,axiom,
! [B: $tType,A: $tType,X: A,A4: set(A),F: fun(A,B),N: B] :
( ~ pp(member2(A,X,A4))
=> ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F,X,N)),A4) = aa(set(A),set(B),image2(A,B,F),A4) ) ) ).
% image_update
tff(fact_2768_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_2769_list_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: list(A)] : aa(list(B),set(B),set2(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),V)) = aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),V)) ).
% list.set_map
tff(fact_2770_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( irrefl(A,R3)
=> ( pp(member2(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),Zs)),lexord(A,R3)))
<=> pp(member2(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),Zs),lexord(A,R3))) ) ) ).
% lexord_same_pref_if_irrefl
tff(fact_2771_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,A4: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
<=> ( inj_on(A,B,F,A4)
& ~ pp(member2(B,aa(A,B,F,A3),aa(set(A),set(B),image2(A,B,F),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))))) ) ) ).
% inj_on_insert
tff(fact_2772_minus__Min__eq__Max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),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_2773_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Conj: fun(A,fun(A,A)),Compl: fun(A,A)] :
( ! [X2: A,Y2: A] : aa(A,A,aa(A,fun(A,A),Xor,X2),Y2) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,Y2))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X2)),Y2))
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
tff(fact_2774_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Xor: fun(A,fun(A,A))] :
( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
<=> ! [X4: A,Y3: A] : aa(A,A,aa(A,fun(A,A),Xor,X4),Y3) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,Y3))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X4)),Y3)) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
tff(fact_2775_imageI,axiom,
! [B: $tType,A: $tType,X: A,A4: set(A),F: fun(A,B)] :
( pp(member2(A,X,A4))
=> pp(member2(B,aa(A,B,F,X),aa(set(A),set(B),image2(A,B,F),A4))) ) ).
% imageI
tff(fact_2776_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: fun(B,A),A4: set(B)] :
( pp(member2(A,Z2,aa(set(B),set(A),image2(B,A,F),A4)))
<=> ? [X4: B] :
( pp(member2(B,X4,A4))
& ( Z2 = aa(B,A,F,X4) ) ) ) ).
% image_iff
tff(fact_2777_bex__imageD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),P: fun(A,bool)] :
( ? [X3: A] :
( pp(member2(A,X3,aa(set(B),set(A),image2(B,A,F),A4)))
& pp(aa(A,bool,P,X3)) )
=> ? [X2: B] :
( pp(member2(B,X2,A4))
& pp(aa(A,bool,P,aa(B,A,F,X2))) ) ) ).
% bex_imageD
tff(fact_2778_image__cong,axiom,
! [B: $tType,A: $tType,M: set(A),N4: set(A),F: fun(A,B),G: fun(A,B)] :
( ( M = N4 )
=> ( ! [X2: A] :
( pp(member2(A,X2,N4))
=> ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
=> ( aa(set(A),set(B),image2(A,B,F),M) = aa(set(A),set(B),image2(A,B,G),N4) ) ) ) ).
% image_cong
tff(fact_2779_ball__imageD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),P: fun(A,bool)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(aa(A,bool,P,X2)) )
=> ! [X3: B] :
( pp(member2(B,X3,A4))
=> pp(aa(A,bool,P,aa(B,A,F,X3))) ) ) ).
% ball_imageD
tff(fact_2780_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A4: set(A),B2: B,F: fun(A,B)] :
( pp(member2(A,X,A4))
=> ( ( B2 = aa(A,B,F,X) )
=> pp(member2(B,B2,aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).
% rev_image_eqI
tff(fact_2781_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),B3: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F),A4) = B3 )
=> ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F)),pow(B,A4)) = pow(A,B3) ) ) ).
% image_Pow_surj
tff(fact_2782_lexord__irrefl,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
=> irrefl(list(A),lexord(A,R)) ) ).
% lexord_irrefl
tff(fact_2783_irrefl__lex,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( irrefl(A,R3)
=> irrefl(list(A),lex(A,R3)) ) ).
% irrefl_lex
tff(fact_2784_image__mono,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B3))) ) ).
% image_mono
tff(fact_2785_image__subsetI,axiom,
! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),B3: set(B)] :
( ! [X2: A] :
( pp(member2(A,X2,A4))
=> pp(member2(B,aa(A,B,F,X2),B3)) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),B3)) ) ).
% image_subsetI
tff(fact_2786_subset__imageE,axiom,
! [A: $tType,B: $tType,B3: set(A),F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F),A4)))
=> ~ ! [C8: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),A4))
=> ( B3 != aa(set(B),set(A),image2(B,A,F),C8) ) ) ) ).
% subset_imageE
tff(fact_2787_image__subset__iff,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),A4)),B3))
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> pp(member2(A,aa(B,A,F,X4),B3)) ) ) ).
% image_subset_iff
tff(fact_2788_subset__image__iff,axiom,
! [A: $tType,B: $tType,B3: set(A),F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F),A4)))
<=> ? [AA: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A4))
& ( B3 = aa(set(B),set(A),image2(B,A,F),AA) ) ) ) ).
% subset_image_iff
tff(fact_2789_image__Un,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = 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,F),A4)),aa(set(B),set(A),image2(B,A,F),B3)) ).
% image_Un
tff(fact_2790_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),A4)),B3))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F)),pow(B,A4))),pow(A,B3))) ) ).
% image_Pow_mono
tff(fact_2791_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F: fun(B,A),A4: set(B)] :
( pp(member2(A,Y,aa(set(B),set(A),image2(B,A,F),A4)))
=> ( aa(B,A,F,aa(A,B,inv_on(B,A,F,A4),Y)) = Y ) ) ).
% f_inv_on_f
tff(fact_2792_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F: fun(B,A),A4: set(B)] :
( pp(member2(A,Y,aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(member2(B,aa(A,B,inv_on(B,A,F,A4),Y),A4)) ) ).
% inv_on_f_range
tff(fact_2793_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),F: fun(B,A),X: A] :
( ( I5 != bot_bot(set(B)) )
=> ( ! [I3: B] :
( pp(member2(B,I3,I5))
=> ( aa(B,A,F,I3) = X ) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),I5)) = X ) ) ) ) ).
% SUP_eq_const
tff(fact_2794_translation__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S4: set(A),T4: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S4),T4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S4)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T4)) ) ).
% translation_Int
tff(fact_2795_inf__Sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B3: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),inf_inf(A),A3)),B3)) ) ).
% inf_Sup
tff(fact_2796_image__Int__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),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,F),A4)),aa(set(B),set(A),image2(B,A,F),B3)))) ).
% image_Int_subset
tff(fact_2797_image__diff__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),minus_minus(set(A),aa(set(B),set(A),image2(B,A,F),A4),aa(set(B),set(A),image2(B,A,F),B3))),aa(set(B),set(A),image2(B,A,F),minus_minus(set(B),A4,B3)))) ).
% image_diff_subset
tff(fact_2798_image__set,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ).
% image_set
tff(fact_2799_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B3: fun(B,set(A)),A4: set(B),A3: B] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))))
=> ( pp(member2(B,A3,A4))
=> pp(aa(set(A),bool,finite_finite2(A),aa(B,set(A),B3,A3))) ) ) ).
% finite_UNION_then_finite
tff(fact_2800_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X,Xs) = Xs ) ) ) ).
% insort_insert_key_triv
tff(fact_2801_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),X: A] :
( ( A4 != bot_bot(set(A)) )
=> ( ! [Y2: A] :
( pp(member2(A,Y2,A4))
=> ( aa(A,B,F,Y2) = aa(A,B,F,X) ) )
=> ( the_elem(B,aa(set(A),set(B),image2(A,B,F),A4)) = aa(A,B,F,X) ) ) ) ).
% the_elem_image_unique
tff(fact_2802_SUP__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),C3: A,F: fun(B,A)] :
( ( I5 != bot_bot(set(B)) )
=> ( ! [I3: B] :
( pp(member2(B,I3,I5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(B,A,F,I3))) )
=> ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),I5)) = C3 )
<=> ! [X4: B] :
( pp(member2(B,X4,I5))
=> ( aa(B,A,F,X4) = C3 ) ) ) ) ) ) ).
% SUP_eq_iff
tff(fact_2803_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),M: A] :
( ( A4 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),M)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),M)) ) ) ) ).
% cSUP_least
tff(fact_2804_inj__on__iff__surj,axiom,
! [B: $tType,A: $tType,A4: set(A),A12: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> ( ? [F5: fun(A,B)] :
( inj_on(A,B,F5,A4)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F5),A4)),A12)) )
<=> ? [G3: fun(B,A)] : aa(set(B),set(A),image2(B,A,G3),A12) = A4 ) ) ).
% inj_on_iff_surj
tff(fact_2805_in__image__insert__iff,axiom,
! [A: $tType,B3: set(set(A)),X: A,A4: set(A)] :
( ! [C8: set(A)] :
( pp(member2(set(A),C8,B3))
=> ~ pp(member2(A,X,C8)) )
=> ( pp(member2(set(A),A4,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),B3)))
<=> ( pp(member2(A,X,A4))
& pp(member2(set(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),B3)) ) ) ) ).
% in_image_insert_iff
tff(fact_2806_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C4: set(A),A4: set(A),B3: set(A)] :
( inj_on(A,B,F,C4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C4))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = 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,F),A4)),aa(set(A),set(B),image2(A,B,F),B3)) ) ) ) ) ).
% inj_on_image_Int
tff(fact_2807_lexl__not__refl,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: list(A)] :
( irrefl(A,R3)
=> ~ pp(member2(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),X),lex(A,R3))) ) ).
% lexl_not_refl
tff(fact_2808_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
tff(fact_2809_Pow__insert,axiom,
! [A: $tType,A3: A,A4: set(A)] : pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A4)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3)),pow(A,A4))) ).
% Pow_insert
tff(fact_2810_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list(A),L3: list(A),F: fun(A,B)] :
( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(A,B,F,aa(nat,A,nth(A,L),I3)) = aa(A,B,F,aa(nat,A,nth(A,L3),I3)) ) )
=> ( aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),L3)) ) ) ) ).
% set_image_eq_pointwiseI
tff(fact_2811_in__set__image__conv__nth,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X: B,L: list(B)] :
( pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),L))))
<=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L)))
& ( aa(B,A,F,aa(nat,B,nth(B,L),I2)) = aa(B,A,F,X) ) ) ) ).
% in_set_image_conv_nth
tff(fact_2812_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set(B),F: fun(B,A),Y: A] :
( ( pp(member2(B,X,A4))
=> ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X,Y)),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(B),set(A),image2(B,A,F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) )
& ( ~ pp(member2(B,X,A4))
=> ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X,Y)),A4) = aa(set(B),set(A),image2(B,A,F),A4) ) ) ) ).
% fun_upd_image
tff(fact_2813_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [H: fun(A,A),N4: set(A)] :
( ! [X2: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X2)),aa(A,A,H,Y2))
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> ( ( N4 != bot_bot(set(A)) )
=> ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_2814_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [H: fun(A,A),N4: set(A)] :
( ! [X2: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X2)),aa(A,A,H,Y2))
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> ( ( N4 != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic5882676163264333800up_fin(A,N4)) = lattic5882676163264333800up_fin(A,aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_2815_hom__Max__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H: fun(A,A),N4: set(A)] :
( ! [X2: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X2)),aa(A,A,H,Y2))
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> ( ( N4 != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic643756798349783984er_Max(A,N4)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).
% hom_Max_commute
tff(fact_2816_hom__Min__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H: fun(A,A),N4: set(A)] :
( ! [X2: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X2)),aa(A,A,H,Y2))
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> ( ( N4 != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic643756798350308766er_Min(A,N4)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).
% hom_Min_commute
tff(fact_2817_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),H: fun(A,A),N4: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ! [X2: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),F,X2),Y2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,H,X2)),aa(A,A,H,Y2))
=> ( pp(aa(set(A),bool,finite_finite2(A),N4))
=> ( ( N4 != bot_bot(set(A)) )
=> ( aa(A,A,H,aa(set(A),A,lattic1715443433743089157tice_F(A,F),N4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).
% semilattice_set.hom_commute
tff(fact_2818_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs: list(A),F: fun(A,B)] :
( ( Xs != nil(A) )
=> ( aa(A,B,F,arg_min_list(A,B,F,Xs)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),Xs))) ) ) ) ).
% f_arg_min_list_f
tff(fact_2819_subset__subseqs,axiom,
! [A: $tType,X5: set(A),Xs: list(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(set(A),X5,aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))) ) ).
% subset_subseqs
tff(fact_2820_finite__set__image,axiom,
! [A: $tType,A4: set(list(A))] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A4)))
=> ( ! [Xs2: list(A)] :
( pp(member2(list(A),Xs2,A4))
=> pp(aa(list(A),bool,distinct(A),Xs2)) )
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),A4)) ) ) ).
% finite_set_image
tff(fact_2821_subseqs__powset,axiom,
! [A: $tType,Xs: list(A)] : aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))) = pow(A,aa(list(A),set(A),set2(A),Xs)) ).
% subseqs_powset
tff(fact_2822_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
<=> ( inj_on(A,B,F,A4)
& inj_on(A,B,F,B3)
& ( 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,F),minus_minus(set(A),A4,B3))),aa(set(A),set(B),image2(A,B,F),minus_minus(set(A),B3,A4))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_2823_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A4: fun(B,set(A)),I: B,B3: set(A),J4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A4,I,B3)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),minus_minus(set(B),J4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),bot_bot(set(B))))))),if(set(A),member2(B,I,J4),B3,bot_bot(set(A)))) ).
% UNION_fun_upd
tff(fact_2824_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(list(A))] :
( inj_on(A,B,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A4)))
=> inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),A4) ) ).
% inj_on_mapI
tff(fact_2825_Union__image__empty,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),bot_bot(set(B))))) = A4 ).
% Union_image_empty
tff(fact_2826_bijective__Empty,axiom,
! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).
% bijective_Empty
tff(fact_2827_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_2828_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R3)) ).
% Field_insert
tff(fact_2829_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
=> ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M2)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
=> ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).
% upt_rec_numeral
tff(fact_2830_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M2) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),aa(nat,A,nth(A,Xs),M2)) ) ) ).
% nth_enumerate_eq
tff(fact_2831_finite__Field__eq__finite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
<=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R)) ) ).
% finite_Field_eq_finite
tff(fact_2832_tl__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),tl(nat),upt(M2,N)) = upt(aa(nat,nat,suc,M2),N) ).
% tl_upt
tff(fact_2833_hd__upt,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( aa(list(nat),nat,hd(nat),upt(I,J)) = I ) ) ).
% hd_upt
tff(fact_2834_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] : aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),drop(nat),M2),upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M2),J) ).
% drop_upt
tff(fact_2835_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] : enumerate(A,N,nil(A)) = nil(product_prod(nat,A)) ).
% enumerate_simps(1)
tff(fact_2836_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_enumerate
tff(fact_2837_Field__empty,axiom,
! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).
% Field_empty
tff(fact_2838_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M2)),N))
=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),take(nat),M2),upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M2)) ) ) ).
% take_upt
tff(fact_2839_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( upt(zero_zero(nat),J) = nil(nat) )
<=> ( J = zero_zero(nat) ) ) ).
% upt_0_eq_Nil_conv
tff(fact_2840_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( upt(I,J) = nil(nat) ) ) ).
% upt_conv_Nil
tff(fact_2841_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K)) )
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,K)) = upt(I,K) ) ) ).
% upt_merge
tff(fact_2842_length__upt,axiom,
! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = minus_minus(nat,J,I) ).
% length_upt
tff(fact_2843_last__upt,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( last(nat,upt(I,J)) = minus_minus(nat,J,one_one(nat)) ) ) ).
% last_upt
tff(fact_2844_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( upt(I,J) = nil(nat) )
<=> ( ( J = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I)) ) ) ).
% upt_eq_Nil_conv
tff(fact_2845_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).
% enumerate_simps(2)
tff(fact_2846_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J))
=> ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).
% nth_upt
tff(fact_2847_map__Suc__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),upt(M2,N)) = upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N)) ).
% map_Suc_upt
tff(fact_2848_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : pp(aa(list(product_prod(nat,A)),bool,distinct(product_prod(nat,A)),enumerate(A,N,Xs))) ).
% distinct_enumerate
tff(fact_2849_distinct__upt,axiom,
! [I: nat,J: nat] : pp(aa(list(nat),bool,distinct(nat),upt(I,J))) ).
% distinct_upt
tff(fact_2850_enumerate__eq__zip,axiom,
! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = aa(list(A),list(product_prod(nat,A)),aa(list(nat),fun(list(A),list(product_prod(nat,A))),zip(nat,A),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))),Xs) ).
% enumerate_eq_zip
tff(fact_2851_total__on__empty,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R3) ).
% total_on_empty
tff(fact_2852_butlast__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),butlast(nat),upt(M2,N)) = upt(M2,minus_minus(nat,N,one_one(nat))) ).
% butlast_upt
tff(fact_2853_atLeast__upt,axiom,
! [N: nat] : set_ord_lessThan(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).
% atLeast_upt
tff(fact_2854_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list(nat),Q2: nat] :
( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M2,Q2) )
<=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M2),Q2) ) ) ).
% upt_conv_Cons_Cons
tff(fact_2855_upt__0,axiom,
! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).
% upt_0
tff(fact_2856_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M2,N)) ).
% sorted_wrt_upt
tff(fact_2857_sorted__upt,axiom,
! [M2: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M2,N)) ).
% sorted_upt
tff(fact_2858_greaterThanAtMost__upt,axiom,
! [N: nat,M2: nat] : set_or3652927894154168847AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M2))) ).
% greaterThanAtMost_upt
tff(fact_2859_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( upt(I,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys) )
<=> ? [K4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),J))
& ( upt(I,K4) = Xs )
& ( upt(K4,J) = Ys ) ) ) ) ).
% upt_eq_append_conv
tff(fact_2860_greaterThanLessThan__upt,axiom,
! [N: nat,M2: nat] : set_or5935395276787703475ssThan(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M2)) ).
% greaterThanLessThan_upt
tff(fact_2861_atMost__upto,axiom,
! [N: nat] : set_ord_atMost(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).
% atMost_upto
tff(fact_2862_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).
% upt_conv_Cons
tff(fact_2863_upt__append,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).
% upt_append
tff(fact_2864_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).
% upt_add_eq_append
tff(fact_2865_bijective__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( bijective(A,B,R)
<=> ( ! [X4: A,Y3: B,Z3: B] :
( ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R))
& pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z3),R)) )
=> ( Y3 = Z3 ) )
& ! [X4: A,Y3: A,Z3: B] :
( ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z3),R))
& pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z3),R)) )
=> ( X4 = Y3 ) ) ) ) ).
% bijective_def
tff(fact_2866_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),minus_minus(nat,N,M2)))
=> ( aa(nat,A,nth(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F),upt(M2,N))),I) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I)) ) ) ).
% nth_map_upt
tff(fact_2867_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list(nat)] :
( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
& ( I = X )
& ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_2868_upt__rec,axiom,
! [I: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = nil(nat) ) ) ) ).
% upt_rec
tff(fact_2869_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).
% enumerate_append_eq
tff(fact_2870_map__upt__eqI,axiom,
! [A: $tType,Xs: list(A),N: nat,M2: nat,F: fun(nat,A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = minus_minus(nat,N,M2) )
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I3)) ) )
=> ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F),upt(M2,N)) = Xs ) ) ) ).
% map_upt_eqI
tff(fact_2871_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N4: nat,L: list(A),M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),nth(A,L)),upt(M,N4)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N4),L)) ) ) ).
% map_nth_upt_drop_take_conv
tff(fact_2872_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
( ( upt(L,H) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),Is2)) )
<=> ( ( Is1 = upt(L,I) )
& ( Is2 = upt(aa(nat,nat,suc,I),H) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),I))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),H)) ) ) ).
% upt_eq_lel_conv
tff(fact_2873_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).
% upt_Suc
tff(fact_2874_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).
% upt_Suc_append
tff(fact_2875_nth__image,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(set(nat),set(A),image2(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),L),Xs)) ) ) ).
% nth_image
tff(fact_2876_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) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L12),L23) ) ) ).
% zipf_zip
tff(fact_2877_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = zero_zero(A) )
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))))
& pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ) ).
% Gcd_fin_0_iff
tff(fact_2878_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),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_2879_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),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_2880_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_2881_atLeastLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastLessThan_empty
tff(fact_2882_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% atLeastLessThan_empty_iff
tff(fact_2883_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% atLeastLessThan_empty_iff2
tff(fact_2884_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_2885_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_2886_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_2887_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_2888_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( pp(dvd_dvd(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A4),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_2889_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,M2,N)) = upt(M2,N) ).
% sorted_list_of_set_range
tff(fact_2890_ceiling__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).
% ceiling_le_one
tff(fact_2891_one__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).
% one_less_ceiling
tff(fact_2892_Int__atLeastLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastLessThan
tff(fact_2893_ceiling__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).
% ceiling_less_zero
tff(fact_2894_zero__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).
% zero_le_ceiling
tff(fact_2895_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_2896_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),minus_minus(A,aa(num,A,numeral_numeral(A),V),one_one(A)))) ) ) ).
% ceiling_less_numeral
tff(fact_2897_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(num,A,numeral_numeral(A),V),one_one(A))),X)) ) ) ).
% numeral_le_ceiling
tff(fact_2898_atLeastLessThan__upt,axiom,
! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).
% atLeastLessThan_upt
tff(fact_2899_ex__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
& pp(aa(nat,bool,P,M3)) )
<=> ? [X4: nat] :
( pp(member2(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
& pp(aa(nat,bool,P,X4)) ) ) ).
% ex_nat_less_eq
tff(fact_2900_all__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
=> pp(aa(nat,bool,P,M3)) )
<=> ! [X4: nat] :
( pp(member2(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,P,X4)) ) ) ).
% all_nat_less_eq
tff(fact_2901_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(3)
tff(fact_2902_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(B,C)),A3: A,As2: list(A),B2: B,Bs: list(B)] : zipf(A,B,C,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),F,A3),B2)),zipf(A,B,C,F,As2,Bs)) ).
% zipf.simps(2)
tff(fact_2903_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(B,C))] : zipf(A,B,C,F,nil(A),nil(B)) = nil(C) ).
% zipf.simps(1)
tff(fact_2904_mult__ceiling__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).
% mult_ceiling_le
tff(fact_2905_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)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(2)
tff(fact_2906_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(1)
tff(fact_2907_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_2908_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).
% floor_le_neg_numeral
tff(fact_2909_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X)) ) ) ).
% neg_numeral_less_floor
tff(fact_2910_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).
% floor_le_numeral
tff(fact_2911_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X)) ) ) ).
% numeral_less_floor
tff(fact_2912_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q2: A,P4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
=> pp(aa(A,bool,aa(A,fun(A,bool),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,P4,Q2))),one_one(A))),Q2)),P4)) ) ) ).
% ceiling_divide_lower
tff(fact_2913_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_2914_zero__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).
% zero_less_floor
tff(fact_2915_floor__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).
% floor_le_zero
tff(fact_2916_one__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).
% one_le_floor
tff(fact_2917_floor__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).
% floor_less_one
tff(fact_2918_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_2919_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T4: A] :
( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T4)))
<=> ! [I2: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),T4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T4),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A)))) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% floor_split
tff(fact_2920_floor__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archim6421214686448440834_floor(A,X) = A3 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).
% floor_eq_iff
tff(fact_2921_floor__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))))
=> ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).
% floor_unique
tff(fact_2922_less__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)) ) ) ).
% less_floor_iff
tff(fact_2923_floor__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).
% floor_le_iff
tff(fact_2924_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_2925_floor__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q2: A,P4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P4,Q2)))),Q2)),P4)) ) ) ).
% floor_divide_lower
tff(fact_2926_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q2: A,P4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P4),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,P4,Q2)))),one_one(A))),Q2))) ) ) ).
% floor_divide_upper
tff(fact_2927_le__mult__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).
% le_mult_floor
tff(fact_2928_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T4: A] :
( pp(aa(int,bool,P,archimedean_ceiling(A,T4)))
<=> ! [I2: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),I2),one_one(A))),T4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T4),aa(int,A,ring_1_of_int(A),I2))) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% ceiling_split
tff(fact_2929_ceiling__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archimedean_ceiling(A,X) = A3 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),A3),one_one(A))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).
% ceiling_eq_iff
tff(fact_2930_ceiling__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),Z2),one_one(A))),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2)))
=> ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).
% ceiling_unique
tff(fact_2931_ceiling__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)),one_one(A))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).
% ceiling_correct
tff(fact_2932_ceiling__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),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_2933_le__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),Z2),one_one(A))),X)) ) ) ).
% le_ceiling_iff
tff(fact_2934_ceiling__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q2: A,P4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P4),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,P4,Q2)))),Q2))) ) ) ).
% ceiling_divide_upper
tff(fact_2935_of__int__1__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ) ).
% of_int_1_less_iff
tff(fact_2936_of__int__less__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),one_one(int))) ) ) ).
% of_int_less_1_iff
tff(fact_2937_of__int__1__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2)) ) ) ).
% of_int_1_le_iff
tff(fact_2938_of__int__le__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),one_one(int))) ) ) ).
% of_int_le_1_iff
tff(fact_2939_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_2940_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_2941_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_2942_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_2943_round__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),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)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))))
=> ( archimedean_round(A,X) = Y ) ) ) ) ).
% round_unique
tff(fact_2944_of__int__round__gt,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2945_of__int__round__ge,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2946_of__int__round__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% of_int_round_le
tff(fact_2947_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_2948_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_2949_round__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
=> ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
=> ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).
% round_altdef
tff(fact_2950_upto__aux__rec,axiom,
! [J: int,I: int,Js: list(int)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto_aux(I,J,Js) = Js ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto_aux(I,J,Js) = upto_aux(I,minus_minus(int,J,one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js)) ) ) ) ).
% upto_aux_rec
tff(fact_2951_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),minus_minus(A,X,aa(int,A,ring_1_of_int(A),N)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
=> ( archimedean_round(A,X) = N ) ) ) ).
% round_unique'
tff(fact_2952_of__int__round__abs__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),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_2953_floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).
% floor_add
tff(fact_2954_abs__idempotent,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_idempotent
tff(fact_2955_abs__0__eq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_0_eq
tff(fact_2956_abs__eq__0,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_eq_0
tff(fact_2957_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_2958_abs__add__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_add_abs
tff(fact_2959_abs__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% abs_mult_self_eq
tff(fact_2960_abs__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_1
tff(fact_2961_abs__minus__cancel,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_minus_cancel
tff(fact_2962_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_le_zero_iff
tff(fact_2963_abs__le__self__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% abs_le_self_iff
tff(fact_2964_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_nonneg
tff(fact_2965_zero__less__abs__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% zero_less_abs_iff
tff(fact_2966_abs__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).
% abs_neg_one
tff(fact_2967_abs__of__nonpos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_nonpos
tff(fact_2968_abs__minus__commute,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),minus_minus(A,A3,B2)) = aa(A,A,abs_abs(A),minus_minus(A,B2,A3)) ) ).
% abs_minus_commute
tff(fact_2969_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_2970_abs__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_mult
tff(fact_2971_abs__le__D1,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% abs_le_D1
tff(fact_2972_abs__ge__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_self
tff(fact_2973_abs__ge__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_zero
tff(fact_2974_abs__of__pos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_pos
tff(fact_2975_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).
% abs_not_less_zero
tff(fact_2976_abs__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).
% abs_triangle_ineq
tff(fact_2977_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,aa(A,A,abs_abs(A),A3),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),minus_minus(A,B2,A3)))) ) ).
% abs_triangle_ineq2_sym
tff(fact_2978_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,aa(A,A,abs_abs(A),A3),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),minus_minus(A,A3,B2)))) ) ).
% abs_triangle_ineq3
tff(fact_2979_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,aa(A,A,abs_abs(A),A3),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),minus_minus(A,A3,B2)))) ) ).
% abs_triangle_ineq2
tff(fact_2980_abs__mult__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C3: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))) ) ) ) ).
% abs_mult_less
tff(fact_2981_abs__ge__minus__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_minus_self
tff(fact_2982_abs__le__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).
% abs_le_iff
tff(fact_2983_abs__le__D2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).
% abs_le_D2
tff(fact_2984_abs__leI,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).
% abs_leI
tff(fact_2985_frac__lt__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).
% frac_lt_1
tff(fact_2986_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_2987_dense__eq0__I,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs(A)
& dense_linorder(A) )
=> ! [X: A] :
( ! [E: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E)) )
=> ( X = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_2988_abs__mult__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).
% abs_mult_pos
tff(fact_2989_abs__eq__mult,axiom,
! [A: $tType] :
( ordered_ring_abs(A)
=> ! [A3: A,B2: A] :
( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
=> ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).
% abs_eq_mult
tff(fact_2990_abs__minus__le__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A))) ) ).
% abs_minus_le_zero
tff(fact_2991_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A,C3: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),minus_minus(A,A3,C3))),aa(A,A,abs_abs(A),minus_minus(A,B2,D3))))) ) ).
% abs_diff_triangle_ineq
tff(fact_2992_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,A3,B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).
% abs_triangle_ineq4
tff(fact_2993_abs__of__neg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_neg
tff(fact_2994_cSup__abs__le,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3)) ) ) ) ).
% cSup_abs_le
tff(fact_2995_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).
% abs_add_one_gt_zero
tff(fact_2996_cSup__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,X2,L))),E3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S),L))),E3)) ) ) ) ).
% cSup_asclose
tff(fact_2997_of__int__leD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).
% of_int_leD
tff(fact_2998_of__int__lessD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).
% of_int_lessD
tff(fact_2999_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) )
<=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_3000_frac__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( archimedean_frac(A,X) = X )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).
% frac_eq
tff(fact_3001_frac__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = 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_3002_abs__square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).
% abs_square_le_1
tff(fact_3003_abs__square__less__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).
% abs_square_less_1
tff(fact_3004_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).
% upto_rec_numeral(2)
tff(fact_3005_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).
% upto_rec_numeral(3)
tff(fact_3006_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).
% upto_rec_numeral(4)
tff(fact_3007_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).
% upto_rec_numeral(1)
tff(fact_3008_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( archim2362893244070406136eiling(B)
& linordered_idom(A) )
=> ! [A3: B,B2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
=> ( pp(member2(B,A3,ring_1_Ints(B)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).
% mult_ceiling_le_Ints
tff(fact_3009_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J))
=> ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).
% nth_upto
tff(fact_3010_upto__Nil,axiom,
! [I: int,J: int] :
( ( upto(I,J) = nil(int) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).
% upto_Nil
tff(fact_3011_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil(int) = upto(I,J) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).
% upto_Nil2
tff(fact_3012_upto__empty,axiom,
! [J: int,I: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto(I,J) = nil(int) ) ) ).
% upto_empty
tff(fact_3013_upto__single,axiom,
! [I: int] : upto(I,I) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),nil(int)) ).
% upto_single
tff(fact_3014_distinct__upto,axiom,
! [I: int,J: int] : pp(aa(list(int),bool,distinct(int),upto(I,J))) ).
% distinct_upto
tff(fact_3015_sorted__upto,axiom,
! [M2: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M2,N)) ).
% sorted_upto
tff(fact_3016_sorted__wrt__upto,axiom,
! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).
% sorted_wrt_upto
tff(fact_3017_Ints__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,B2: A] :
( pp(member2(A,A3,ring_1_Ints(A)))
=> ( pp(member2(A,B2,ring_1_Ints(A)))
=> pp(member2(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),ring_1_Ints(A))) ) ) ) ).
% Ints_mult
tff(fact_3018_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(member2(A,one_one(A),ring_1_Ints(A))) ) ).
% Ints_1
tff(fact_3019_upto__code,axiom,
! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).
% upto_code
tff(fact_3020_upto__aux__def,axiom,
! [I: int,J: int,Js: list(int)] : upto_aux(I,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),Js) ).
% upto_aux_def
tff(fact_3021_greaterThanAtMost__upto,axiom,
! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).
% greaterThanAtMost_upto
tff(fact_3022_atLeastLessThan__upto,axiom,
! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,minus_minus(int,J,one_one(int)))) ).
% atLeastLessThan_upto
tff(fact_3023_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [A3: A] :
( pp(member2(A,A3,ring_1_Ints(A)))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).
% Ints_odd_nonzero
tff(fact_3024_upto__split2,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).
% upto_split2
tff(fact_3025_upto__split1,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,minus_minus(int,J,one_one(int)))),upto(J,K)) ) ) ) ).
% upto_split1
tff(fact_3026_greaterThanLessThan__upto,axiom,
! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),minus_minus(int,J,one_one(int)))) ).
% greaterThanLessThan_upto
tff(fact_3027_Ints__odd__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(member2(A,A3,ring_1_Ints(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% Ints_odd_less_0
tff(fact_3028_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(member2(A,X,ring_1_Ints(A)))
=> ( ( X != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_3029_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(member2(A,X,ring_1_Ints(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
=> ( X = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_3030_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(member2(A,X,ring_1_Ints(A)))
=> ( pp(member2(A,Y,ring_1_Ints(A)))
=> ( ( X = Y )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),minus_minus(A,X,Y))),one_one(A))) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_3031_upto_Osimps,axiom,
! [I: int,J: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = nil(int) ) ) ) ).
% upto.simps
tff(fact_3032_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list(int)] :
( ( upto(X,Xa) = Y )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
=> ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
=> ( Y = nil(int) ) ) ) ) ).
% upto.elims
tff(fact_3033_upto__rec1,axiom,
! [I: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).
% upto_rec1
tff(fact_3034_upto__rec2,axiom,
! [I: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,minus_minus(int,J,one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).
% upto_rec2
tff(fact_3035_frac__neg,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( pp(member2(A,X,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
& ( ~ pp(member2(A,X,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = minus_minus(A,one_one(A),archimedean_frac(A,X)) ) ) ) ) ).
% frac_neg
tff(fact_3036_upto__split3,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,minus_minus(int,J,one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).
% upto_split3
tff(fact_3037_frac__unique__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: A] :
( ( archimedean_frac(A,X) = A3 )
<=> ( pp(member2(A,minus_minus(A,X,A3),ring_1_Ints(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).
% frac_unique_iff
tff(fact_3038_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( archim2362893244070406136eiling(B)
& linordered_idom(A) )
=> ! [A3: B,B2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
=> ( pp(member2(B,A3,ring_1_Ints(B)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).
% le_mult_floor_Ints
tff(fact_3039_fact__split,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,minus_minus(nat,N,K),N)))),semiring_char_0_fact(A,minus_minus(nat,N,K))) ) ) ) ).
% fact_split
tff(fact_3040_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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,R3)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N3)),aa(nat,B,nth(B,Ys),N3)),R3)) ) ) ) ).
% listrel_iff_nth
tff(fact_3041_fact__binomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( 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),binomial(N,K))) = divide_divide(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,minus_minus(nat,N,K))) ) ) ) ).
% fact_binomial
tff(fact_3042_binomial__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,A,semiring_1_of_nat(A),binomial(N,K)) = divide_divide(A,semiring_char_0_fact(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,minus_minus(nat,N,K)))) ) ) ) ).
% binomial_fact
tff(fact_3043_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
( pp(member2(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)),F),Fs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
| ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
& pp(member2(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_3044_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] : ~ pp(member2(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_3045_prod_OatMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = 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_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% prod.atMost_Suc
tff(fact_3046_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = 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_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).
% prod.lessThan_Suc
tff(fact_3047_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M2: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = 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,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).
% prod.op_ivl_Suc
tff(fact_3048_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list(A),R3: set(product_prod(A,B))] :
( pp(member2(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),nil(B)),listrel(A,B,R3)))
=> ( Xs = nil(A) ) ) ).
% listrel_Nil2
tff(fact_3049_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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)),nil(A)),Xs),listrel(A,B,R3)))
=> ( Xs = nil(B) ) ) ).
% listrel_Nil1
tff(fact_3050_listrel_ONil,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : pp(member2(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)),nil(A)),nil(B)),listrel(A,B,R3))) ).
% listrel.Nil
tff(fact_3051_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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,R3)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).
% listrel_eq_len
tff(fact_3052_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,P4: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P4))
=> ( 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,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,P4)) ) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_3053_listrel__mono,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S4: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S4))
=> pp(aa(set(product_prod(list(A),list(B))),bool,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),bool),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R3)),listrel(A,B,S4))) ) ).
% listrel_mono
tff(fact_3054_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R3: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R3))
=> ( pp(member2(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,R3)))
=> pp(member2(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,R3))) ) ) ).
% listrel.Cons
tff(fact_3055_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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,R3)))
=> ~ ! [Y2: B,Ys2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y2),R3))
=> ~ pp(member2(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,R3))) ) ) ) ).
% listrel_Cons1
tff(fact_3056_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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,R3)))
=> ~ ! [X2: A,Xs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),R3))
=> ~ pp(member2(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys),listrel(A,B,R3))) ) ) ) ).
% listrel_Cons2
tff(fact_3057_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,N)) ) ).
% prod.atLeast0_lessThan_Suc
tff(fact_3058_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_3059_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: nat,B2: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_3060_measures__less,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
=> pp(member2(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)),F),Fs)))) ) ).
% measures_less
tff(fact_3061_measures__lesseq,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,Fs)))
=> pp(member2(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)),F),Fs)))) ) ) ).
% measures_lesseq
tff(fact_3062_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
( pp(member2(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R3)))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2),R3))
=> ~ pp(member2(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys2),listrel(A,B,R3))) ) ) ) ) ) ).
% listrel.cases
tff(fact_3063_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
( pp(member2(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R3)))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs3: list(A),Ys3: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
& pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R3))
& pp(member2(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),Ys3),listrel(A,B,R3))) ) ) ) ).
% listrel.simps
tff(fact_3064_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ~ pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)) ) ) ) ) ).
% prod.insert
tff(fact_3065_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( ~ pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).
% prod.infinite
tff(fact_3066_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_3067_prod__Un,axiom,
! [A: $tType,B: $tType] :
( field(A)
=> ! [A4: set(B),B3: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)))
=> ( aa(B,A,F,X2) != zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = divide_divide(A,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),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ) ).
% prod_Un
tff(fact_3068_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( semidom_divide(A)
=> ! [A4: set(B),F: fun(B,A),A3: B] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( aa(B,A,F,A3) != zero_zero(A) )
=> ( ( pp(member2(B,A3,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4),aa(B,A,F,A3)) ) )
& ( ~ pp(member2(B,A3,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4) ) ) ) ) ) ) ).
% prod_diff1
tff(fact_3069_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A4: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) != one_one(A) )
=> ~ ! [A5: B] :
( pp(member2(B,A5,A4))
=> ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_3070_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( ! [X2: B] :
( pp(member2(B,X2,A4))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).
% prod.neutral
tff(fact_3071_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4))) ) ) ).
% prod_ge_1
tff(fact_3072_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member2(B,X2,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),one_one(A))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),one_one(A))) ) ) ).
% prod_le_1
tff(fact_3073_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,one_one(A)),one_one(A)))
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X23))
& pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23))) )
=> ( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S))) ) ) ) ) ) ).
% prod.related
tff(fact_3074_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) ) )
& ( ~ pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)) ) ) ) ) ) ).
% prod.insert_if
tff(fact_3075_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),T5: set(C),S: set(B),I: fun(C,B),J: fun(B,C),T: set(C),G: fun(B,A),H: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( pp(aa(set(C),bool,finite_finite2(C),T5))
=> ( ! [A5: B] :
( pp(member2(B,A5,minus_minus(set(B),S,S3)))
=> ( aa(C,B,I,aa(B,C,J,A5)) = A5 ) )
=> ( ! [A5: B] :
( pp(member2(B,A5,minus_minus(set(B),S,S3)))
=> pp(member2(C,aa(B,C,J,A5),minus_minus(set(C),T,T5))) )
=> ( ! [B4: C] :
( pp(member2(C,B4,minus_minus(set(C),T,T5)))
=> ( aa(B,C,J,aa(C,B,I,B4)) = B4 ) )
=> ( ! [B4: C] :
( pp(member2(C,B4,minus_minus(set(C),T,T5)))
=> pp(member2(B,aa(C,B,I,B4),minus_minus(set(B),S,S3))) )
=> ( ! [A5: B] :
( pp(member2(B,A5,S3))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B4: C] :
( pp(member2(C,B4,T5))
=> ( aa(C,A,H,B4) = one_one(A) ) )
=> ( ! [A5: B] :
( pp(member2(B,A5,S))
=> ( aa(C,A,H,aa(B,C,J,A5)) = aa(B,A,G,A5) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_3076_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I5: set(A),I: A,F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),I5))
=> ( pp(member2(A,I,I5))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I)))
=> ( ! [I3: A] :
( pp(member2(A,I3,I5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,I3))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I5))) ) ) ) ) ) ).
% less_1_prod2
tff(fact_3077_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I5: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),I5))
=> ( ( I5 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I3))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I5))) ) ) ) ) ).
% less_1_prod
tff(fact_3078_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B3: set(B),A4: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = 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),minus_minus(set(B),A4,B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ).
% prod.subset_diff
tff(fact_3079_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_3080_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,H,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_3081_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S) ) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_3082_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T) ) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_3083_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C4: set(B),A4: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C4))
=> ( ! [A5: B] :
( pp(member2(B,A5,minus_minus(set(B),C4,A4)))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B4: B] :
( pp(member2(B,B4,minus_minus(set(B),C4,B3)))
=> ( aa(B,A,H,B4) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C4) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_3084_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C4: set(B),A4: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C4))
=> ( ! [A5: B] :
( pp(member2(B,A5,minus_minus(set(B),C4,A4)))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B4: B] :
( pp(member2(B,B4,minus_minus(set(B),C4,B3)))
=> ( aa(B,A,H,B4) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) )
<=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C4) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_3085_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) = 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),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ).
% prod.union_inter
tff(fact_3086_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),minus_minus(set(B),A4,B3))) ) ) ) ).
% prod.Int_Diff
tff(fact_3087_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H,I3) = one_one(A) ) )
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),S,T)))
=> ( aa(B,A,G,I3) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T)))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_3088_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(A)
=> ! [A4: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ! [I3: B] :
( pp(member2(B,I3,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I3)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I3)),aa(B,A,G,I3))) ) )
=> ( ( A4 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4))) ) ) ) ) ).
% prod_mono_strict
tff(fact_3089_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),X: B] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ).
% prod.insert_remove
tff(fact_3090_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% prod.remove
tff(fact_3091_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = 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),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ) ).
% prod.union_inter_neutral
tff(fact_3092_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = 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),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ) ).
% prod.union_disjoint
tff(fact_3093_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),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),minus_minus(set(B),A4,B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),minus_minus(set(B),B3,A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% prod.union_diff2
tff(fact_3094_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [B3: set(A),A4: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ! [B4: A] :
( pp(member2(A,B4,minus_minus(set(A),B3,A4)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,B4))) )
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A5))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B3))) ) ) ) ) ) ).
% prod_mono2
tff(fact_3095_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(A)
=> ! [A4: set(B),F: fun(B,A),N: A,K: nat] :
( ! [I3: B] :
( pp(member2(B,I3,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I3)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I3)),N)) ) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(B,A4)),K))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).
% prod_le_power
tff(fact_3096_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),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)) = 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_3097_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Xs: list(B),G: fun(B,A)] :
( pp(aa(list(B),bool,distinct(B),Xs))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups5270119922927024881d_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) ) ) ) ).
% prod.distinct_set_conv_list
tff(fact_3098_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M2: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = 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,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_3099_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_3100_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( ~ pp(dvd_dvd(A,A3,one_one(A)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,B2)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_3101_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( pp(member2(list(A),nil(A),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
% Nil_in_shuffles
tff(fact_3102_finite__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys))) ).
% finite_shuffles
tff(fact_3103_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% atLeastatMost_empty_iff2
tff(fact_3104_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% atLeastatMost_empty_iff
tff(fact_3105_atLeastatMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastatMost_empty
tff(fact_3106_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A,C3: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),bot_bot(set(A))) )
<=> ( ( A3 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_3107_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ).
% atLeastAtMost_singleton
tff(fact_3108_euclidean__size__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).
% euclidean_size_1
tff(fact_3109_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_3110_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_3111_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_3112_Int__atLeastAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastAtMost
tff(fact_3113_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,C3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastAtMostR1
tff(fact_3114_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_ord_atMost(A,D3)) = set_or1337092689740270186AtMost(A,A3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastAtMostL1
tff(fact_3115_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).
% image_mult_atLeastAtMost
tff(fact_3116_shuffles__commutes,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Ys),Xs) ).
% shuffles_commutes
tff(fact_3117_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z2: A] :
( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Ys)))) ) ).
% Cons_in_shuffles_rightI
tff(fact_3118_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z2: A] :
( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Xs)),Ys))) ) ).
% Cons_in_shuffles_leftI
tff(fact_3119_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(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_3120_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(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_3121_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = nil(A) )
=> ( ( Ys = nil(A) )
=> pp(member2(list(A),nil(A),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys))) ) ) ).
% Nil_in_shufflesI
tff(fact_3122_refl__on__Int,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),B3: set(A),S4: set(product_prod(A,A))] :
( refl_on(A,A4,R3)
=> ( refl_on(A,B3,S4)
=> refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3),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),S4)) ) ) ).
% refl_on_Int
tff(fact_3123_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).
% atLeastAtMost_singleton'
tff(fact_3124_all__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
=> pp(aa(nat,bool,P,M3)) )
<=> ! [X4: nat] :
( pp(member2(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,P,X4)) ) ) ).
% all_nat_less
tff(fact_3125_ex__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
& pp(aa(nat,bool,P,M3)) )
<=> ? [X4: nat] :
( pp(member2(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
& pp(aa(nat,bool,P,X4)) ) ) ).
% ex_nat_less
tff(fact_3126_shufflesE,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( ( ( Zs = Xs )
=> ( Ys != nil(A) ) )
=> ( ( ( Zs = Ys )
=> ( Xs != nil(A) ) )
=> ( ! [X2: A,Xs4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs4) )
=> ! [Z4: A,Zs4: list(A)] :
( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Zs4) )
=> ( ( X2 = Z4 )
=> ~ pp(member2(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs4),Ys))) ) ) )
=> ~ ! [Y2: A,Ys4: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys4) )
=> ! [Z4: A,Zs4: list(A)] :
( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Zs4) )
=> ( ( Y2 = Z4 )
=> ~ pp(member2(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys4))) ) ) ) ) ) ) ) ).
% shufflesE
tff(fact_3127_length__shuffles,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).
% length_shuffles
tff(fact_3128_set__shuffles,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ) ) ).
% set_shuffles
tff(fact_3129_euclidean__size__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) ) ) ) ).
% euclidean_size_unit
tff(fact_3130_euclidean__size__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A] : euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)) ) ).
% euclidean_size_mult
tff(fact_3131_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_3132_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] : aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))) ).
% shuffles.simps(3)
tff(fact_3133_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys))),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))) ).
% Cons_shuffles_subset1
tff(fact_3134_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys))),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ).
% Cons_shuffles_subset2
tff(fact_3135_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(7)
tff(fact_3136_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)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(4)
tff(fact_3137_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(8)
tff(fact_3138_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(5)
tff(fact_3139_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(4)
tff(fact_3140_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
<=> ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) )
& ( A3 != zero_zero(A) ) ) ) ) ).
% unit_iff_euclidean_size
tff(fact_3141_size__mult__mono_H,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).
% size_mult_mono'
tff(fact_3142_size__mult__mono,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).
% size_mult_mono
tff(fact_3143_euclidean__size__times__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = euclid6346220572633701492n_size(A,B2) ) ) ) ).
% euclidean_size_times_unit
tff(fact_3144_atLeastAtMost__upt,axiom,
! [N: nat,M2: nat] : set_or1337092689740270186AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M2))) ).
% atLeastAtMost_upt
tff(fact_3145_shuffles_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),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)))) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [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),X2)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs2),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)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys2))) ) ) ) ) ) ) ).
% shuffles.elims
tff(fact_3146_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3147_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_3148_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).
% atLeastAtMost_diff_ends
tff(fact_3149_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% prod.atLeast0_atMost_Suc
tff(fact_3150_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_3151_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_3152_prod_Olast__plus,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))) ) ) ) ).
% prod.last_plus
tff(fact_3153_prod_Ohead,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M2,N))) ) ) ) ).
% prod.head
tff(fact_3154_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(list(A),bool,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)) )
=> ( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> pp(aa(list(A),bool,distinct(A),Zs)) ) ) ) ) ).
% distinct_disjoint_shuffles
tff(fact_3155_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z2: A,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
<=> ( ( ( Xs != nil(A) )
& ( aa(list(A),A,hd(A),Xs) = Z2 )
& pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),tl(A),Xs)),Ys))) )
| ( ( Ys != nil(A) )
& ( aa(list(A),A,hd(A),Ys) = Z2 )
& pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),tl(A),Ys)))) ) ) ) ).
% Cons_in_shuffles_iff
tff(fact_3156_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(6)
tff(fact_3157_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),times_times(A)),Xs),one_one(A)) ) ).
% prod_list.eq_foldr
tff(fact_3158_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_3159_prod_Ohead__if,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M2: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N)) = 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,M2,N))),aa(nat,A,G,N)) ) ) ) ) ).
% prod.head_if
tff(fact_3160_prod_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A),P4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P4))) = 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,M2,N))),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),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P4)))) ) ) ) ).
% prod.ub_add_nat
tff(fact_3161_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),X),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C3),X)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_3162_divmod__cases,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,A3: A] :
( ( ( B2 != zero_zero(A) )
=> ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
=> ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2) ) ) )
=> ( ( ( B2 != zero_zero(A) )
=> ! [Q3: A,R5: A] :
( ( euclid7384307370059645450egment(A,R5) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R5)),euclid6346220572633701492n_size(A,B2)))
=> ( ( R5 != zero_zero(A) )
=> ( ( divide_divide(A,A3,B2) = Q3 )
=> ( ( modulo_modulo(A,A3,B2) = R5 )
=> ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R5) ) ) ) ) ) ) )
=> ( B2 = zero_zero(A) ) ) ) ) ).
% divmod_cases
tff(fact_3163_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R3: A,Q2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R3) = A3 )
=> ( modulo_modulo(A,A3,B2) = R3 ) ) ) ) ) ) ).
% mod_eqI
tff(fact_3164_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R3: A,Q2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R3) = A3 )
=> ( divide_divide(A,A3,B2) = Q2 ) ) ) ) ) ) ).
% div_eqI
tff(fact_3165_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R3: A,Q2: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
=> ( 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),Q2),B2)),R3),B2) = Q2 ) ) ) ) ) ).
% div_bounded
tff(fact_3166_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N))),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,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_aa(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))),N)),one_one(nat)))) ) ).
% pochhammer_times_pochhammer_half
tff(fact_3167_image__ident,axiom,
! [A: $tType,Y5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ab(A,A)),Y5) = Y5 ).
% image_ident
tff(fact_3168_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A4: set(B),C3: multiset(A)] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_ac(multiset(A),fun(B,multiset(A)),C3)),A4)) = C3 ) ) ).
% subset_mset.cSUP_const
tff(fact_3169_map__ident,axiom,
! [A: $tType,X3: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)),X3) = X3 ).
% map_ident
tff(fact_3170_division__segment__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).
% division_segment_1
tff(fact_3171_singleton__conv2,axiom,
! [A: $tType,A3: A] : collect(A,aa(A,fun(A,bool),fequal(A),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).
% singleton_conv2
tff(fact_3172_singleton__conv,axiom,
! [A: $tType,A3: A] : collect(A,aTP_Lamp_ad(A,fun(A,bool),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).
% singleton_conv
tff(fact_3173_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ae(B,A)),A4) = one_one(A) ) ).
% prod.neutral_const
tff(fact_3174_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_3175_division__segment__of__nat,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),N)) = one_one(A) ) ).
% division_segment_of_nat
tff(fact_3176_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_af(B,A)),A4)) = bot_bot(A) ) ).
% SUP_bot
tff(fact_3177_SUP__bot__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B3: fun(B,A),A4: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A4)) = bot_bot(A) )
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> ( aa(B,A,B3,X4) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(1)
tff(fact_3178_SUP__bot__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B3: fun(B,A),A4: set(B)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A4)) )
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> ( aa(B,A,B3,X4) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(2)
tff(fact_3179_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),C3: A] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ag(A,fun(B,A),C3)),A4)) = C3 ) ) ) ).
% cSUP_const
tff(fact_3180_SUP__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F: A] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),F)),A4)) = F ) ) ) ).
% SUP_const
tff(fact_3181_foldl__length,axiom,
! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),aa(fun(nat,fun(A,nat)),fun(nat,fun(list(A),nat)),foldl(nat,A),aTP_Lamp_ai(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).
% foldl_length
tff(fact_3182_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( 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_aj(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
& ( ~ pp(member2(B,A3,S))
=> ( 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_aj(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = one_one(A) ) ) ) ) ) ).
% prod.delta'
tff(fact_3183_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( 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_ak(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = aa(B,A,B2,A3) ) )
& ( ~ pp(member2(B,A3,S))
=> ( 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_ak(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = one_one(A) ) ) ) ) ) ).
% prod.delta
tff(fact_3184_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: fun(B,bool),F: 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_al(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),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,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),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),collect(B,aTP_Lamp_am(fun(B,bool),fun(B,bool),P))))) ).
% if_image_distrib
tff(fact_3185_UN__constant,axiom,
! [B: $tType,A: $tType,A4: set(B),C3: set(A)] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),C3)),A4)) = bot_bot(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),C3)),A4)) = C3 ) ) ) ).
% UN_constant
tff(fact_3186_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,aTP_Lamp_ao(A,fun(A,bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),L)) ) ) ).
% sorted_wrt_rev_linord
tff(fact_3187_division__segment__euclidean__size,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),aa(nat,A,semiring_1_of_nat(A),euclid6346220572633701492n_size(A,A3))) = A3 ) ).
% division_segment_euclidean_size
tff(fact_3188_UN__singleton,axiom,
! [A: $tType,A4: 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_ap(A,set(A))),A4)) = A4 ).
% UN_singleton
tff(fact_3189_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set(B),A3: A,B3: fun(B,set(A))] :
( ( ( C4 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_aq(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C4)) = bot_bot(set(A)) ) )
& ( ( C4 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_aq(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C4))) ) ) ) ).
% UN_simps(1)
tff(fact_3190_UN__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set(F6),A4: set(E4),B3: fun(F6,set(E4))] :
( ( ( C4 = bot_bot(set(F6)) )
=> ( aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),aa(fun(F6,set(E4)),fun(F6,set(E4)),aTP_Lamp_ar(set(E4),fun(fun(F6,set(E4)),fun(F6,set(E4))),A4),B3)),C4)) = bot_bot(set(E4)) ) )
& ( ( C4 != bot_bot(set(F6)) )
=> ( aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),aa(fun(F6,set(E4)),fun(F6,set(E4)),aTP_Lamp_ar(set(E4),fun(fun(F6,set(E4)),fun(F6,set(E4))),A4),B3)),C4)) = aa(set(E4),set(E4),aa(set(E4),fun(set(E4),set(E4)),sup_sup(set(E4)),A4),aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),B3),C4))) ) ) ) ).
% UN_simps(3)
tff(fact_3191_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set(C),A4: fun(C,set(D)),B3: set(D)] :
( ( ( C4 = bot_bot(set(C)) )
=> ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_as(fun(C,set(D)),fun(set(D),fun(C,set(D))),A4),B3)),C4)) = bot_bot(set(D)) ) )
& ( ( C4 != bot_bot(set(C)) )
=> ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_as(fun(C,set(D)),fun(set(D),fun(C,set(D))),A4),B3)),C4)) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A4),C4))),B3) ) ) ) ).
% UN_simps(2)
tff(fact_3192_Max__const,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [A4: set(B),C3: A] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( A4 != bot_bot(set(B)) )
=> ( lattic643756798349783984er_Max(A,aa(set(B),set(A),image2(B,A,aTP_Lamp_at(A,fun(B,A),C3)),A4)) = C3 ) ) ) ) ).
% Max_const
tff(fact_3193_Min__const,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [A4: set(B),C3: A] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( A4 != bot_bot(set(B)) )
=> ( lattic643756798350308766er_Min(A,aa(set(B),set(A),image2(B,A,aTP_Lamp_at(A,fun(B,A),C3)),A4)) = C3 ) ) ) ) ).
% Min_const
tff(fact_3194_foldr__length,axiom,
! [A: $tType,L: list(A)] : aa(nat,nat,aa(list(A),fun(nat,nat),aa(fun(A,fun(nat,nat)),fun(list(A),fun(nat,nat)),foldr(A,nat),aTP_Lamp_au(A,fun(nat,nat))),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).
% foldr_length
tff(fact_3195_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),aTP_Lamp_av(fun(B,A),fun(B,list(A)),F)),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) ).
% concat_map_singleton
tff(fact_3196_set__concat,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).
% set_concat
tff(fact_3197_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_aw(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_distinct_length_eq
tff(fact_3198_sorted__wrt__true,axiom,
! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_ax(A,fun(A,bool)),Xs) ).
% sorted_wrt_true
tff(fact_3199_sorted__wrt__rev,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
<=> sorted_wrt(A,aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P),Xs) ) ).
% sorted_wrt_rev
tff(fact_3200_insort__insert__triv,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( linord329482645794927042rt_key(A,A,aTP_Lamp_az(A,A),X,Xs) = Xs ) ) ) ).
% insort_insert_triv
tff(fact_3201_tl__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),tl(A),Ys)),aTP_Lamp_ba(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).
% tl_append
tff(fact_3202_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F: fun(B,A),Xs: list(B)] :
( sorted_wrt(A,R,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
<=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_bb(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F),Xs) ) ).
% sorted_wrt_map
tff(fact_3203_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),A4: set(B)] :
( pp(member2(A,B2,aa(set(B),set(A),image2(B,A,F),A4)))
=> ~ ! [X2: B] :
( ( B2 = aa(B,A,F,X2) )
=> ~ pp(member2(B,X2,A4)) ) ) ).
% imageE
tff(fact_3204_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A4: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(C),set(B),image2(C,B,G),A4)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_bc(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A4) ).
% image_image
tff(fact_3205_Compr__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),P: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_bd(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F),A4),P)) = aa(set(B),set(A),image2(B,A,F),collect(B,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_be(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F),A4),P))) ).
% Compr_image_eq
tff(fact_3206_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ~ ! [F4: fun(A,nat)] :
( ? [N7: nat] : aa(set(A),set(nat),image2(A,nat,F4),A4) = collect(nat,aTP_Lamp_bf(nat,fun(nat,bool),N7))
=> ~ inj_on(A,nat,F4,A4) ) ) ).
% finite_imp_inj_to_nat_seg'
tff(fact_3207_less__set__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).
% less_set_def
tff(fact_3208_Collect__imp__eq,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),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)),collect(A,P))),collect(A,Q)) ).
% Collect_imp_eq
tff(fact_3209_Compl__eq,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aTP_Lamp_bh(set(A),fun(A,bool),A4)) ).
% Compl_eq
tff(fact_3210_Collect__neg__eq,axiom,
! [A: $tType,P: fun(A,bool)] : collect(A,aTP_Lamp_bi(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).
% Collect_neg_eq
tff(fact_3211_uminus__set__def,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4))) ).
% uminus_set_def
tff(fact_3212_list_Omap__ident,axiom,
! [A: $tType,T4: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)),T4) = T4 ).
% list.map_ident
tff(fact_3213_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A3: A,F: fun(C,B),Xs: list(C)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),A3),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),aa(fun(A,fun(C,A)),fun(A,fun(list(C),A)),foldl(A,C),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_bj(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F)),A3),Xs) ).
% foldl_map
tff(fact_3214_Pow__def,axiom,
! [A: $tType,A4: set(A)] : pow(A,A4) = collect(set(A),aTP_Lamp_bk(set(A),fun(set(A),bool),A4)) ).
% Pow_def
tff(fact_3215_Collect__subset,axiom,
! [A: $tType,A4: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bl(set(A),fun(fun(A,bool),fun(A,bool)),A4),P))),A4)) ).
% Collect_subset
tff(fact_3216_less__eq__set__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).
% less_eq_set_def
tff(fact_3217_insert__compr,axiom,
! [A: $tType,A3: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_bm(A,fun(set(A),fun(A,bool)),A3),B3)) ).
% insert_compr
tff(fact_3218_insert__Collect,axiom,
! [A: $tType,A3: A,P: fun(A,bool)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),collect(A,P)) = collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bn(A,fun(fun(A,bool),fun(A,bool)),A3),P)) ).
% insert_Collect
tff(fact_3219_Set_Oempty__def,axiom,
! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_bo(A,bool)) ).
% Set.empty_def
tff(fact_3220_insert__def,axiom,
! [A: $tType,A3: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_ad(A,fun(A,bool),A3))),B3) ).
% insert_def
tff(fact_3221_inj__singleton,axiom,
! [A: $tType,A4: set(A)] : inj_on(A,set(A),aTP_Lamp_ap(A,set(A)),A4) ).
% inj_singleton
tff(fact_3222_Collect__conv__if2,axiom,
! [A: $tType,P: fun(A,bool),A3: A] :
( ( pp(aa(A,bool,P,A3))
=> ( collect(A,aa(A,fun(A,bool),aTP_Lamp_bp(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) )
& ( ~ pp(aa(A,bool,P,A3))
=> ( collect(A,aa(A,fun(A,bool),aTP_Lamp_bp(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).
% Collect_conv_if2
tff(fact_3223_Collect__conv__if,axiom,
! [A: $tType,P: fun(A,bool),A3: A] :
( ( pp(aa(A,bool,P,A3))
=> ( collect(A,aa(A,fun(A,bool),aTP_Lamp_bq(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) )
& ( ~ pp(aa(A,bool,P,A3))
=> ( collect(A,aa(A,fun(A,bool),aTP_Lamp_bq(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).
% Collect_conv_if
tff(fact_3224_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)) = aa(list(A),C,aa(fun(A,fun(list(A),C)),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),C)),fun(list(A),C)),case_list(C,A),aa(B,C,H,F1)),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_br(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F22)),List) ).
% list.case_distrib
tff(fact_3225_Un__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_bs(set(A),fun(set(A),fun(A,bool)),A4),B3)) ).
% Un_def
tff(fact_3226_Collect__disj__eq,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bt(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,P)),collect(A,Q)) ).
% Collect_disj_eq
tff(fact_3227_sup__set__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).
% sup_set_def
tff(fact_3228_Collect__conj__eq,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bu(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),collect(A,Q)) ).
% Collect_conj_eq
tff(fact_3229_inf__Int__eq,axiom,
! [A: $tType,R: set(A),S: set(A),X3: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),R)),aTP_Lamp_a(set(A),fun(A,bool),S)),X3))
<=> pp(member2(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R),S))) ) ).
% inf_Int_eq
tff(fact_3230_inf__set__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A4)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).
% inf_set_def
tff(fact_3231_Int__Collect,axiom,
! [A: $tType,X: A,A4: set(A),P: fun(A,bool)] :
( pp(member2(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),collect(A,P))))
<=> ( pp(member2(A,X,A4))
& pp(aa(A,bool,P,X)) ) ) ).
% Int_Collect
tff(fact_3232_Int__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_bv(set(A),fun(set(A),fun(A,bool)),A4),B3)) ).
% Int_def
tff(fact_3233_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X3: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),S)),X3),Xa3))
<=> pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),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))),R),S))) ) ).
% inf_Int_eq2
tff(fact_3234_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C3: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_bx(fun(A,B),fun(A,product_prod(B,A)),C3),S) ).
% inj_Pair(2)
tff(fact_3235_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C3: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_by(fun(A,B),fun(A,product_prod(A,B)),C3),S) ).
% inj_Pair(1)
tff(fact_3236_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_bz(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_3237_lambda__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ( aTP_Lamp_ca(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).
% lambda_zero
tff(fact_3238_set__diff__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : minus_minus(set(A),A4,B3) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_cb(set(A),fun(set(A),fun(A,bool)),A4),B3)) ).
% set_diff_eq
tff(fact_3239_minus__set__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : minus_minus(set(A),A4,B3) = collect(A,minus_minus(fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool),A4),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).
% minus_set_def
tff(fact_3240_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list(A)] :
( ( List = nil(A) )
<=> pp(aa(list(A),bool,aa(fun(A,fun(list(A),bool)),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),bool)),fun(list(A),bool)),case_list(bool,A),fTrue),aTP_Lamp_cc(A,fun(list(A),bool))),List)) ) ).
% list.disc_eq_case(1)
tff(fact_3241_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list(A)] :
( ( List != nil(A) )
<=> pp(aa(list(A),bool,aa(fun(A,fun(list(A),bool)),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),bool)),fun(list(A),bool)),case_list(bool,A),fFalse),aTP_Lamp_cd(A,fun(list(A),bool))),List)) ) ).
% list.disc_eq_case(2)
tff(fact_3242_tl__def,axiom,
! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ce(A,fun(list(A),list(A)))),List) ).
% tl_def
tff(fact_3243_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H: fun(B,A),A4: 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_cf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4) = 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),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A4)) ) ).
% prod.distrib
tff(fact_3244_Shift__def,axiom,
! [A: $tType,Kl2: set(list(A)),K: A] : bNF_Greatest_Shift(A,Kl2,K) = collect(list(A),aa(A,fun(list(A),bool),aTP_Lamp_cg(set(list(A)),fun(A,fun(list(A),bool)),Kl2),K)) ).
% Shift_def
tff(fact_3245_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
=> pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ci(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_3246_image__constant__conv,axiom,
! [B: $tType,A: $tType,A4: set(B),C3: A] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(B),set(A),image2(B,A,aTP_Lamp_cj(A,fun(B,A),C3)),A4) = bot_bot(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(B),set(A),image2(B,A,aTP_Lamp_cj(A,fun(B,A),C3)),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),bot_bot(set(A))) ) ) ) ).
% image_constant_conv
tff(fact_3247_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A4: set(A),C3: B] :
( pp(member2(A,X,A4))
=> ( aa(set(A),set(B),image2(A,B,aTP_Lamp_ck(B,fun(A,B),C3)),A4) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C3),bot_bot(set(B))) ) ) ).
% image_constant
tff(fact_3248_translation__subtract__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S4: set(A),T4: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cl(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S4),T4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_cl(A,fun(A,A),A3)),S4)),aa(set(A),set(A),image2(A,A,aTP_Lamp_cl(A,fun(A,A),A3)),T4)) ) ).
% translation_subtract_Int
tff(fact_3249_SUP__inf__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),A4: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cn(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B3)),A4)) ) ).
% SUP_inf_distrib2
tff(fact_3250_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),B3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_co(A,fun(fun(B,A),fun(B,A)),A3),F)),B3)) ) ).
% inf_SUP
tff(fact_3251_Sup__inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B3: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_cp(A,fun(A,A),A3)),B3)) ) ).
% Sup_inf
tff(fact_3252_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),B3: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),B3))),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),F),A3)),B3)) ) ).
% SUP_inf
tff(fact_3253_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_cr(set(B),fun(fun(B,bool),fun(B,bool)),A4),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_cs(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A4) ) ) ) ).
% prod.inter_filter
tff(fact_3254_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),aa(fun(nat,fun(A,nat)),fun(nat,fun(list(A),nat)),foldl(nat,A),aTP_Lamp_ai(nat,fun(A,nat))),A3),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).
% foldl_length_aux
tff(fact_3255_UNION__empty__conv_I2_J,axiom,
! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: set(B)] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4)) = bot_bot(set(A)) )
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> ( aa(B,set(A),B3,X4) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(2)
tff(fact_3256_UNION__empty__conv_I1_J,axiom,
! [B: $tType,A: $tType,B3: fun(B,set(A)),A4: 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),B3),A4)) )
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> ( aa(B,set(A),B3,X4) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(1)
tff(fact_3257_UN__empty,axiom,
! [B: $tType,A: $tType,B3: 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),B3),bot_bot(set(B)))) = bot_bot(set(A)) ).
% UN_empty
tff(fact_3258_UN__empty2,axiom,
! [B: $tType,A: $tType,A4: 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_ct(B,set(A))),A4)) = bot_bot(set(A)) ).
% UN_empty2
tff(fact_3259_sorted__map,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
<=> sorted_wrt(B,aTP_Lamp_cu(fun(B,A),fun(B,fun(B,bool)),F),Xs) ) ) ).
% sorted_map
tff(fact_3260_UN__extend__simps_I5_J,axiom,
! [I7: $tType,J5: $tType,A4: set(I7),B3: fun(J5,set(I7)),C4: set(J5)] : aa(set(I7),set(I7),aa(set(I7),fun(set(I7),set(I7)),inf_inf(set(I7)),A4),aa(set(set(I7)),set(I7),complete_Sup_Sup(set(I7)),aa(set(J5),set(set(I7)),image2(J5,set(I7),B3),C4))) = aa(set(set(I7)),set(I7),complete_Sup_Sup(set(I7)),aa(set(J5),set(set(I7)),image2(J5,set(I7),aa(fun(J5,set(I7)),fun(J5,set(I7)),aTP_Lamp_cv(set(I7),fun(fun(J5,set(I7)),fun(J5,set(I7))),A4),B3)),C4)) ).
% UN_extend_simps(5)
tff(fact_3261_UN__extend__simps_I4_J,axiom,
! [H2: $tType,G4: $tType,A4: fun(G4,set(H2)),C4: set(G4),B3: set(H2)] : aa(set(H2),set(H2),aa(set(H2),fun(set(H2),set(H2)),inf_inf(set(H2)),aa(set(set(H2)),set(H2),complete_Sup_Sup(set(H2)),aa(set(G4),set(set(H2)),image2(G4,set(H2),A4),C4))),B3) = aa(set(set(H2)),set(H2),complete_Sup_Sup(set(H2)),aa(set(G4),set(set(H2)),image2(G4,set(H2),aa(set(H2),fun(G4,set(H2)),aTP_Lamp_cw(fun(G4,set(H2)),fun(set(H2),fun(G4,set(H2))),A4),B3)),C4)) ).
% UN_extend_simps(4)
tff(fact_3262_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B3: set(A),A4: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_cx(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A4)),I5)) ).
% Int_UN_distrib
tff(fact_3263_Int__UN__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType,A4: fun(B,set(A)),I5: set(B),B3: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_cz(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A4),B3),J4)),I5)) ).
% Int_UN_distrib2
tff(fact_3264_Int__Union,axiom,
! [A: $tType,A4: set(A),B3: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4)),B3)) ).
% Int_Union
tff(fact_3265_Int__Union2,axiom,
! [A: $tType,B3: set(set(A)),A4: 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)),B3)),A4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_da(set(A),fun(set(A),set(A)),A4)),B3)) ).
% Int_Union2
tff(fact_3266_zip__same__conv__map,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xs),Xs) = aa(list(A),list(product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_db(A,product_prod(A,A))),Xs) ).
% zip_same_conv_map
tff(fact_3267_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set(A),S5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> pp(aa(set(set(A)),bool,finite_finite2(set(A)),collect(set(A),aa(set(A),fun(set(A),bool),aTP_Lamp_dc(set(A),fun(set(A),fun(set(A),bool)),S),S5)))) ) ).
% finite_if_eq_beyond_finite
tff(fact_3268_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).
% arg_min_list.simps(2)
tff(fact_3269_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),A3: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),Xs),A3) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),aTP_Lamp_dd(fun(B,fun(A,A)),fun(A,fun(B,A)),F)),A3),aa(list(B),list(B),rev(B),Xs)) ).
% foldr_conv_foldl
tff(fact_3270_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),A3),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aTP_Lamp_de(fun(A,fun(B,A)),fun(B,fun(A,A)),F)),aa(list(B),list(B),rev(B),Xs)),A3) ).
% foldl_conv_foldr
tff(fact_3271_distinct__finite__set,axiom,
! [A: $tType,X: set(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aTP_Lamp_df(set(A),fun(list(A),bool),X)))) ).
% distinct_finite_set
tff(fact_3272_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_az(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_3273_sorted__insort__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),X: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_az(A,A),X,Xs)) ) ) ).
% sorted_insort_insert
tff(fact_3274_product__concat__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xs),Ys) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_dg(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).
% product_concat_map
tff(fact_3275_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_dh(nat,fun(nat,nat),Ofs)),upt(A3,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).
% map_add_upt'
tff(fact_3276_numeral__code_I3_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),bit1(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).
% numeral_code(3)
tff(fact_3277_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).
% SUP_empty
tff(fact_3278_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),C3: A] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),C3)),A4)) = bot_bot(A) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),C3)),A4)) = C3 ) ) ) ) ).
% SUP_constant
tff(fact_3279_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))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W))) ) ).
% power_numeral_even
tff(fact_3280_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))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W))) ) ).
% power_numeral_odd
tff(fact_3281_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_di(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A4) ) ) ) ).
% prod.inter_restrict
tff(fact_3282_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),minus_minus(set(B),A4,collect(B,aTP_Lamp_dj(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) ) ) ) ).
% prod.setdiff_irrelevant
tff(fact_3283_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set(B),A3: A,B3: fun(B,set(A))] :
( ( ( C4 = bot_bot(set(B)) )
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C4))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) )
& ( ( C4 != bot_bot(set(B)) )
=> ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_aq(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C4)) ) ) ) ).
% UN_extend_simps(1)
tff(fact_3284_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set(F6),A4: set(E4),B3: fun(F6,set(E4))] :
( ( ( C4 = bot_bot(set(F6)) )
=> ( aa(set(E4),set(E4),aa(set(E4),fun(set(E4),set(E4)),sup_sup(set(E4)),A4),aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),B3),C4))) = A4 ) )
& ( ( C4 != bot_bot(set(F6)) )
=> ( aa(set(E4),set(E4),aa(set(E4),fun(set(E4),set(E4)),sup_sup(set(E4)),A4),aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),B3),C4))) = aa(set(set(E4)),set(E4),complete_Sup_Sup(set(E4)),aa(set(F6),set(set(E4)),image2(F6,set(E4),aa(fun(F6,set(E4)),fun(F6,set(E4)),aTP_Lamp_ar(set(E4),fun(fun(F6,set(E4)),fun(F6,set(E4))),A4),B3)),C4)) ) ) ) ).
% UN_extend_simps(3)
tff(fact_3285_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C4: set(C),A4: fun(C,set(D)),B3: set(D)] :
( ( ( C4 = bot_bot(set(C)) )
=> ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A4),C4))),B3) = B3 ) )
& ( ( C4 != bot_bot(set(C)) )
=> ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A4),C4))),B3) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_as(fun(C,set(D)),fun(set(D),fun(C,set(D))),A4),B3)),C4)) ) ) ) ).
% UN_extend_simps(2)
tff(fact_3286_length__transpose,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_dk(list(A),fun(nat,nat))),Xs),zero_zero(nat)) ).
% length_transpose
tff(fact_3287_foldr__length__aux,axiom,
! [A: $tType,L: list(A),A3: nat] : aa(nat,nat,aa(list(A),fun(nat,nat),aa(fun(A,fun(nat,nat)),fun(list(A),fun(nat,nat)),foldr(A,nat),aTP_Lamp_au(A,fun(nat,nat))),L),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).
% foldr_length_aux
tff(fact_3288_Succ__def,axiom,
! [A: $tType,Kl2: set(list(A)),Kl: list(A)] : bNF_Greatest_Succ(A,Kl2,Kl) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_dl(set(list(A)),fun(list(A),fun(A,bool)),Kl2),Kl)) ).
% Succ_def
tff(fact_3289_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_dm(fun(B,list(A)),fun(B,set(A)),F)),aa(list(B),set(B),set2(B),Xs))) ).
% set_list_bind
tff(fact_3290_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X3: list(A),Xa3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),R3)),X3),Xa3))
<=> pp(member2(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)),X3),Xa3),listrel(A,B,R3))) ) ).
% listrelp_listrel_eq
tff(fact_3291_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aTP_Lamp_do(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).
% n_lists.simps(2)
tff(fact_3292_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: fun(nat,A),M2: nat] : enumerate(A,N,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F),upt(N,M2))) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_dp(fun(nat,A),fun(nat,product_prod(nat,A)),F)),upt(N,M2)) ).
% enumerate_map_upt
tff(fact_3293_map__add__upt,axiom,
! [N: nat,M2: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_dh(nat,fun(nat,nat),N)),upt(zero_zero(nat),M2)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).
% map_add_upt
tff(fact_3294_min__list_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_dq(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).
% min_list.simps
tff(fact_3295_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A),Xss2: list(list(A))] : aa(list(list(A)),list(list(A)),product_lists(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss2)) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(A),list(list(list(A))),aa(fun(A,list(list(A))),fun(list(A),list(list(list(A)))),map(A,list(list(A))),aTP_Lamp_dr(list(list(A)),fun(A,list(list(A))),Xss2)),Xs)) ).
% product_lists.simps(2)
tff(fact_3296_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: 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_ds(fun(B,A),fun(B,set(A)),F)),A4)) = aa(set(B),set(A),image2(B,A,F),A4) ).
% UNION_singleton_eq_range
tff(fact_3297_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),G: fun(A,B),B3: set(A)] :
( inj_on(A,B,F,A4)
=> ( inj_on(A,B,G,B3)
=> ( ( 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,F),A4)),aa(set(A),set(B),image2(A,B,G),B3)) = 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_dt(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) ) ) ) ).
% inj_on_disjoint_Un
tff(fact_3298_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S: set(B),F: fun(B,A),K: A] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( S != bot_bot(set(B)) )
=> ( lattic643756798349783984er_Max(A,aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_du(fun(B,A),fun(A,fun(B,A)),F),K)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798349783984er_Max(A,aa(set(B),set(A),image2(B,A,F),S))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_3299_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S: set(B),F: fun(B,A),K: A] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( S != bot_bot(set(B)) )
=> ( lattic643756798350308766er_Min(A,aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_du(fun(B,A),fun(A,fun(B,A)),F),K)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798350308766er_Min(A,aa(set(B),set(A),image2(B,A,F),S))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_3300_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( 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,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dv(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_3301_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_dw(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A4) = 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),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% prod.If_cases
tff(fact_3302_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = 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_dv(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).
% prod.atMost_Suc_shift
tff(fact_3303_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = 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_dv(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.lessThan_Suc_shift
tff(fact_3304_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(list(A)),subseqs(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X)),aa(list(A),list(list(A)),subseqs(A),Xs))),aa(list(A),list(list(A)),subseqs(A),Xs)) ).
% subseqs.simps(2)
tff(fact_3305_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: fun(A,nat),X21: A,X22: list(A)] : aa(list(A),nat,size_list(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),aa(list(A),nat,size_list(A,X),X22))),aa(nat,nat,suc,zero_zero(nat))) ).
% list.size_gen(2)
tff(fact_3306_map__upt__Suc,axiom,
! [A: $tType,F: fun(nat,A),N: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F,zero_zero(nat))),aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_dx(fun(nat,A),fun(nat,A),F)),upt(zero_zero(nat),N))) ).
% map_upt_Suc
tff(fact_3307_finite__lists__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_length_eq
tff(fact_3308_horner__sum__foldr,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_dz(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3)),Xs),zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_3309_map__nth,axiom,
! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).
% map_nth
tff(fact_3310_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).
% slice_prepend
tff(fact_3311_distinct__finite__subset,axiom,
! [A: $tType,X: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),X))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aTP_Lamp_ea(set(A),fun(list(A),bool),X)))) ) ).
% distinct_finite_subset
tff(fact_3312_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,fun(list(B),list(product_prod(A,B)))),fun(list(B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(B,fun(list(B),list(product_prod(A,B)))),fun(list(B),list(product_prod(A,B)))),case_list(list(product_prod(A,B)),B),nil(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_eb(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).
% zip_Cons1
tff(fact_3313_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,fun(list(A),list(product_prod(A,B)))),fun(list(A),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(A,fun(list(A),list(product_prod(A,B)))),fun(list(A),list(product_prod(A,B)))),case_list(list(product_prod(A,B)),A),nil(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_ec(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).
% zip_Cons
tff(fact_3314_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = collect(list(A),aa(list(A),fun(list(A),bool),aTP_Lamp_ed(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).
% set_n_lists
tff(fact_3315_map__decr__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_ee(nat,nat)),upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = upt(M2,N) ).
% map_decr_upt
tff(fact_3316_atLeastAtMost__upto,axiom,
! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).
% atLeastAtMost_upto
tff(fact_3317_map__by__foldl,axiom,
! [B: $tType,A: $tType,F: fun(A,B),L: list(A)] : aa(list(A),list(B),aa(list(B),fun(list(A),list(B)),aa(fun(list(B),fun(A,list(B))),fun(list(B),fun(list(A),list(B))),foldl(list(B),A),aTP_Lamp_ef(fun(A,B),fun(list(B),fun(A,list(B))),F)),nil(B)),L) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),L) ).
% map_by_foldl
tff(fact_3318_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_eg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))) ) )
& ( ~ pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_eg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).
% prod.delta_remove
tff(fact_3319_prod_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I5))
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I5))
=> ( ( X2 != Xa4 )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa4)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_eh(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A4),G)),I5) ) ) ) ) ) ).
% prod.UNION_disjoint
tff(fact_3320_division__segment__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B2)) ) ) ) ) ).
% division_segment_mult
tff(fact_3321_prod_OatMost__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,N)) = 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_dv(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.atMost_shift
tff(fact_3322_is__unit__division__segment,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A] : pp(dvd_dvd(A,euclid7384307370059645450egment(A,A3),one_one(A))) ) ).
% is_unit_division_segment
tff(fact_3323_finite__lists__length__le,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ei(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_length_le
tff(fact_3324_foldl__set,axiom,
! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),aa(fun(set(A),fun(set(A),set(A))),fun(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)),collect(set(A),aTP_Lamp_ej(list(set(A)),fun(set(A),bool),L))) ).
% foldl_set
tff(fact_3325_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list(B)] : pow(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X)),pow(B,aa(list(B),set(B),set2(B),Xs)))) ).
% Pow_set(2)
tff(fact_3326_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,C3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ek(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ek(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3),aa(A,A,aa(A,fun(A,A),times_times(A),X),C3)) ) ) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ek(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_3327_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_el(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_el(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_el(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
tff(fact_3328_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_em(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_em(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3),C3),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_em(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2),C3),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_3329_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_en(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_en(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_en(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
tff(fact_3330_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_eo(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_eo(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,divide_divide(A,A3,M2),C3),minus_minus(A,divide_divide(A,B2,M2),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_eo(A,fun(A,fun(A,A)),M2),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,divide_divide(A,B2,M2),C3),minus_minus(A,divide_divide(A,A3,M2),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
tff(fact_3331_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A),C3: A] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ep(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3)),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),minus_minus(nat,finite_card(B,S),one_one(nat)))) ) )
& ( ~ pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ep(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3)),S) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),finite_card(B,S)) ) ) ) ) ) ).
% prod_gen_delta
tff(fact_3332_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)),aa(fun(A,list(A)),fun(list(A),list(list(A))),map(A,list(A)),aTP_Lamp_eq(A,list(A))),Xs)) ) ).
% mergesort_remdups_def
tff(fact_3333_card__lists__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),A4),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),finite_card(A,A4)),N) ) ) ).
% card_lists_length_eq
tff(fact_3334_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),Xss2: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss2)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_er(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss2))))) ).
% transpose.simps(3)
tff(fact_3335_transpose_Oelims,axiom,
! [A: $tType,X: list(list(A)),Y: list(list(A))] :
( ( transpose(A,X) = Y )
=> ( ( ( X = nil(list(A)) )
=> ( Y != nil(list(A)) ) )
=> ( ! [Xss: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss) )
=> ( Y != transpose(A,Xss) ) )
=> ~ ! [X2: A,Xs2: list(A),Xss: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss) )
=> ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_er(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ) ) ) ).
% transpose.elims
tff(fact_3336_prod_Oin__pairs,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M2: nat,N: 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))),M2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).
% prod.in_pairs
tff(fact_3337_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% prod.in_pairs_0
tff(fact_3338_signed__take__bit__code,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = if(A,bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A)))),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)) ) ).
% signed_take_bit_code
tff(fact_3339_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),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_eu(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact'
tff(fact_3340_gbinomial__mult__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eu(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact
tff(fact_3341_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P4: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P4))
=> ( 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_ev(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P4)) = 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_ew(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,minus_minus(nat,P4,aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% prod.zero_middle
tff(fact_3342_transpose__rectangle,axiom,
! [A: $tType,Xs: list(list(A)),N: nat] :
( ( ( Xs = nil(list(A)) )
=> ( N = zero_zero(nat) ) )
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I3)) = N ) )
=> ( transpose(A,Xs) = aa(list(nat),list(list(A)),aa(fun(nat,list(A)),fun(list(nat),list(list(A))),map(nat,list(A)),aTP_Lamp_ey(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).
% transpose_rectangle
tff(fact_3343_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),finite_card(A,A4)))
=> ( finite_card(list(A),collect(list(A),aa(set(A),fun(list(A),bool),aTP_Lamp_ez(nat,fun(set(A),fun(list(A),bool)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fa(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),minus_minus(nat,finite_card(A,A4),K)),one_one(nat)),finite_card(A,A4))) ) ) ).
% card_lists_distinct_length_eq'
tff(fact_3344_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set(A),K: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),finite_card(A,A4)))
=> ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_aw(set(A),fun(nat,fun(list(A),bool)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fa(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),minus_minus(nat,finite_card(A,A4),K)),one_one(nat)),finite_card(A,A4))) ) ) ) ).
% card_lists_distinct_length_eq
tff(fact_3345_set__Cons__sing__Nil,axiom,
! [A: $tType,A4: set(A)] : aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),A4),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_fb(A,list(A))),A4) ).
% set_Cons_sing_Nil
tff(fact_3346_gbinomial__code,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( ( ( K = zero_zero(nat) )
=> ( aa(nat,A,gbinomial(A,A3),K) = one_one(A) ) )
& ( ( K != zero_zero(nat) )
=> ( aa(nat,A,gbinomial(A,A3),K) = divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_fc(A,fun(nat,fun(A,A)),A3),zero_zero(nat),minus_minus(nat,K,one_one(nat)),one_one(A)),semiring_char_0_fact(A,K)) ) ) ) ) ).
% gbinomial_code
tff(fact_3347_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)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_dg(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).
% product_code
tff(fact_3348_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fd(nat,fun(list(A),bool),I)),Xs))))
=> ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).
% nth_nth_transpose_sorted
tff(fact_3349_pochhammer__code,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fe(A,fun(nat,fun(A,A)),A3),zero_zero(nat),minus_minus(nat,N,one_one(nat)),one_one(A)) ) ) ) ) ).
% pochhammer_code
tff(fact_3350_predicate1I,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).
% predicate1I
tff(fact_3351_inf1I,axiom,
! [A: $tType,A4: fun(A,bool),X: A,B3: fun(A,bool)] :
( pp(aa(A,bool,A4,X))
=> ( pp(aa(A,bool,B3,X))
=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3),X)) ) ) ).
% inf1I
tff(fact_3352_inf2I,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X: A,Y: B,B3: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B3),X),Y)) ) ) ).
% inf2I
tff(fact_3353_filter__filter,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ff(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),Xs) ).
% filter_filter
tff(fact_3354_sup1CI,axiom,
! [A: $tType,B3: fun(A,bool),X: A,A4: fun(A,bool)] :
( ( ~ pp(aa(A,bool,B3,X))
=> pp(aa(A,bool,A4,X)) )
=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A4),B3),X)) ) ).
% sup1CI
tff(fact_3355_filter__True,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = Xs ) ) ).
% filter_True
tff(fact_3356_filter__append,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys)) ).
% filter_append
tff(fact_3357_remove1__filter__not,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).
% remove1_filter_not
tff(fact_3358_removeAll__filter__not,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).
% removeAll_filter_not
tff(fact_3359_set__filter,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_fg(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).
% set_filter
tff(fact_3360_filter__False,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = nil(A) ) ) ).
% filter_False
tff(fact_3361_predicate1D,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool),X: A] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
=> ( pp(aa(A,bool,P,X))
=> pp(aa(A,bool,Q,X)) ) ) ).
% predicate1D
tff(fact_3362_rev__predicate1D,axiom,
! [A: $tType,P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
=> pp(aa(A,bool,Q,X)) ) ) ).
% rev_predicate1D
tff(fact_3363_sup1I2,axiom,
! [A: $tType,B3: fun(A,bool),X: A,A4: fun(A,bool)] :
( pp(aa(A,bool,B3,X))
=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A4),B3),X)) ) ).
% sup1I2
tff(fact_3364_sup1I1,axiom,
! [A: $tType,A4: fun(A,bool),X: A,B3: fun(A,bool)] :
( pp(aa(A,bool,A4,X))
=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A4),B3),X)) ) ).
% sup1I1
tff(fact_3365_sup1E,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),X: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A4),B3),X))
=> ( ~ pp(aa(A,bool,A4,X))
=> pp(aa(A,bool,B3,X)) ) ) ).
% sup1E
tff(fact_3366_inf2D2,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool)),X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B3),X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y)) ) ).
% inf2D2
tff(fact_3367_inf2D1,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool)),X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B3),X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y)) ) ).
% inf2D1
tff(fact_3368_inf1D2,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),X: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3),X))
=> pp(aa(A,bool,B3,X)) ) ).
% inf1D2
tff(fact_3369_inf1D1,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),X: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3),X))
=> pp(aa(A,bool,A4,X)) ) ).
% inf1D1
tff(fact_3370_inf2E,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool)),X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B3),X),Y))
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y))
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y)) ) ) ).
% inf2E
tff(fact_3371_inf1E,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),X: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3),X))
=> ~ ( pp(aa(A,bool,A4,X))
=> ~ pp(aa(A,bool,B3,X)) ) ) ).
% inf1E
tff(fact_3372_filter_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),nil(A)) = nil(A) ).
% filter.simps(1)
tff(fact_3373_rev__filter,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),rev(A),Xs)) ).
% rev_filter
tff(fact_3374_filter_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ) ).
% filter.simps(2)
tff(fact_3375_distinct__filter,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))) ) ).
% distinct_filter
tff(fact_3376_filter__cong,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Ys)))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) ) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Ys) ) ) ) ).
% filter_cong
tff(fact_3377_filter__id__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = Xs )
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X4)) ) ) ).
% filter_id_conv
tff(fact_3378_sorted__wrt__filter,axiom,
! [A: $tType,F: fun(A,fun(A,bool)),Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,F,Xs)
=> sorted_wrt(A,F,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) ).
% sorted_wrt_filter
tff(fact_3379_filter__remove1,axiom,
! [A: $tType,Q: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Xs)) ).
% filter_remove1
tff(fact_3380_partition__in__shuffles,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] : pp(member2(list(A),Xs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bi(fun(A,bool),fun(A,bool),P)),Xs)))) ).
% partition_in_shuffles
tff(fact_3381_removeAll__filter__not__eq,axiom,
! [A: $tType,X: A] : aa(A,fun(list(A),list(A)),removeAll(A),X) = aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_fh(A,fun(A,bool)),X)) ).
% removeAll_filter_not_eq
tff(fact_3382_empty__filter__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( nil(A) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) )
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(aa(A,bool,P,X4)) ) ) ).
% empty_filter_conv
tff(fact_3383_filter__empty__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = nil(A) )
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(aa(A,bool,P,X4)) ) ) ).
% filter_empty_conv
tff(fact_3384_filter__is__subset,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),set(A),set2(A),Xs))) ).
% filter_is_subset
tff(fact_3385_length__filter__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_filter_le
tff(fact_3386_sorted__filter_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),L)) ) ) ).
% sorted_filter'
tff(fact_3387_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),P: fun(B,bool)] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
=> pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)))) ) ).
% distinct_map_filter
tff(fact_3388_last__filter,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ( Xs != nil(A) )
=> ( pp(aa(A,bool,P,last(A,Xs)))
=> ( last(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = last(A,Xs) ) ) ) ).
% last_filter
tff(fact_3389_sum__length__filter__compl,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bi(fun(A,bool),fun(A,bool),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).
% sum_length_filter_compl
tff(fact_3390_filter__concat,axiom,
! [A: $tType,P4: fun(A,bool),Xs: list(list(A))] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P4),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P4)),Xs)) ).
% filter_concat
tff(fact_3391_inter__set__filter,axiom,
! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_a(set(A),fun(A,bool),A4)),Xs)) ).
% inter_set_filter
tff(fact_3392_sorted__same,axiom,
! [A: $tType] :
( linorder(A)
=> ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,bool),aTP_Lamp_fi(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs)),Xs)) ) ).
% sorted_same
tff(fact_3393_filter__shuffles,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys)) ).
% filter_shuffles
tff(fact_3394_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fj(list(A),bool)),Xs)) = aa(list(list(A)),list(A),concat(A),Xs) ).
% concat_filter_neq_Nil
tff(fact_3395_union__coset__filter,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A4) = coset(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bh(set(A),fun(A,bool),A4)),Xs)) ).
% union_coset_filter
tff(fact_3396_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ~ pp(aa(A,bool,P,X))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% length_filter_less
tff(fact_3397_filter__eq__Cons__iff,axiom,
! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
<=> ? [Us: list(A),Vs2: list(A)] :
( ( 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),X),Vs2)) )
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Us)))
=> ~ pp(aa(A,bool,P,X4)) )
& pp(aa(A,bool,P,X))
& ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs2) ) ) ) ).
% filter_eq_Cons_iff
tff(fact_3398_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) )
<=> ? [Us: list(A),Vs2: list(A)] :
( ( 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),X),Vs2)) )
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Us)))
=> ~ pp(aa(A,bool,P,X4)) )
& pp(aa(A,bool,P,X))
& ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs2) ) ) ) ).
% Cons_eq_filter_iff
tff(fact_3399_filter__eq__ConsD,axiom,
! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
=> ? [Us3: list(A),Vs3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
& ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Us3)))
=> ~ pp(aa(A,bool,P,X3)) )
& pp(aa(A,bool,P,X))
& ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs3) ) ) ) ).
% filter_eq_ConsD
tff(fact_3400_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) )
=> ? [Us3: list(A),Vs3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
& ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Us3)))
=> ~ pp(aa(A,bool,P,X3)) )
& pp(aa(A,bool,P,X))
& ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs3) ) ) ) ).
% Cons_eq_filterD
tff(fact_3401_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),P: fun(B,bool)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) ) ) ).
% sorted_filter
tff(fact_3402_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Y: A,Xs: list(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_fk(fun(A,B),fun(A,fun(A,bool)),F),Y)),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),Y)),Xs) ) ) ).
% inj_on_filter_key_eq
tff(fact_3403_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_fl(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F),G),Xs)),Xs))) ) ).
% sorted_map_same
tff(fact_3404_length__filter__conv__card,axiom,
! [A: $tType,P4: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P4),Xs)) = finite_card(nat,collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_fm(fun(A,bool),fun(list(A),fun(nat,bool)),P4),Xs))) ).
% length_filter_conv_card
tff(fact_3405_minus__coset__filter,axiom,
! [A: $tType,A4: set(A),Xs: list(A)] : minus_minus(set(A),A4,coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_a(set(A),fun(A,bool),A4)),Xs)) ).
% minus_coset_filter
tff(fact_3406_filter__eq__snocD,axiom,
! [A: $tType,P: fun(A,bool),L: list(A),L3: list(A),X: A] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),L) = 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),X),nil(A))) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),L)))
& pp(aa(A,bool,P,X)) ) ) ).
% filter_eq_snocD
tff(fact_3407_filter__conv__foldr,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),foldr(A,list(A)),aTP_Lamp_fn(fun(A,bool),fun(A,fun(list(A),list(A))),P)),Xs),nil(A)) ).
% filter_conv_foldr
tff(fact_3408_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),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),Y)),Xs)) ).
% set_minus_filter_out
tff(fact_3409_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_fp(list(A),fun(A,bool),Ys)),Zs) = Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
tff(fact_3410_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_fq(list(A),fun(A,bool),Ys)),Zs) = Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
tff(fact_3411_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_fp(list(A),fun(A,bool),Xs)),Zs) = Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
tff(fact_3412_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( pp(member2(list(A),Zs,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_fq(list(A),fun(A,bool),Xs)),Zs) = Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
tff(fact_3413_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))))
=> ? [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)),N) = aa(nat,A,nth(A,Xs),M4) )
& ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M4),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) ) ) ).
% filter_nth_ex_nth
tff(fact_3414_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list(A),Xss2: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_fr(list(B),fun(nat,nat))),Xss2),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_fs(list(B),fun(nat,nat))),aa(list(list(B)),list(list(B)),aa(fun(list(B),bool),fun(list(list(B)),list(list(B))),filter2(list(B)),aTP_Lamp_ft(list(B),bool)),Xss2)),zero_zero(nat)))) ).
% transpose_aux_max
tff(fact_3415_distinct__length__filter,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).
% distinct_length_filter
tff(fact_3416_transpose__max__length,axiom,
! [A: $tType,Xs: list(list(A))] : aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_dk(list(A),fun(nat,nat))),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fj(list(A),bool)),Xs)) ).
% transpose_max_length
tff(fact_3417_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss2)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),tl(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fj(list(A),bool)),Xss2)) ).
% transpose_aux_filter_tail
tff(fact_3418_transpose__aux__filter__head,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_er(A,fun(list(A),list(A))))),Xss2)) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),hd(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fj(list(A),bool)),Xss2)) ).
% transpose_aux_filter_head
tff(fact_3419_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fu(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_3420_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list(list(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_fv(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fd(nat,fun(list(A),bool),I)),Xs)) ) ) ).
% nth_transpose
tff(fact_3421_distinct__concat_H,axiom,
! [A: $tType,Xs: list(list(A))] :
( pp(aa(list(list(A)),bool,distinct(list(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fj(list(A),bool)),Xs)))
=> ( ! [Ys2: list(A)] :
( pp(member2(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> pp(aa(list(A),bool,distinct(A),Ys2)) )
=> ( ! [Ys2: list(A),Zs3: list(A)] :
( pp(member2(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( pp(member2(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( ( Ys2 != 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),Ys2)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) )
=> pp(aa(list(A),bool,distinct(A),aa(list(list(A)),list(A),concat(A),Xs))) ) ) ) ).
% distinct_concat'
tff(fact_3422_transpose__column__length,axiom,
! [A: $tType,Xs: list(list(A)),I: nat] :
( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fd(nat,fun(list(A),bool),I)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).
% transpose_column_length
tff(fact_3423_transpose__column,axiom,
! [A: $tType,Xs: list(list(A)),I: nat] :
( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_fv(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_fd(nat,fun(list(A),bool),I)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).
% transpose_column
tff(fact_3424_choose__odd__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fw(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).
% choose_odd_sum
tff(fact_3425_choose__even__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fx(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).
% choose_even_sum
tff(fact_3426_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fy(A,fun(nat,A),A3)),set_ord_atMost(nat,M2)) = 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),M2)),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),one_one(nat)))) ) ).
% gbinomial_partial_row_sum
tff(fact_3427_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M2: 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),M2))),one_one(A)))),set_ord_atMost(nat,M2)) = 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))),M2)) ) ).
% gbinomial_r_part_sum
tff(fact_3428_sum__gp,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [N: nat,M2: nat,X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( 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,M2,N)) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
=> ( ( ( 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)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(nat,A,semiring_1_of_nat(A),minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M2)) ) )
& ( ( 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)),set_or1337092689740270186AtMost(nat,M2,N)) = divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),minus_minus(A,one_one(A),X)) ) ) ) ) ) ) ).
% sum_gp
tff(fact_3429_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_3430_sum__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_fz(A,fun(B,A),Y)),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A4))),Y) ) ).
% sum_constant
tff(fact_3431_sum__zero__power,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: set(nat),C3: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A4))
& pp(member2(nat,zero_zero(nat),A4)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,A),C3)),A4) = aa(nat,A,C3,zero_zero(nat)) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A4))
& pp(member2(nat,zero_zero(nat),A4)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,A),C3)),A4) = zero_zero(A) ) ) ) ) ).
% sum_zero_power
tff(fact_3432_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A4: set(B),F: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_gb(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P)),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P))) ) ) ) ).
% sum_mult_of_bool_eq
tff(fact_3433_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A4: set(B),P: fun(B,bool),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( 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_gc(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F)),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P))) ) ) ) ).
% sum_of_bool_mult_eq
tff(fact_3434_sum__zero__power_H,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: set(nat),C3: fun(nat,A),D3: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A4))
& pp(member2(nat,zero_zero(nat),A4)) )
=> ( 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_gd(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A4) = divide_divide(A,aa(nat,A,C3,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A4))
& pp(member2(nat,zero_zero(nat),A4)) )
=> ( 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_gd(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A4) = zero_zero(A) ) ) ) ) ).
% sum_zero_power'
tff(fact_3435_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A4: set(B),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ge(fun(B,bool),fun(B,A),P)),A4) = aa(nat,A,semiring_1_of_nat(A),finite_card(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))) ) ) ) ) ).
% sum_of_bool_eq
tff(fact_3436_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [R3: A,F: fun(B,A),A4: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)) = 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_gf(A,fun(fun(B,A),fun(B,A)),R3),F)),A4) ) ).
% sum_distrib_left
tff(fact_3437_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),A4: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),R3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_gg(fun(B,A),fun(A,fun(B,A)),F),R3)),A4) ) ).
% sum_distrib_right
tff(fact_3438_sum__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_0(B)
=> ! [F: fun(A,B),A4: set(A),G: fun(C,B),B3: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_gi(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F),G),B3)),A4) ) ).
% sum_product
tff(fact_3439_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A4: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( A4 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4))) ) ) ) ) ).
% sum_strict_mono
tff(fact_3440_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_gj(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A4) ) ) ) ).
% sum.inter_restrict
tff(fact_3441_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gk(A,fun(nat,fun(nat,A)),X),M2)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),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)),I5)) ) ).
% sum_power_add
tff(fact_3442_filter__upt__take__conv,axiom,
! [A: $tType,P: fun(A,bool),M2: nat,L: list(A),N: nat] : aa(list(nat),list(nat),aa(fun(nat,bool),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_gl(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),P),M2),L)),upt(N,M2)) = aa(list(nat),list(nat),aa(fun(nat,bool),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,bool),aTP_Lamp_gm(fun(A,bool),fun(list(A),fun(nat,bool)),P),L)),upt(N,M2)) ).
% filter_upt_take_conv
tff(fact_3443_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [I5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I5))
=> ( ( I5 != bot_bot(set(B)) )
=> ( ! [I3: B] :
( pp(member2(B,I3,I5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,I3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),I5))) ) ) ) ) ).
% sum_pos
tff(fact_3444_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),F: fun(B,A),K5: A] :
( ! [I3: B] :
( pp(member2(B,I3,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I3)),K5)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A4))),K5))) ) ) ).
% sum_bounded_above
tff(fact_3445_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),K5: A,F: fun(B,A)] :
( ! [I3: B] :
( pp(member2(B,I3,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F,I3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A4))),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4))) ) ) ).
% sum_bounded_below
tff(fact_3446_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H,I3) = zero_zero(A) ) )
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),S,T)))
=> ( aa(B,A,G,I3) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T)))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
tff(fact_3447_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ).
% sum.union_inter
tff(fact_3448_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),minus_minus(set(B),A4,B3))) ) ) ) ).
% sum.Int_Diff
tff(fact_3449_upt__filter__extend,axiom,
! [U: nat,U3: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),U3))
=> ( ! [I3: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),I3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),U3)) )
=> ~ pp(aa(nat,bool,P,I3)) )
=> ( aa(list(nat),list(nat),aa(fun(nat,bool),fun(list(nat),list(nat)),filter2(nat),P),upt(zero_zero(nat),U)) = aa(list(nat),list(nat),aa(fun(nat,bool),fun(list(nat),list(nat)),filter2(nat),P),upt(zero_zero(nat),U3)) ) ) ) ).
% upt_filter_extend
tff(fact_3450_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_gn(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% sum.If_cases
tff(fact_3451_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ) ).
% sum.union_inter_neutral
tff(fact_3452_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),G: fun(B,A),X: B] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ).
% sum.insert_remove
tff(fact_3453_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% sum.remove
tff(fact_3454_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [A4: set(B),A3: B,F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( pp(member2(B,A3,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) = minus_minus(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4),aa(B,A,F,A3)) ) )
& ( ~ pp(member2(B,A3,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4) ) ) ) ) ) ).
% sum_diff1
tff(fact_3455_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [A4: set(B),B3: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% sum_Un
tff(fact_3456_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ) ).
% sum.union_disjoint
tff(fact_3457_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A4: set(A),B3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),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),F),minus_minus(set(A),A4,B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),minus_minus(set(A),B3,A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).
% sum_Un2
tff(fact_3458_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),minus_minus(set(B),A4,B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),minus_minus(set(B),B3,A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% sum.union_diff2
tff(fact_3459_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_go(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))) ) )
& ( ~ pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_go(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).
% sum.delta_remove
tff(fact_3460_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: set(B),F: fun(B,A),B2: A] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_gp(fun(B,A),fun(A,fun(B,A)),F),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,aa(A,fun(B,bool),aTP_Lamp_gq(fun(B,A),fun(A,fun(B,bool)),F),B2))))),divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,aa(A,fun(B,bool),aTP_Lamp_gr(fun(B,A),fun(A,fun(B,bool)),F),B2)))),B2)) ) ) ) ).
% sum_div_partition
tff(fact_3461_sum_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I5: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I5))
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I5))
=> ( ( X2 != Xa4 )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa4)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_gs(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A4),G)),I5) ) ) ) ) ) ).
% sum.UNION_disjoint
tff(fact_3462_power__diff__1__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N),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)),set_ord_lessThan(nat,N))) ) ).
% power_diff_1_eq
tff(fact_3463_one__diff__power__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = 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_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq
tff(fact_3464_geometric__sum,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N: 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)),set_ord_lessThan(nat,N)) = divide_divide(A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N),one_one(A)),minus_minus(A,X,one_one(A))) ) ) ) ).
% geometric_sum
tff(fact_3465_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gt(A,fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).
% gbinomial_parallel_sum
tff(fact_3466_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [I: C,A4: set(C),F: fun(C,B)] :
( pp(member2(C,I,A4))
=> ( ! [X2: C] :
( pp(member2(C,X2,minus_minus(set(C),A4,aa(set(C),set(C),aa(C,fun(set(C),set(C)),insert2(C),I),bot_bot(set(C))))))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F,X2))) )
=> ( pp(aa(set(C),bool,finite_finite2(C),A4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F,I)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A4))) ) ) ) ) ).
% member_le_sum
tff(fact_3467_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( linordered_field(A)
=> ! [A4: set(B),F: fun(B,A),K5: A] :
( ! [I3: B] :
( pp(member2(B,I3,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I3)),divide_divide(A,K5,aa(nat,A,semiring_1_of_nat(A),finite_card(B,A4))))) )
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ( A4 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),K5)) ) ) ) ) ).
% sum_bounded_above_divide
tff(fact_3468_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),F: fun(B,A),K5: A] :
( ! [I3: B] :
( pp(member2(B,I3,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I3)),K5)) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),finite_card(B,A4)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A4))),K5))) ) ) ) ).
% sum_bounded_above_strict
tff(fact_3469_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
( ! [I3: A] :
( pp(member2(A,I3,I5))
=> pp(aa(B,bool,aa(B,fun(B,bool),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),I5) = one_one(B) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),minus_minus(B,aa(A,B,A3,I3),B2))),Delta)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),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_gu(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5),B2))),Delta)) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_3470_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S: set(A),R: set(B),G: fun(A,B),F: fun(B,C)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( pp(aa(set(B),bool,finite_finite2(B),R))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),R))
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_gv(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),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_gx(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F)),R) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_3471_sum__gp__basic,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),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_ord_atMost(nat,N))) = minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ).
% sum_gp_basic
tff(fact_3472_power__diff__sumr2,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat,Y: A] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = 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_gy(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,N))) ) ).
% power_diff_sumr2
tff(fact_3473_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat,Y: A] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N))) = 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_gz(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).
% diff_power_eq_sum
tff(fact_3474_sum__gp__strict,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: 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)),set_ord_lessThan(nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
& ( ( 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)),set_ord_lessThan(nat,N)) = divide_divide(A,minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),minus_minus(A,one_one(A),X)) ) ) ) ) ).
% sum_gp_strict
tff(fact_3475_sum__power__shift,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M2: nat,N: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( 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,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),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_ord_atMost(nat,minus_minus(nat,N,M2)))) ) ) ) ).
% sum_power_shift
tff(fact_3476_mask__eq__sum__exp,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] : minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N),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)))),collect(nat,aTP_Lamp_bf(nat,fun(nat,bool),N))) ) ).
% mask_eq_sum_exp
tff(fact_3477_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M2: nat,N: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( 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,M2,N))) = minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).
% sum_gp_multiplied
tff(fact_3478_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = 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_ha(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq'
tff(fact_3479_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hb(A,fun(nat,A),A3)),set_ord_atMost(nat,M2)) = 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))),M2)),aa(nat,A,gbinomial(A,minus_minus(A,A3,one_one(A))),M2)) ) ).
% gbinomial_sum_lower_neg
tff(fact_3480_binomial__ring,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = 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_hc(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% binomial_ring
tff(fact_3481_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = 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_hd(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% pochhammer_binomial_sum
tff(fact_3482_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),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_he(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum
tff(fact_3483_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hf(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).
% gbinomial_sum_up_index
tff(fact_3484_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M2: nat,A3: 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_hg(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),set_ord_atMost(nat,M2)) = 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_hh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),set_ord_atMost(nat,M2)) ) ).
% gbinomial_partial_sum_poly
tff(fact_3485_filter__upt__last,axiom,
! [A: $tType,P: fun(A,bool),L: list(A),Js: list(nat),J: nat,I: nat] :
( ( aa(list(nat),list(nat),aa(fun(nat,bool),fun(list(nat),list(nat)),filter2(nat),aa(list(A),fun(nat,bool),aTP_Lamp_gm(fun(A,bool),fun(list(A),fun(nat,bool)),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),J),nil(nat))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I))) ) ) ) ).
% filter_upt_last
tff(fact_3486_double__gauss__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).
% double_gauss_sum
tff(fact_3487_double__arith__series,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,D3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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_hi(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ).
% double_arith_series
tff(fact_3488_sum__gp0,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: 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)),set_ord_atMost(nat,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
& ( ( X != one_one(A) )
=> ( 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_ord_atMost(nat,N)) = 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,N))),minus_minus(A,one_one(A),X)) ) ) ) ) ).
% sum_gp0
tff(fact_3489_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M2: nat,A3: 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_hg(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),set_ord_atMost(nat,M2)) = 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_hj(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),set_ord_atMost(nat,M2)) ) ).
% gbinomial_partial_sum_poly_xpos
tff(fact_3490_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( ( N != one_one(nat) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hk(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_3491_card__lists__length__le,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ei(set(A),fun(nat,fun(list(A),bool)),A4),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),finite_card(A,A4))),set_ord_atMost(nat,N)) ) ) ).
% card_lists_length_le
tff(fact_3492_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).
% double_gauss_sum_from_Suc_0
tff(fact_3493_gauss__sum,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: 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),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% gauss_sum
tff(fact_3494_arith__series,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,D3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_hl(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% arith_series
tff(fact_3495_sum__gp__offset,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,N: 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)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
& ( ( X != one_one(A) )
=> ( 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,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = 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),M2)),minus_minus(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),minus_minus(A,one_one(A),X)) ) ) ) ) ).
% sum_gp_offset
tff(fact_3496_choose__alternating__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hm(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_3497_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: 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)),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% gauss_sum_from_Suc_0
tff(fact_3498_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [R3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fy(A,fun(nat,A),R3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M2)) = 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,M2)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M2))) ) ).
% gchoose_row_sum_weighted
tff(fact_3499_of__nat__code,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_hn(A,A),N,zero_zero(A)) ) ).
% of_nat_code
tff(fact_3500_remove__rev__alt__def,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remove_rev(A,X),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),X)),aa(list(A),list(A),rev(A),Xs)) ).
% remove_rev_alt_def
tff(fact_3501_divmod__step__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_ho(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_3502_Pow__fold,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pow(A,A4) = finite_fold(A,set(set(A)),aTP_Lamp_hp(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)))),A4) ) ) ).
% Pow_fold
tff(fact_3503_length__remdups__concat,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),aa(list(list(A)),list(A),concat(A),Xss2))) = finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss2)))) ).
% length_remdups_concat
tff(fact_3504_remdups__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),remdups(nat),upt(M2,N)) = upt(M2,N) ).
% remdups_upt
tff(fact_3505_fold__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Z2: A] : finite_fold(B,A,F,Z2,bot_bot(set(B))) = Z2 ).
% fold_empty
tff(fact_3506_remdups__eq__nil__iff,axiom,
! [A: $tType,X: list(A)] :
( ( aa(list(A),list(A),remdups(A),X) = nil(A) )
<=> ( X = nil(A) ) ) ).
% remdups_eq_nil_iff
tff(fact_3507_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X: list(A)] :
( ( nil(A) = aa(list(A),list(A),remdups(A),X) )
<=> ( X = nil(A) ) ) ).
% remdups_eq_nil_right_iff
tff(fact_3508_set__remdups,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),remdups(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_remdups
tff(fact_3509_length__remdups__eq,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
<=> ( aa(list(A),list(A),remdups(A),Xs) = Xs ) ) ).
% length_remdups_eq
tff(fact_3510_distinct__remdups,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),remdups(A),Xs))) ).
% distinct_remdups
tff(fact_3511_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),remdups(A),Xs) = Xs )
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ).
% remdups_id_iff_distinct
tff(fact_3512_length__remdups__leq,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_remdups_leq
tff(fact_3513_nested__case__prod__simp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: 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),product_case_prod(B,C,fun(D,A),F),X),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_hq(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),X) ).
% nested_case_prod_simp
tff(fact_3514_remdups__remdups,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),remdups(A),aa(list(A),list(A),remdups(A),Xs)) = aa(list(A),list(A),remdups(A),Xs) ).
% remdups_remdups
tff(fact_3515_remdups_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),remdups(A),nil(A)) = nil(A) ).
% remdups.simps(1)
tff(fact_3516_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),remdups(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),remdups(B),Xs))) = aa(list(A),list(A),remdups(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ).
% remdups_map_remdups
tff(fact_3517_remdups__append2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),remdups(A),Ys))) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).
% remdups_append2
tff(fact_3518_distinct__remdups__id,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),remdups(A),Xs) = Xs ) ) ).
% distinct_remdups_id
tff(fact_3519_remdups__filter,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),remdups(A),Xs)) ).
% remdups_filter
tff(fact_3520_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : aa(list(product_prod(B,C)),list(product_prod(A,product_prod(B,C))),aa(list(A),fun(list(product_prod(B,C)),list(product_prod(A,product_prod(B,C)))),zip(A,product_prod(B,C)),Xs),aa(list(C),list(product_prod(B,C)),aa(list(B),fun(list(C),list(product_prod(B,C))),zip(B,C),Ys),Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),aa(fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),fun(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))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C)),aTP_Lamp_hs(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),aa(list(product_prod(A,C)),list(product_prod(B,product_prod(A,C))),aa(list(B),fun(list(product_prod(A,C)),list(product_prod(B,product_prod(A,C)))),zip(B,product_prod(A,C)),Ys),aa(list(C),list(product_prod(A,C)),aa(list(A),fun(list(C),list(product_prod(A,C))),zip(A,C),Xs),Zs))) ).
% zip_left_commute
tff(fact_3521_insert__remdups,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),aa(list(A),list(A),remdups(A),Xs)) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)) ).
% insert_remdups
tff(fact_3522_remdups_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),remdups(A),Xs) ) )
& ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),remdups(A),Xs)) ) ) ) ).
% remdups.simps(2)
tff(fact_3523_zip__commute,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(list(product_prod(B,A)),list(product_prod(A,B))),map(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B),aTP_Lamp_ht(B,fun(A,product_prod(A,B))))),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Ys),Xs)) ).
% zip_commute
tff(fact_3524_sorted__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),remdups(A),Xs)) ) ) ).
% sorted_remdups
tff(fact_3525_map2__map__map,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),product_case_prod(B,C,A,H)),aa(list(C),list(product_prod(B,C)),aa(list(B),fun(list(C),list(product_prod(B,C))),zip(B,C),aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),F),Xs)),aa(list(D),list(C),aa(fun(D,C),fun(list(D),list(C)),map(D,C),G),Xs))) = aa(list(D),list(A),aa(fun(D,A),fun(list(D),list(A)),map(D,A),aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_hu(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F),G)),Xs) ).
% map2_map_map
tff(fact_3526_remove1__remdups,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),aa(list(A),list(A),remdups(A),Xs)) = aa(list(A),list(A),remdups(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)) ) ) ).
% remove1_remdups
tff(fact_3527_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F),Xs)),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(list(product_prod(C,B)),list(product_prod(A,B))),map(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B),aTP_Lamp_hv(fun(C,A),fun(C,fun(B,product_prod(A,B))),F))),aa(list(B),list(product_prod(C,B)),aa(list(C),fun(list(B),list(product_prod(C,B))),zip(C,B),Xs),Ys)) ).
% zip_map1
tff(fact_3528_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),aa(fun(product_prod(A,C),product_prod(A,B)),fun(list(product_prod(A,C)),list(product_prod(A,B))),map(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B),aTP_Lamp_hw(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),aa(list(C),list(product_prod(A,C)),aa(list(A),fun(list(C),list(product_prod(A,C))),zip(A,C),Xs),Ys)) ).
% zip_map2
tff(fact_3529_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),aa(fun(product_prod(C,D),product_prod(A,B)),fun(list(product_prod(C,D)),list(product_prod(A,B))),map(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_hx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),aa(list(D),list(product_prod(C,D)),aa(list(C),fun(list(D),list(product_prod(C,D))),zip(C,D),Xs),Ys)) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F),Xs)),aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),G),Ys)) ).
% map_prod_fun_zip
tff(fact_3530_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_3531_length__remdups__card__conv,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xs)) = finite_card(A,aa(list(A),set(A),set2(A),Xs)) ).
% length_remdups_card_conv
tff(fact_3532_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F),aa(list(C),list(product_prod(B,C)),aa(list(B),fun(list(C),list(product_prod(B,C))),zip(B,C),aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),G),Xs)),Ys)) = aa(list(product_prod(D,C)),list(A),aa(fun(product_prod(D,C),A),fun(list(product_prod(D,C)),list(A)),map(product_prod(D,C),A),product_case_prod(D,C,A,aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_hy(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G))),aa(list(C),list(product_prod(D,C)),aa(list(D),fun(list(C),list(product_prod(D,C))),zip(D,C),Xs),Ys)) ).
% map_zip_map
tff(fact_3533_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F),aa(list(C),list(product_prod(B,C)),aa(list(B),fun(list(C),list(product_prod(B,C))),zip(B,C),Xs),aa(list(D),list(C),aa(fun(D,C),fun(list(D),list(C)),map(D,C),G),Ys))) = aa(list(product_prod(B,D)),list(A),aa(fun(product_prod(B,D),A),fun(list(product_prod(B,D)),list(A)),map(product_prod(B,D),A),product_case_prod(B,D,A,aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_hz(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G))),aa(list(D),list(product_prod(B,D)),aa(list(B),fun(list(D),list(product_prod(B,D))),zip(B,D),Xs),Ys)) ).
% map_zip_map2
tff(fact_3534_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A4: set(A),F: fun(A,nat)] :
( ( pp(member2(A,A3,A4))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4),aa(A,nat,F,A3)) ) )
& ( ~ pp(member2(A,A3,A4))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4) ) ) ) ).
% sum_diff1_nat
tff(fact_3535_Sup__fold__sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,complete_Sup_Sup(A),A4) = finite_fold(A,A,sup_sup(A),bot_bot(A),A4) ) ) ) ).
% Sup_fold_sup
tff(fact_3536_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = finite_fold(A,A,inf_inf(A),X,A4) ) ) ) ).
% Inf_fin.eq_fold
tff(fact_3537_image__fold__insert,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),set(B),image2(A,B,F),A4) = finite_fold(A,set(B),aTP_Lamp_ia(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A4) ) ) ).
% image_fold_insert
tff(fact_3538_sum__Un__nat,axiom,
! [A: $tType,A4: set(A),B3: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),B3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).
% sum_Un_nat
tff(fact_3539_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F: fun(C,fun(A,C)),B2: C] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),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),aa(fun(C,fun(product_prod(B,A),C)),fun(C,fun(list(product_prod(B,A)),C)),foldl(C,product_prod(B,A)),aTP_Lamp_ic(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F)),B2),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xs),Ys)) = aa(list(A),C,aa(C,fun(list(A),C),aa(fun(C,fun(A,C)),fun(C,fun(list(A),C)),foldl(C,A),F),B2),Ys) ) ) ).
% foldl_snd_zip
tff(fact_3540_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups5270119922927024881d_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xs))) ) ).
% prod.set_conv_list
tff(fact_3541_sum__count__set,axiom,
! [A: $tType,Xs: list(A),X5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5))
=> ( pp(aa(set(A),bool,finite_finite2(A),X5))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X5) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).
% sum_count_set
tff(fact_3542_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B))] :
( pp(aa(set(A),bool,finite_finite2(A),I5))
=> ( ! [X2: A] :
( pp(member2(A,X2,I5))
=> pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A4,X2))) )
=> ( ! [X2: A] :
( pp(member2(A,X2,I5))
=> ! [Xa4: A] :
( pp(member2(A,Xa4,I5))
=> ( ( X2 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X2)),aa(A,set(B),A4,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),A4),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_id(fun(A,set(B)),fun(A,nat),A4)),I5) ) ) ) ) ).
% card_UN_disjoint
tff(fact_3543_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] : unique8689654367752047608divmod(A,bit1(M2),bit0(N)) = aa(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,M2,N)) ) ).
% divmod_algorithm_code(6)
tff(fact_3544_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] : unique8689654367752047608divmod(A,bit0(M2),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_if(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).
% divmod_algorithm_code(5)
tff(fact_3545_listset_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(set(A)),set(list(A)),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_3546_transpose__transpose,axiom,
! [A: $tType,Xs: list(list(A))] :
( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
=> ( transpose(A,transpose(A,Xs)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),takeWhile(list(A)),aTP_Lamp_fj(list(A),bool)),Xs) ) ) ).
% transpose_transpose
tff(fact_3547_Set__filter__fold,axiom,
! [A: $tType,A4: set(A),P: fun(A,bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( filter3(A,P,A4) = finite_fold(A,set(A),aTP_Lamp_ig(fun(A,bool),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A4) ) ) ).
% Set_filter_fold
tff(fact_3548_takeWhile__idem,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ).
% takeWhile_idem
tff(fact_3549_member__filter,axiom,
! [A: $tType,X: A,P: fun(A,bool),A4: set(A)] :
( pp(member2(A,X,filter3(A,P,A4)))
<=> ( pp(member2(A,X,A4))
& pp(aa(A,bool,P,X)) ) ) ).
% member_filter
tff(fact_3550_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = Xs )
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X4)) ) ) ).
% takeWhile_eq_all_conv
tff(fact_3551_takeWhile__append1,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ) ).
% takeWhile_append1
tff(fact_3552_takeWhile__append2,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Ys)) ) ) ).
% takeWhile_append2
tff(fact_3553_Set_Ofilter__def,axiom,
! [A: $tType,P: fun(A,bool),A4: set(A)] : filter3(A,P,A4) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ih(fun(A,bool),fun(set(A),fun(A,bool)),P),A4)) ).
% Set.filter_def
tff(fact_3554_distinct__takeWhile,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))) ) ).
% distinct_takeWhile
tff(fact_3555_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),nil(A)) = nil(A) ).
% takeWhile.simps(1)
tff(fact_3556_set__takeWhileD,axiom,
! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X)) ) ) ).
% set_takeWhileD
tff(fact_3557_takeWhile__cong,axiom,
! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
( ( L = K )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),L)))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) ) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),Q),K) ) ) ) ).
% takeWhile_cong
tff(fact_3558_inj__split__Cons,axiom,
! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_dn(list(A),fun(A,list(A)))),X5) ).
% inj_split_Cons
tff(fact_3559_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) ) ) ).
% takeWhile.simps(2)
tff(fact_3560_takeWhile__tail,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A),L: list(A)] :
( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),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),L))) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ).
% takeWhile_tail
tff(fact_3561_length__takeWhile__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_takeWhile_le
tff(fact_3562_sorted__takeWhile,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) ) ) ).
% sorted_takeWhile
tff(fact_3563_takeWhile__eq__take,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),Xs) ).
% takeWhile_eq_take
tff(fact_3564_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = nil(A) )
<=> ( ( Xs = nil(A) )
| ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).
% takeWhile_eq_Nil_iff
tff(fact_3565_nth__length__takeWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))) ) ).
% nth_length_takeWhile
tff(fact_3566_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).
% takeWhile_nth
tff(fact_3567_drop__takeWhile,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L))))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),L)) ) ) ).
% drop_takeWhile
tff(fact_3568_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : aa(list(product_prod(B,C)),list(product_prod(A,product_prod(B,C))),aa(list(A),fun(list(product_prod(B,C)),list(product_prod(A,product_prod(B,C)))),zip(A,product_prod(B,C)),Xs),aa(list(C),list(product_prod(B,C)),aa(list(B),fun(list(C),list(product_prod(B,C))),zip(B,C),Ys),Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),aa(fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),fun(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))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C)),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_ii(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),aa(list(C),list(product_prod(product_prod(A,B),C)),aa(list(product_prod(A,B)),fun(list(C),list(product_prod(product_prod(A,B),C))),zip(product_prod(A,B),C),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)),Zs)) ).
% zip_assoc
tff(fact_3569_listrel__def,axiom,
! [B: $tType,A: $tType,X3: set(product_prod(A,B))] : listrel(A,B,X3) = collect(product_prod(list(A),list(B)),product_case_prod(list(A),list(B),bool,listrelp(A,B,aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),X3)))) ).
% listrel_def
tff(fact_3570_takeWhile__not__last,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aTP_Lamp_ij(list(A),fun(A,bool),Xs)),Xs) = aa(list(A),list(A),butlast(A),Xs) ) ) ).
% takeWhile_not_last
tff(fact_3571_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)))) ) ) ).
% length_takeWhile_less_P_nth
tff(fact_3572_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( ( I = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L)) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),L)))
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),J2))) )
& ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I))) ) ) ) ).
% eq_len_takeWhile_conv
tff(fact_3573_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),J2))) ) ) ) ).
% less_length_takeWhile_conv
tff(fact_3574_filter__set,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter3(A,P,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ).
% filter_set
tff(fact_3575_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) ) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ) ).
% takeWhile_eq_take_P_nth
tff(fact_3576_listset_Osimps_I2_J,axiom,
! [A: $tType,A4: set(A),As3: list(set(A))] : aa(list(set(A)),set(list(A)),listset(A),aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),A4),As3)) = aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),A4),aa(list(set(A)),set(list(A)),listset(A),As3)) ).
% listset.simps(2)
tff(fact_3577_lexordp__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
<=> pp(member2(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,collect(product_prod(A,A),product_case_prod(A,A,bool,ord_less(A)))))) ) ) ).
% lexordp_conv_lexord
tff(fact_3578_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F: fun(A,fun(C,C)),B2: C] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs)))
=> ( aa(C,C,aa(list(product_prod(B,A)),fun(C,C),aa(fun(product_prod(B,A),fun(C,C)),fun(list(product_prod(B,A)),fun(C,C)),foldr(product_prod(B,A),C),product_case_prod(B,A,fun(C,C),aTP_Lamp_ik(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F))),aa(list(A),list(product_prod(B,A)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xs),Ys)),B2) = aa(C,C,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),foldr(A,C),F),Ys),B2) ) ) ).
% foldr_snd_zip
tff(fact_3579_inter__Set__filter,axiom,
! [A: $tType,B3: set(A),A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = filter3(A,aTP_Lamp_a(set(A),fun(A,bool),A4),B3) ) ) ).
% inter_Set_filter
tff(fact_3580_lenlex__conv,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_il(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).
% lenlex_conv
tff(fact_3581_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),T4: A] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_im(fun(B,A),fun(A,fun(B,bool)),F),T4)),Xs) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),aa(A,fun(B,bool),aTP_Lamp_im(fun(B,A),fun(A,fun(B,bool)),F),T4)),Xs) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
tff(fact_3582_List_Olexordp__def,axiom,
! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( lexordp(A,R3,Xs,Ys)
<=> pp(member2(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,collect(product_prod(A,A),product_case_prod(A,A,bool,R3))))) ) ).
% List.lexordp_def
tff(fact_3583_listrel1p__def,axiom,
! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( listrel1p(A,R3,Xs,Ys)
<=> pp(member2(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,collect(product_prod(A,A),product_case_prod(A,A,bool,R3))))) ) ).
% listrel1p_def
tff(fact_3584_transpose_Opelims,axiom,
! [A: $tType,X: list(list(A)),Y: list(list(A))] :
( ( transpose(A,X) = Y )
=> ( accp(list(list(A)),transpose_rel(A),X)
=> ( ( ( X = nil(list(A)) )
=> ( ( Y = nil(list(A)) )
=> ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
=> ( ! [Xss: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss) )
=> ( ( Y = transpose(A,Xss) )
=> ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) ) )
=> ~ ! [X2: A,Xs2: list(A),Xss: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss) )
=> ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_er(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss))))) )
=> ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss)) ) ) ) ) ) ) ).
% transpose.pelims
tff(fact_3585_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),Xss2: list(list(A))] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss2))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss2)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_er(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ).
% transpose.psimps(3)
tff(fact_3586_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_in(A,fun(A,bool),X)),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),ord_less_eq(A),X)),Xs)))) ) ).
% quicksort.simps(2)
tff(fact_3587_set__quicksort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),linorder_quicksort(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).
% set_quicksort
tff(fact_3588_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( accp(list(list(A)),transpose_rel(A),nil(list(A)))
=> ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).
% transpose.psimps(1)
tff(fact_3589_quicksort_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(list(A),list(A),linorder_quicksort(A),nil(A)) = nil(A) ) ) ).
% quicksort.simps(1)
tff(fact_3590_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss2: list(list(A))] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) = transpose(A,Xss2) ) ) ).
% transpose.psimps(2)
tff(fact_3591_sorted__quicksort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_quicksort(A),Xs)) ) ).
% sorted_quicksort
tff(fact_3592_transpose_Opinduct,axiom,
! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
( accp(list(list(A)),transpose_rel(A),A0)
=> ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
=> pp(aa(list(list(A)),bool,P,nil(list(A)))) )
=> ( ! [Xss: list(list(A))] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss))
=> ( pp(aa(list(list(A)),bool,P,Xss))
=> pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss))) ) )
=> ( ! [X2: A,Xs2: list(A),Xss: list(list(A))] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss))
=> ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))))),Xss)))))
=> pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss))) ) )
=> pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).
% transpose.pinduct
tff(fact_3593_quicksort_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Y: list(A)] :
( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != nil(A) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_in(A,fun(A,bool),X2)),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),ord_less_eq(A),X2)),Xs2)))) ) ) ) ) ) ).
% quicksort.elims
tff(fact_3594_quicksort_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Y: list(A)] :
( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
=> ( accp(list(A),linord6200660962353139674rt_rel(A),X)
=> ( ( ( X = nil(A) )
=> ( ( Y = nil(A) )
=> ~ accp(list(A),linord6200660962353139674rt_rel(A),nil(A)) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_in(A,fun(A,bool),X2)),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),ord_less_eq(A),X2)),Xs2)))) )
=> ~ accp(list(A),linord6200660962353139674rt_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) ) ) ) ) ) ).
% quicksort.pelims
tff(fact_3595_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [F2: set(A),I5: set(A),F: fun(A,B),I: A] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,bool)),I5),F))),F2))
=> ( ( pp(member2(A,I,I5))
=> ( groups1027152243600224163dd_sum(A,B,F,minus_minus(set(A),I5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = minus_minus(B,groups1027152243600224163dd_sum(A,B,F,I5),aa(A,B,F,I)) ) )
& ( ~ pp(member2(A,I,I5))
=> ( groups1027152243600224163dd_sum(A,B,F,minus_minus(set(A),I5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F,I5) ) ) ) ) ) ) ).
% sum_diff1'_aux
tff(fact_3596_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs) = aa(list(B),list(B),aa(fun(B,fun(list(B),list(B))),fun(list(B),list(B)),aa(list(B),fun(fun(B,fun(list(B),list(B))),fun(list(B),list(B))),case_list(list(B),B),nil(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_it(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),F),Xs)),Xs) ) ).
% sort_key_by_quicksort_code
tff(fact_3597_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [A3: B,Xs: list(B),F: fun(B,A)] :
( pp(member2(B,A3,aa(list(B),set(B),set2(B),Xs)))
=> ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> ( ( aa(list(B),B,hd(B),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_iu(B,fun(fun(B,A),fun(B,bool)),A3),F)),Xs)) = A3 )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),A3),Xs)) = Xs ) ) ) ) ) ).
% insort_key_remove1
tff(fact_3598_sort__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),linorder_sort_key(nat,nat),aTP_Lamp_fa(nat,nat)),upt(M2,N)) = upt(M2,N) ).
% sort_upt
tff(fact_3599_sort__upto,axiom,
! [I: int,J: int] : aa(list(int),list(int),aa(fun(int,int),fun(list(int),list(int)),linorder_sort_key(int,int),aTP_Lamp_iv(int,int)),upto(I,J)) = upto(I,J) ).
% sort_upto
tff(fact_3600_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),nil(B)) = nil(B) ) ).
% sort_key_simps(1)
tff(fact_3601_set__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs)) = aa(list(B),set(B),set2(B),Xs) ) ).
% set_sort
tff(fact_3602_length__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).
% length_sort
tff(fact_3603_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] :
( pp(aa(list(B),bool,distinct(B),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs)))
<=> pp(aa(list(B),bool,distinct(B),Xs)) ) ) ).
% distinct_sort
tff(fact_3604_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [X: B,F: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = Xs ) ).
% remove1_insort_key
tff(fact_3605_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [P4: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P4,bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty'
tff(fact_3606_length__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).
% length_insort
tff(fact_3607_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs)) ) ).
% sort_key_simps(2)
tff(fact_3608_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ~ pp(member2(A,X,A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A4)) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_3609_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),minus_minus(set(A),A4,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_3610_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [P: fun(B,bool),X: B,F: fun(B,A),Xs: list(B)] :
( ~ pp(aa(B,bool,P,X))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs) ) ) ) ).
% filter_insort_triv
tff(fact_3611_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Y: B,Xs: list(B)] :
( ( aa(B,A,F,X) != aa(B,A,F,Y) )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),Y),Xs)) ) ) ) ).
% insort_key_left_comm
tff(fact_3612_insort__left__comm,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),Xs)) ) ).
% insort_left_comm
tff(fact_3613_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [C3: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),aTP_Lamp_iw(B,fun(A,B),C3)),Xs) = Xs ) ).
% sort_key_const
tff(fact_3614_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),A3: B,Xs: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),A3),Xs) != nil(B) ) ).
% insort_not_Nil
tff(fact_3615_filter__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [P: fun(B,bool),F: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs)) = aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) ) ).
% filter_sort
tff(fact_3616_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(fun(B,fun(list(B),list(B))),fun(list(B),fun(list(B),list(B))),foldr(B,list(B)),linorder_insort_key(B,A,F)),Xs),nil(B)) ) ).
% sort_key_def
tff(fact_3617_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(B,fun(A,bool),aTP_Lamp_ix(fun(A,B),fun(B,fun(A,bool)),F),K)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(B,fun(A,bool),aTP_Lamp_ix(fun(A,B),fun(B,fun(A,bool)),F),K)),Xs) ) ).
% sort_key_stable
tff(fact_3618_sort__quicksort,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)) = linorder_quicksort(A) ) ) ).
% sort_quicksort
tff(fact_3619_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Y: B,Ys: list(B)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y)))
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y)))
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Ys)) ) ) ) ) ).
% insort_key.simps(2)
tff(fact_3620_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),nil(B)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)) ) ).
% insort_key.simps(1)
tff(fact_3621_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),aa(list(B),set(B),set2(B),Xs)) ) ).
% set_insort_key
tff(fact_3622_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( pp(aa(list(B),bool,distinct(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)))
<=> ( ~ pp(member2(B,X,aa(list(B),set(B),set2(B),Xs)))
& pp(aa(list(B),bool,distinct(B),Xs)) ) ) ) ).
% distinct_insort
tff(fact_3623_sorted__insort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),Xs))
<=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% sorted_insort
tff(fact_3624_sorted__sort__id,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)),Xs) = Xs ) ) ) ).
% sorted_sort_id
tff(fact_3625_sorted__sort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)),Xs)) ) ).
% sorted_sort
tff(fact_3626_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),nil(A),A4) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_3627_sort__mergesort,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)) = mergesort(A) ) ) ).
% sort_mergesort
tff(fact_3628_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% sort_mergesort_by_rel
tff(fact_3629_insort__insert__insort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( linord329482645794927042rt_key(A,A,aTP_Lamp_az(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),Xs) ) ) ) ).
% insort_insert_insort
tff(fact_3630_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),F: fun(B,A),A3: B] :
( ! [X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,A3)),aa(B,A,F,X2))) )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),A3),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xs) ) ) ) ).
% insort_is_Cons
tff(fact_3631_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)))
<=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ) ) ).
% sorted_insort_key
tff(fact_3632_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs))) ) ).
% sorted_sort_key
tff(fact_3633_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( ( pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X,Xs) = Xs ) )
& ( ~ pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs) ) ) ) ) ).
% insort_insert_key_def
tff(fact_3634_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( ~ pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs) ) ) ) ).
% insort_insert_insort_key
tff(fact_3635_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)),aa(list(A),list(A),remdups(A),Xs)) ) ).
% sorted_list_of_set_sort_remdups
tff(fact_3636_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs))))
<=> ( ~ pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))))
& pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))) ) ) ) ).
% distinct_insort_key
tff(fact_3637_filter__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))
=> ( pp(aa(B,bool,P,X))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),X),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) ) ) ) ) ).
% filter_insort
tff(fact_3638_insort__remove1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,Xs: list(A)] :
( pp(member2(A,A3,aa(list(A),set(A),set2(A),Xs)))
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),A3),Xs)) = Xs ) ) ) ) ).
% insort_remove1
tff(fact_3639_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Pivot: A] : linorder_part(B,A,F,Pivot,nil(B)) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),nil(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B))) ) ).
% part_code(1)
tff(fact_3640_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),minus_minus(set(A),A4,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_3641_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),A3: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3)) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),A3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).
% sorted_insort_is_snoc
tff(fact_3642_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [I5: set(A),F: fun(A,B),I: A] :
( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,bool)),I5),F))))
=> ( ( pp(member2(A,I,I5))
=> ( groups1027152243600224163dd_sum(A,B,F,minus_minus(set(A),I5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = minus_minus(B,groups1027152243600224163dd_sum(A,B,F,I5),aa(A,B,F,I)) ) )
& ( ~ pp(member2(A,I,I5))
=> ( groups1027152243600224163dd_sum(A,B,F,minus_minus(set(A),I5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F,I5) ) ) ) ) ) ).
% sum_diff1'
tff(fact_3643_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Pivot: A,X: B,Xs: list(B)] : linorder_part(B,A,F,Pivot,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B))),product_case_prod(list(B),product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_iz(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),F),Pivot),X)),linorder_part(B,A,F,Pivot,Xs)) ) ).
% part_code(2)
tff(fact_3644_Bleast__code,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P)),aTP_Lamp_ja(A,fun(list(A),A))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)),Xs))) ) ).
% Bleast_code
tff(fact_3645_Fpow__Pow__finite,axiom,
! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A4)),collect(set(A),finite_finite2(A))) ).
% Fpow_Pow_finite
tff(fact_3646_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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_jb(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).
% set_zip
tff(fact_3647_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
( pp(member2(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,R3)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X4: product_prod(A,B)] :
( pp(member2(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
=> pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),R3)),X4)) ) ) ) ).
% listrel_iff_zip
tff(fact_3648_ball__empty,axiom,
! [A: $tType,P: fun(A,bool),X3: A] :
( pp(member2(A,X3,bot_bot(set(A))))
=> pp(aa(A,bool,P,X3)) ) ).
% ball_empty
tff(fact_3649_eq__or__mem__image__simp,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A3: B,B3: set(B)] : collect(A,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_jc(fun(B,A),fun(B,fun(set(B),fun(A,bool))),F),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F,A3)),collect(A,aa(set(B),fun(A,bool),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(A,bool)),F),B3))) ).
% eq_or_mem_image_simp
tff(fact_3650_set__Cons__def,axiom,
! [A: $tType,A4: set(A),XS: set(list(A))] : aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),A4),XS) = collect(list(A),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_je(set(A),fun(set(list(A)),fun(list(A),bool)),A4),XS)) ).
% set_Cons_def
tff(fact_3651_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,bool)] : collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_jf(fun(B,A),fun(fun(B,bool),fun(A,bool)),F),P)) = aa(set(B),set(A),image2(B,A,F),collect(B,P)) ).
% setcompr_eq_image
tff(fact_3652_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] : collect(A,aa(set(B),fun(A,bool),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(A,bool)),F),A4)) = aa(set(B),set(A),image2(B,A,F),A4) ).
% Setcompr_eq_image
tff(fact_3653_empty__in__Fpow,axiom,
! [A: $tType,A4: set(A)] : pp(member2(set(A),bot_bot(set(A)),finite_Fpow(A,A4))) ).
% empty_in_Fpow
tff(fact_3654_Field__not__elem,axiom,
! [A: $tType,V: A,R: set(product_prod(A,A))] :
( ~ pp(member2(A,V,field2(A,R)))
=> ! [X3: product_prod(A,A)] :
( pp(member2(product_prod(A,A),X3,R))
=> pp(aa(product_prod(A,A),bool,product_case_prod(A,A,bool,aTP_Lamp_jg(A,fun(A,fun(A,bool)),V)),X3)) ) ) ).
% Field_not_elem
tff(fact_3655_listrel1__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : listrel1(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_jh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).
% listrel1_def
tff(fact_3656_lexord__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : lexord(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_ji(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).
% lexord_def
tff(fact_3657_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Ys: list(A)] :
( sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
<=> ( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Ys)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X),X4)) )
& sorted_wrt(A,P,Ys) ) ) ).
% sorted_wrt.simps(2)
tff(fact_3658_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ sorted_wrt(A,X,Xa)
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ( ! [Xa4: A] :
( pp(member2(A,Xa4,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa4)) )
& sorted_wrt(A,X,Ys2) ) ) ) ).
% sorted_wrt.elims(3)
tff(fact_3659_Un__interval,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [B1: A,B22: A,B32: A,F: fun(A,B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B1),B22))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B22),B32))
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),collect(B,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_jj(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B22),F))),collect(B,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_jj(A,fun(A,fun(fun(A,B),fun(B,bool))),B22),B32),F))) = collect(B,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_jj(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B32),F)) ) ) ) ) ).
% Un_interval
tff(fact_3660_lex__conv,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_jk(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).
% lex_conv
tff(fact_3661_takeWhile__append,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Ys)) ) )
& ( ~ ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X3)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ) ).
% takeWhile_append
tff(fact_3662_list__collect__set__def,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),collect(set(A),aa(list(B),fun(set(A),bool),aTP_Lamp_jl(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F),L))) ).
% list_collect_set_def
tff(fact_3663_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( sorted_wrt(A,X,Xa)
=> ( ( Xa != nil(A) )
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ~ ( ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa3)) )
& sorted_wrt(A,X,Ys2) ) ) ) ) ).
% sorted_wrt.elims(2)
tff(fact_3664_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( sorted_wrt(A,X,Xa)
<=> pp(Y) )
=> ( ( ( Xa = nil(A) )
=> ~ pp(Y) )
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ( pp(Y)
<=> ~ ( ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa2)) )
& sorted_wrt(A,X,Ys2) ) ) ) ) ) ).
% sorted_wrt.elims(1)
tff(fact_3665_set__conv__nth,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_jm(list(A),fun(A,bool),Xs)) ).
% set_conv_nth
tff(fact_3666_inf__Sup1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic5882676163264333800up_fin(A,A4)) = lattic5882676163264333800up_fin(A,collect(A,aa(A,fun(A,bool),aTP_Lamp_jn(set(A),fun(A,fun(A,bool)),A4),X))) ) ) ) ) ).
% inf_Sup1_distrib
tff(fact_3667_inf__Sup2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic5882676163264333800up_fin(A,A4)),lattic5882676163264333800up_fin(A,B3)) = lattic5882676163264333800up_fin(A,collect(A,aa(set(A),fun(A,bool),aTP_Lamp_jo(set(A),fun(set(A),fun(A,bool)),A4),B3))) ) ) ) ) ) ) ).
% inf_Sup2_distrib
tff(fact_3668_sup__Inf2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( B3 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_jp(set(A),fun(set(A),fun(A,bool)),A4),B3))) ) ) ) ) ) ) ).
% sup_Inf2_distrib
tff(fact_3669_sup__Inf1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_jq(set(A),fun(A,fun(A,bool)),A4),X))) ) ) ) ) ).
% sup_Inf1_distrib
tff(fact_3670_list__eq__iff__zip__eq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = Ys )
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
& ! [X4: product_prod(A,A)] :
( pp(member2(product_prod(A,A),X4,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Xs),Ys))))
=> pp(aa(product_prod(A,A),bool,product_case_prod(A,A,bool,fequal(A)),X4)) ) ) ) ).
% list_eq_iff_zip_eq
tff(fact_3671_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
( ! [X2: product_prod(list(A),list(A))] :
( pp(member2(product_prod(list(A),list(A)),X2,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),aa(list(list(A)),list(product_prod(list(A),list(A))),aa(list(list(A)),fun(list(list(A)),list(product_prod(list(A),list(A)))),zip(list(A),list(A)),Xs),Ys))))
=> pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_jr(list(A),fun(list(A),bool))),X2)) )
=> ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
=> ( ( aa(list(list(A)),list(A),concat(A),Xs) = aa(list(list(A)),list(A),concat(A),Ys) )
<=> ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
tff(fact_3672_concat__injective,axiom,
! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
( ( aa(list(list(A)),list(A),concat(A),Xs) = aa(list(list(A)),list(A),concat(A),Ys) )
=> ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
=> ( ! [X2: product_prod(list(A),list(A))] :
( pp(member2(product_prod(list(A),list(A)),X2,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),aa(list(list(A)),list(product_prod(list(A),list(A))),aa(list(list(A)),fun(list(list(A)),list(product_prod(list(A),list(A)))),zip(list(A),list(A)),Xs),Ys))))
=> pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_jr(list(A),fun(list(A),bool))),X2)) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
tff(fact_3673_list__collect__set__alt,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),collect(set(A),aa(list(B),fun(set(A),bool),aTP_Lamp_js(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F),L))) ).
% list_collect_set_alt
tff(fact_3674_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),L)) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_jt(nat,fun(list(A),fun(A,bool)),N),L)) ).
% set_drop_conv
tff(fact_3675_lexn__conv,axiom,
! [A: $tType,R3: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_ju(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R3),N))) ).
% lexn_conv
tff(fact_3676_finite__inf__Sup,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [A3: A,A4: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_jv(A,fun(set(A),fun(A,bool)),A3),A4))) ) ).
% finite_inf_Sup
tff(fact_3677_pairself__image__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,fun(B,bool))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),collect(product_prod(B,B),product_case_prod(B,B,bool,P))) = collect(product_prod(A,A),aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_jw(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),F),P)) ).
% pairself_image_eq
tff(fact_3678_brk__rel__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : brk_rel(A,B,R) = aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B))),aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),fun(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B)))),sup_sup(set(product_prod(product_prod(bool,A),product_prod(bool,B)))),collect(product_prod(product_prod(bool,A),product_prod(bool,B)),aTP_Lamp_jx(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),R))),collect(product_prod(product_prod(bool,A),product_prod(bool,B)),aTP_Lamp_jy(product_prod(product_prod(bool,A),product_prod(bool,B)),bool))) ).
% brk_rel_def
tff(fact_3679_lexn_Osimps_I1_J,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).
% lexn.simps(1)
tff(fact_3680_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,B2: A] : aa(product_prod(A,A),product_prod(B,B),pairself(A,B,F),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,A3)),aa(A,B,F,B2)) ).
% pairself.simps
tff(fact_3681_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X: fun(A,B),Xa: product_prod(A,A),Y: product_prod(B,B)] :
( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa) = Y )
=> ~ ! [A5: A,B4: A] :
( ( Xa = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4) )
=> ( Y != aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A5)),aa(A,B,X,B4)) ) ) ) ).
% pairself.elims
tff(fact_3682_Pow__Compl,axiom,
! [A: $tType,A4: set(A)] : pow(A,aa(set(A),set(A),uminus_uminus(set(A)),A4)) = collect(set(A),aTP_Lamp_jz(set(A),fun(set(A),bool),A4)) ).
% Pow_Compl
tff(fact_3683_lexn__length,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),N: nat] :
( pp(member2(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),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)))
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
& ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).
% lexn_length
tff(fact_3684_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X: fun(A,B),Xa: product_prod(A,A),Y: product_prod(B,B)] :
( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa) = Y )
=> ( accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X),Xa))
=> ~ ! [A5: A,B4: A] :
( ( Xa = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4) )
=> ( ( Y = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A5)),aa(A,B,X,B4)) )
=> ~ accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4))) ) ) ) ) ).
% pairself.pelims
tff(fact_3685_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ord(A)
=> ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_ka(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).
% ord_class.lexordp_def
tff(fact_3686_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),B3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),S))
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
=> ( finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = finite_fold(A,B,F,finite_fold(A,B,F,Z2,A4),B3) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
tff(fact_3687_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),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_kb(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum_funpow
tff(fact_3688_funpow__times__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [F: fun(A,nat),X: A] : compow(fun(A,A),aa(A,nat,F,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,F,X))) ) ).
% funpow_times_power
tff(fact_3689_rotate__def,axiom,
! [A: $tType,N: nat] : aa(nat,fun(list(A),list(A)),rotate(A),N) = compow(fun(list(A),list(A)),N,rotate1(A)) ).
% rotate_def
tff(fact_3690_of__nat__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,compow(fun(A,A),N,aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).
% of_nat_def
tff(fact_3691_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = 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))),A3) ) ).
% numeral_add_unfold_funpow
tff(fact_3692_butlast__power,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),compow(fun(list(A),list(A)),N,butlast(A)),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs),N)),Xs) ).
% butlast_power
tff(fact_3693_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_3694_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,A4: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),S))
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,minus_minus(set(A),A4,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_3695_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),X: A,Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S))
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( finite_fold(A,B,F,Z2,A4) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,minus_minus(set(A),A4,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_3696_ord_Olexordp__def,axiom,
! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_kc(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).
% ord.lexordp_def
tff(fact_3697_enumerate__Suc,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = 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_kd(set(A),fun(A,bool),S))),bot_bot(set(A)))),N) ) ).
% enumerate_Suc
tff(fact_3698_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_ke(A,fun(A,bool)),aTP_Lamp_kf(A,fun(A,bool))) ) ).
% sup_bot.semilattice_neutr_order_axioms
tff(fact_3699_min__list_Oelims,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: list(A),Y: A] :
( ( min_list(A,X) = Y )
=> ( ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X2),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_dq(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) ) )
=> ~ ( ( X = nil(A) )
=> ( Y != undefined(A) ) ) ) ) ) ).
% min_list.elims
tff(fact_3700_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Xs: list(A),Y: A,Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),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)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Ys)) ) ) ) ).
% ord.lexordp_simps(3)
tff(fact_3701_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),nil(A)),Ys))
<=> ( Ys != nil(A) ) ) ).
% ord.lexordp_simps(1)
tff(fact_3702_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),nil(A))) ).
% ord.lexordp_simps(2)
tff(fact_3703_LeastI,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),K: A] :
( pp(aa(A,bool,P,K))
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% LeastI
tff(fact_3704_LeastI2__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),Q: fun(A,bool)] :
( ? [X_13: A] : pp(aa(A,bool,P,X_13))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_ex
tff(fact_3705_LeastI__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool)] :
( ? [X_13: A] : pp(aa(A,bool,P,X_13))
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% LeastI_ex
tff(fact_3706_LeastI2,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,A3))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2
tff(fact_3707_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] :
( ! [X2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Xs)) ) ).
% ord.lexordp_irreflexive
tff(fact_3708_ord_Olexordp_Ocong,axiom,
! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = lexordp2(A,Less) ).
% ord.lexordp.cong
tff(fact_3709_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( Z2 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
tff(fact_3710_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
tff(fact_3711_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),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))) ) ).
% ord.lexordp.Cons
tff(fact_3712_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Ys))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),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))) ) ) ) ).
% ord.lexordp.Cons_eq
tff(fact_3713_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),Q: fun(A,bool)] :
( ? [X_13: A] : pp(aa(A,bool,P,X_13))
=> ( ! [A5: A] :
( pp(aa(A,bool,P,A5))
=> ( ! [B11: A] :
( pp(aa(A,bool,P,B11))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B11)) )
=> pp(aa(A,bool,Q,A5)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_wellorder_ex
tff(fact_3714_LeastI2__wellorder,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,A3))
=> ( ! [A5: A] :
( pp(aa(A,bool,P,A5))
=> ( ! [B11: A] :
( pp(aa(A,bool,P,B11))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B11)) )
=> pp(aa(A,bool,Q,A5)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_wellorder
tff(fact_3715_Least__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A] :
( pp(aa(A,bool,P,X))
=> ( ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
=> ( ord_Least(A,P) = X ) ) ) ) ).
% Least_equality
tff(fact_3716_LeastI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4)) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).
% LeastI2_order
tff(fact_3717_Least1__le,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),Z2: A] :
( ? [X3: A] :
( pp(aa(A,bool,P,X3))
& ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y2)) )
& ! [Y2: A] :
( ( pp(aa(A,bool,P,Y2))
& ! [Ya2: A] :
( pp(aa(A,bool,P,Ya2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Ya2)) ) )
=> ( Y2 = X3 ) ) )
=> ( pp(aa(A,bool,P,Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),Z2)) ) ) ) ).
% Least1_le
tff(fact_3718_Least1I,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool)] :
( ? [X3: A] :
( pp(aa(A,bool,P,X3))
& ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y2)) )
& ! [Y2: A] :
( ( pp(aa(A,bool,P,Y2))
& ! [Ya2: A] :
( pp(aa(A,bool,P,Ya2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Ya2)) ) )
=> ( Y2 = X3 ) ) )
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% Least1I
tff(fact_3719_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Us2: list(A),Vs: list(A),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us2),Vs))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ).
% ord.lexordp_append_leftI
tff(fact_3720_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A),Us2: list(A),Vs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
=> ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),A5))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us2),Vs)) ) ) ).
% ord.lexordp_append_leftD
tff(fact_3721_ord_Olexordp__into__lexordp__eq,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Ys))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp_eq(A,Less),Xs),Ys)) ) ).
% ord.lexordp_into_lexordp_eq
tff(fact_3722_Least__le,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),K: A] :
( pp(aa(A,bool,P,K))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),K)) ) ) ).
% Least_le
tff(fact_3723_not__less__Least,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [K: A,P: fun(A,bool)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),ord_Least(A,P)))
=> ~ pp(aa(A,bool,P,K)) ) ) ).
% not_less_Least
tff(fact_3724_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Y: A,Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ).
% ord.lexordp.Nil
tff(fact_3725_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),A1),A22))
=> ( ( ( A1 = nil(A) )
=> ! [Y2: A,Ys2: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys2: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X2),Y2)) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Ys2: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X2),Y2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs2),Ys2)) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
tff(fact_3726_ord_Olexordp_Osimps,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),A1),A22))
<=> ( ? [Y3: A,Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y3),X4))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs3),Ys3)) ) ) ) ).
% ord.lexordp.simps
tff(fact_3727_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Us2: list(A),Xs: list(A),Ys: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),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),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),Y),Ys)))) ) ).
% ord.lexordp_append_left_rightI
tff(fact_3728_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list(A),Less: fun(A,fun(A,bool)),Xs: list(A)] :
( ( Ys != nil(A) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))) ) ).
% ord.lexordp_append_rightI
tff(fact_3729_hd__def,axiom,
! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),undefined(A)),aTP_Lamp_kg(A,fun(list(A),A))),List) ).
% hd_def
tff(fact_3730_Bleast__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [S: set(A),P: fun(A,bool)] : bleast(A,S,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_kh(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).
% Bleast_def
tff(fact_3731_abort__Bleast__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [S: set(A),P: fun(A,bool)] : abort_Bleast(A,S,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_kh(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).
% abort_Bleast_def
tff(fact_3732_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X: fun(A,fun(B,C)),Xa: list(A),Xb: list(B),Y: list(C)] :
( ( zipf(A,B,C,X,Xa,Xb) = Y )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( Y != nil(C) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( Y != aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A5),B4)),zipf(A,B,C,X,As,Bs2)) ) ) )
=> ( ( ? [V2: A,Va2: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)
=> ( ( Xb = nil(B) )
=> ( Y != undefined(list(C)) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V2: B,Va2: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)
=> ( Y != undefined(list(C)) ) ) ) ) ) ) ) ).
% zipf.elims
tff(fact_3733_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: fun(A,B),Xa: list(A),Y: A] :
( ( arg_min_list(A,B,X,Xa) = Y )
=> ( ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ( Y != X2 ) )
=> ( ! [X2: A,Y2: A,Zs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)) )
=> ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)))),X2,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3))) ) )
=> ~ ( ( Xa = nil(A) )
=> ( Y != undefined(A) ) ) ) ) ) ) ).
% arg_min_list.elims
tff(fact_3734_min__list_Opelims,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: list(A),Y: A] :
( ( min_list(A,X) = Y )
=> ( accp(list(A),min_list_rel(A),X)
=> ( ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X2),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_dq(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) )
=> ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) )
=> ~ ( ( X = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).
% min_list.pelims
tff(fact_3735_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X: fun(A,fun(B,C)),Xa: list(A),Xb: list(B),Y: list(C)] :
( ( zipf(A,B,C,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( ( Y = nil(C) )
=> ~ accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( ( Y = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A5),B4)),zipf(A,B,C,X,As,Bs2)) )
=> ~ accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))) ) ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xb = nil(B) )
=> ( ( Y = undefined(list(C)) )
=> ~ accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) )
=> ( ( Y = undefined(list(C)) )
=> ~ accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)))) ) ) ) ) ) ) ) ) ).
% zipf.pelims
tff(fact_3736_filter__rev__alt,axiom,
! [A: $tType,P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),P),L) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),rev(A),L)) ).
% filter_rev_alt
tff(fact_3737_filter__rev__aux__alt,axiom,
! [A: $tType,A3: list(A),P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),rev(A),L))),A3) ).
% filter_rev_aux_alt
tff(fact_3738_filter__rev__def,axiom,
! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).
% filter_rev_def
tff(fact_3739_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,A3: list(A),Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A3)),P),Xs) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),Xs) ) ) ) ).
% filter_rev_aux.simps(2)
tff(fact_3740_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A3: list(A),P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),nil(A)) = A3 ).
% filter_rev_aux.simps(1)
tff(fact_3741_precise__extr__pure_I2_J,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),P: bool] :
( precise(A,B,aa(bool,fun(A,fun(B,assn)),aTP_Lamp_ki(fun(A,fun(B,assn)),fun(bool,fun(A,fun(B,assn))),R),P))
<=> ( pp(P)
=> precise(A,B,R) ) ) ).
% precise_extr_pure(2)
tff(fact_3742_precise__extr__pure_I1_J,axiom,
! [B: $tType,A: $tType,P: bool,R: fun(A,fun(B,assn))] :
( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_kj(bool,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),P),R))
<=> ( pp(P)
=> precise(A,B,R) ) ) ).
% precise_extr_pure(1)
tff(fact_3743_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),P4: fun(B,A),I: B] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),P4))))
=> ( ( pp(member2(B,I,I5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),I5) ) )
& ( ~ pp(member2(B,I,I5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P4,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),I5)) ) ) ) ) ) ).
% prod.insert'
tff(fact_3744_set__nths,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5)) = collect(A,aa(set(nat),fun(A,bool),aTP_Lamp_kk(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).
% set_nths
tff(fact_3745_nths__nil,axiom,
! [A: $tType,A4: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),nil(A)),A4) = nil(A) ).
% nths_nil
tff(fact_3746_nths__empty,axiom,
! [A: $tType,Xs: list(A)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),bot_bot(set(nat))) = nil(A) ).
% nths_empty
tff(fact_3747_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P4: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty'
tff(fact_3748_nths__upt__eq__take,axiom,
! [A: $tType,L: list(A),N: nat] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),set_ord_lessThan(nat,N)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L) ).
% nths_upt_eq_take
tff(fact_3749_nths__singleton,axiom,
! [A: $tType,A4: set(nat),X: A] :
( ( pp(member2(nat,zero_zero(nat),A4))
=> ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),A4) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
& ( ~ pp(member2(nat,zero_zero(nat),A4))
=> ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),A4) = nil(A) ) ) ) ).
% nths_singleton
tff(fact_3750_notin__set__nthsI,axiom,
! [A: $tType,X: A,Xs: list(A),I5: set(nat)] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5)))) ) ).
% notin_set_nthsI
tff(fact_3751_in__set__nthsD,axiom,
! [A: $tType,X: A,Xs: list(A),I5: set(nat)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_nthsD
tff(fact_3752_nths__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),I5: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)),I5) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(set(nat),list(B),aa(list(B),fun(set(nat),list(B)),nths(B),Xs),I5)) ).
% nths_map
tff(fact_3753_distinct__nthsI,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5))) ) ).
% distinct_nthsI
tff(fact_3754_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_kl(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5) ) ).
% prod.non_neutral'
tff(fact_3755_set__nths__subset,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5))),aa(list(A),set(A),set2(A),Xs))) ).
% set_nths_subset
tff(fact_3756_nths__all,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member2(nat,I3,I5)) )
=> ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5) = Xs ) ) ).
% nths_all
tff(fact_3757_sorted__nths,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I5: set(nat)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5)) ) ) ).
% sorted_nths
tff(fact_3758_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I5: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),I5) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I5)) ).
% nths_drop
tff(fact_3759_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ).
% prod.distrib_triv'
tff(fact_3760_snga__prec,axiom,
! [A: $tType] :
( heap(A)
=> precise(list(A),array(A),aTP_Lamp_km(list(A),fun(array(A),assn))) ) ).
% snga_prec
tff(fact_3761_drop__eq__nths,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),collect(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).
% drop_eq_nths
tff(fact_3762_sngr__prec,axiom,
! [A: $tType] :
( heap(A)
=> precise(A,ref(A),aTP_Lamp_kn(A,fun(ref(A),assn))) ) ).
% sngr_prec
tff(fact_3763_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),T: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T) ) ) ) ) ).
% prod.mono_neutral_left'
tff(fact_3764_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),T: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S) ) ) ) ) ).
% prod.mono_neutral_right'
tff(fact_3765_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),T: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H,I3) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),T) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_3766_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),T: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T))
=> ( ! [X2: B] :
( pp(member2(B,X2,minus_minus(set(B),T,S)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),S) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_3767_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ) ).
% prod.distrib'
tff(fact_3768_nths__append,axiom,
! [A: $tType,L: list(A),L3: list(A),A4: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3)),A4) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),A4)),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L3),collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_ko(list(A),fun(set(nat),fun(nat,bool)),L),A4)))) ).
% nths_append
tff(fact_3769_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [I5: set(B),P4: fun(B,A)] :
( ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),P4))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P4),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),P4))) ) )
& ( ~ pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),I5),P4))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P4),I5) = one_one(A) ) ) ) ) ).
% prod.G_def
tff(fact_3770_filter__in__nths,axiom,
! [A: $tType,Xs: list(A),S4: set(nat)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(set(nat),fun(A,bool),aTP_Lamp_kp(list(A),fun(set(nat),fun(A,bool)),Xs),S4)),Xs) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),S4) ) ) ).
% filter_in_nths
tff(fact_3771_length__nths,axiom,
! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I5)) = finite_card(nat,collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_kq(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).
% length_nths
tff(fact_3772_filter__eq__nths,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_fm(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).
% filter_eq_nths
tff(fact_3773_nths__Cons,axiom,
! [A: $tType,X: A,L: list(A),A4: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L)),A4) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),if(list(A),member2(nat,zero_zero(nat),A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A))),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),L),collect(nat,aTP_Lamp_kr(set(nat),fun(nat,bool),A4)))) ).
% nths_Cons
tff(fact_3774_sorted__list__of__set__def,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_az(A,A)) ) ) ).
% sorted_list_of_set_def
tff(fact_3775_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A3: A] : enumerate(A,N,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),A3)) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_ks(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ).
% enumerate_replicate_eq
tff(fact_3776_finite__def,axiom,
! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),bool),aTP_Lamp_kt(fun(set(A),bool),fun(set(A),bool))) ).
% finite_def
tff(fact_3777_listrel1__subset__listrel,axiom,
! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
=> ( refl_on(A,top_top(set(A)),R6)
=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel(A,A,R6))) ) ) ).
% listrel1_subset_listrel
tff(fact_3778_top__apply,axiom,
! [D: $tType,C: $tType] :
( top(C)
=> ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).
% top_apply
tff(fact_3779_UNIV__I,axiom,
! [A: $tType,X: A] : pp(member2(A,X,top_top(set(A)))) ).
% UNIV_I
tff(fact_3780_inf__top__left,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),X) = X ) ).
% inf_top_left
tff(fact_3781_inf__top__right,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).
% inf_top_right
tff(fact_3782_inf__eq__top__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = top_top(A) )
<=> ( ( X = top_top(A) )
& ( Y = top_top(A) ) ) ) ) ).
% inf_eq_top_iff
tff(fact_3783_top__eq__inf__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A,Y: A] :
( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) )
<=> ( ( X = top_top(A) )
& ( Y = top_top(A) ) ) ) ) ).
% top_eq_inf_iff
tff(fact_3784_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = top_top(A) )
<=> ( ( A3 = top_top(A) )
& ( B2 = top_top(A) ) ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_3785_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A3) = A3 ) ).
% inf_top.left_neutral
tff(fact_3786_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A,B2: A] :
( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
<=> ( ( A3 = top_top(A) )
& ( B2 = top_top(A) ) ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_3787_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),top_top(A)) = A3 ) ).
% inf_top.right_neutral
tff(fact_3788_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).
% boolean_algebra.disj_one_right
tff(fact_3789_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).
% boolean_algebra.disj_one_left
tff(fact_3790_sup__top__right,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).
% sup_top_right
tff(fact_3791_sup__top__left,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).
% sup_top_left
tff(fact_3792_Int__UNIV,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = top_top(set(A)) )
<=> ( ( A4 = top_top(set(A)) )
& ( B3 = top_top(set(A)) ) ) ) ).
% Int_UNIV
tff(fact_3793_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_3794_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_3795_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_3796_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_3797_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_3798_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_3799_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat,Y: A] :
( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),X) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
<=> ( ( M2 = N )
& ( ( M2 != zero_zero(nat) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
tff(fact_3800_length__replicate,axiom,
! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = N ).
% length_replicate
tff(fact_3801_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] : aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),I),nil(A))) = nil(A) ).
% concat_replicate_trivial
tff(fact_3802_map__replicate,axiom,
! [A: $tType,B: $tType,F: fun(B,A),N: nat,X: B] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),aa(B,A,F,X)) ).
% map_replicate
tff(fact_3803_Pow__UNIV,axiom,
! [A: $tType] : pow(A,top_top(set(A))) = top_top(set(set(A))) ).
% Pow_UNIV
tff(fact_3804_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] : aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ).
% rev_replicate
tff(fact_3805_Collect__const,axiom,
! [A: $tType,P: bool] :
( ( pp(P)
=> ( collect(A,aTP_Lamp_ku(bool,fun(A,bool),P)) = top_top(set(A)) ) )
& ( ~ pp(P)
=> ( collect(A,aTP_Lamp_ku(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).
% Collect_const
tff(fact_3806_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_3807_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_3808_sup__compl__top__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).
% sup_compl_top_left1
tff(fact_3809_sup__compl__top__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).
% sup_compl_top_left2
tff(fact_3810_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).
% boolean_algebra.disj_cancel_left
tff(fact_3811_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).
% boolean_algebra.disj_cancel_right
tff(fact_3812_Diff__UNIV,axiom,
! [A: $tType,A4: set(A)] : minus_minus(set(A),A4,top_top(set(A))) = bot_bot(set(A)) ).
% Diff_UNIV
tff(fact_3813_card__ge__UNIV,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,top_top(set(A)))),finite_card(A,S)))
<=> ( S = top_top(set(A)) ) ) ) ).
% card_ge_UNIV
tff(fact_3814_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A),Ys: list(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Ys) )
<=> ( Xs = Ys ) ) ) ).
% inj_map_eq_map
tff(fact_3815_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( nil(A) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) )
<=> ( N = zero_zero(nat) ) ) ).
% empty_replicate
tff(fact_3816_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) = nil(A) )
<=> ( N = zero_zero(nat) ) ) ).
% replicate_empty
tff(fact_3817_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y))))
<=> ( ( X = Y )
& ( N != zero_zero(nat) ) ) ) ).
% in_set_replicate
tff(fact_3818_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A3))))
& pp(aa(A,bool,P,X4)) )
<=> ( pp(aa(A,bool,P,A3))
& ( N != zero_zero(nat) ) ) ) ).
% Bex_set_replicate
tff(fact_3819_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A3))))
=> pp(aa(A,bool,P,X4)) )
<=> ( pp(aa(A,bool,P,A3))
| ( N = zero_zero(nat) ) ) ) ).
% Ball_set_replicate
tff(fact_3820_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_3821_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),I) = X ) ) ).
% nth_replicate
tff(fact_3822_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( aa(list(A),A,hd(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = X ) ) ).
% hd_replicate
tff(fact_3823_drop__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),minus_minus(nat,K,I)),X) ).
% drop_replicate
tff(fact_3824_takeWhile__replicate,axiom,
! [A: $tType,P: fun(A,bool),X: A,N: nat] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) ) ) ).
% takeWhile_replicate
tff(fact_3825_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = X ) ) ).
% last_replicate
tff(fact_3826_foldr__replicate,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),N: nat,X: B] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = compow(fun(A,A),N,aa(B,fun(A,A),F,X)) ).
% foldr_replicate
tff(fact_3827_take__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),K)),X) ).
% take_replicate
tff(fact_3828_inj__mult__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A] :
( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% inj_mult_left
tff(fact_3829_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] : aa(list(A),list(A),tl(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),minus_minus(nat,N,one_one(nat))),X) ).
% tl_replicate
tff(fact_3830_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X)),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),J),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),replicate(product_prod(A,B)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_replicate
tff(fact_3831_range__constant,axiom,
! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aTP_Lamp_cj(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_3832_inj__mapI,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),top_top(set(list(A)))) ) ).
% inj_mapI
tff(fact_3833_inj__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),top_top(set(list(A))))
<=> inj_on(A,B,F,top_top(set(A))) ) ).
% inj_map
tff(fact_3834_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ).
% set_replicate
tff(fact_3835_rangeI,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X: B] : pp(member2(A,aa(B,A,F,X),aa(set(B),set(A),image2(B,A,F),top_top(set(B))))) ).
% rangeI
tff(fact_3836_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),X: B] :
( ( B2 = aa(B,A,F,X) )
=> pp(member2(A,B2,aa(set(B),set(A),image2(B,A,F),top_top(set(B))))) ) ).
% range_eqI
tff(fact_3837_Un__UNIV__left,axiom,
! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B3) = top_top(set(A)) ).
% Un_UNIV_left
tff(fact_3838_Un__UNIV__right,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),top_top(set(A))) = top_top(set(A)) ).
% Un_UNIV_right
tff(fact_3839_Int__UNIV__left,axiom,
! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B3) = B3 ).
% Int_UNIV_left
tff(fact_3840_Int__UNIV__right,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),top_top(set(A))) = A4 ).
% Int_UNIV_right
tff(fact_3841_UNIV__eq__I,axiom,
! [A: $tType,A4: set(A)] :
( ! [X2: A] : pp(member2(A,X2,A4))
=> ( top_top(set(A)) = A4 ) ) ).
% UNIV_eq_I
tff(fact_3842_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : pp(member2(A,X2,top_top(set(A)))) ).
% UNIV_witness
tff(fact_3843_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ).
% append_replicate_commute
tff(fact_3844_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_3845_total__lexord,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( total_on(A,top_top(set(A)),R3)
=> total_on(list(A),top_top(set(list(A))),lexord(A,R3)) ) ).
% total_lexord
tff(fact_3846_subset__UNIV,axiom,
! [A: $tType,A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),top_top(set(A)))) ).
% subset_UNIV
tff(fact_3847_top_Oextremum__strict,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).
% top.extremum_strict
tff(fact_3848_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( ( A3 != top_top(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).
% top.not_eq_extremum
tff(fact_3849_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).
% boolean_algebra.conj_one_right
tff(fact_3850_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_uniqueI
tff(fact_3851_top_Oextremum__unique,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
<=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_unique
tff(fact_3852_top__greatest,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),top_top(A))) ) ).
% top_greatest
tff(fact_3853_total__lenlex,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( total_on(A,top_top(set(A)),R3)
=> total_on(list(A),top_top(set(list(A))),lenlex(A,R3)) ) ).
% total_lenlex
tff(fact_3854_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_3855_empty__not__UNIV,axiom,
! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).
% empty_not_UNIV
tff(fact_3856_UNIV__def,axiom,
! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_kv(A,bool)) ).
% UNIV_def
tff(fact_3857_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_kw(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).
% range_composition
tff(fact_3858_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
( pp(member2(A,B2,aa(set(B),set(A),image2(B,A,F),top_top(set(B)))))
=> ~ ! [X2: B] : B2 != aa(B,A,F,X2) ) ).
% rangeE
tff(fact_3859_infinite__UNIV__listI,axiom,
! [A: $tType] : ~ pp(aa(set(list(A)),bool,finite_finite2(list(A)),top_top(set(list(A))))) ).
% infinite_UNIV_listI
tff(fact_3860_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( linorder(A)
=> inj_on(A,A,aTP_Lamp_az(A,A),top_top(set(A))) ) ).
% sorted_list_of_set.inj_on
tff(fact_3861_inj__mapD,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),top_top(set(list(A))))
=> inj_on(A,B,F,top_top(set(A))) ) ).
% inj_mapD
tff(fact_3862_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),X) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ).
% replicate_Suc
tff(fact_3863_replicate__0,axiom,
! [A: $tType,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),zero_zero(nat)),X) = nil(A) ).
% replicate_0
tff(fact_3864_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),Xs)) ).
% replicate_app_Cons_same
tff(fact_3865_replicate__length__same,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( X2 = X ) )
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),X) = Xs ) ) ).
% replicate_length_same
tff(fact_3866_replicate__eqI,axiom,
! [A: $tType,Xs: list(A),N: nat,X: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
=> ( ! [Y2: A] :
( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( Y2 = X ) )
=> ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) ) ) ).
% replicate_eqI
tff(fact_3867_sup__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).
% sup_cancel_left1
tff(fact_3868_sup__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).
% sup_cancel_left2
tff(fact_3869_sorted__replicate,axiom,
! [A: $tType] :
( linorder(A)
=> ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ) ).
% sorted_replicate
tff(fact_3870_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),X)) ).
% replicate_add
tff(fact_3871_filter__replicate,axiom,
! [A: $tType,P: fun(A,bool),X: A,N: nat] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) ) ) ).
% filter_replicate
tff(fact_3872_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
=> ? [M4: nat,N7: nat,Zs3: list(A)] :
( ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),M4),Zs3)) = Xs )
& ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N7),Zs3)) = Ys ) ) ) ).
% comm_append_are_replicate
tff(fact_3873_range__subsetD,axiom,
! [B: $tType,A: $tType,F: fun(B,A),B3: set(A),I: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))),B3))
=> pp(member2(A,aa(B,A,F,I),B3)) ) ).
% range_subsetD
tff(fact_3874_map__injective,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),Ys: list(B)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
=> ( inj_on(B,A,F,top_top(set(B)))
=> ( Xs = Ys ) ) ) ).
% map_injective
tff(fact_3875_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_3876_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_3877_Compl__partition,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = top_top(set(A)) ).
% Compl_partition
tff(fact_3878_Compl__partition2,axiom,
! [A: $tType,A4: 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)),A4)),A4) = top_top(set(A)) ).
% Compl_partition2
tff(fact_3879_Compl__eq__Diff__UNIV,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = minus_minus(set(A),top_top(set(A)),A4) ).
% Compl_eq_Diff_UNIV
tff(fact_3880_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_cj(A,fun(B,A),K)),Lst) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),Lst)),K) ).
% map_replicate_const
tff(fact_3881_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),X)),Xs))),X) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),X)),Xs) ).
% replicate_length_filter
tff(fact_3882_UNIV__coset,axiom,
! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).
% UNIV_coset
tff(fact_3883_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] : collect(A,aTP_Lamp_kx(fun(B,A),fun(A,bool),F)) = aa(set(B),set(A),image2(B,A,F),top_top(set(B))) ).
% full_SetCompr_eq
tff(fact_3884_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(B,A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( inj_on(B,A,F,top_top(set(B)))
=> pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ky(set(A),fun(fun(B,A),fun(B,bool)),S),F)))) ) ) ).
% finite_Collect
tff(fact_3885_sup__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% sup_shunt
tff(fact_3886_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X)),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),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X)) ).
% replicate_append_same
tff(fact_3887_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [A3: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),X) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),X) = top_top(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Y) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Y) = top_top(A) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_3888_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A3: A,X: B] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) )
=> ( aa(B,A,F,X) = A3 ) ) ).
% range_eq_singletonD
tff(fact_3889_image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B3: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = 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,F),A4)),aa(set(A),set(B),image2(A,B,F),B3)) ) ) ).
% image_Int
tff(fact_3890_lex__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image2(nat,set(product_prod(list(A),list(A))),lexn(A,R3)),top_top(set(nat)))) ).
% lex_def
tff(fact_3891_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X: A,Xs: list(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),removeAll(A),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),removeAll(B),aa(A,B,F,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)) ) ) ).
% map_removeAll_inj
tff(fact_3892_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_kz(A,fun(nat,A),X)),upt(zero_zero(nat),I)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X) ).
% map_replicate_trivial
tff(fact_3893_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N = zero_zero(nat) )
=> ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = bot_bot(set(A)) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = 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_3894_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),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_3895_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).
% replicate_Suc_conv_snoc
tff(fact_3896_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_az(A,A)) ) ).
% sorted_list_of_set.folding_insort_key_axioms
tff(fact_3897_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_3898_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_3899_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ).
% zip_replicate1
tff(fact_3900_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K)),L) ).
% map_zip2
tff(fact_3901_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ht(B,fun(A,product_prod(A,B))),K)),L) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),aa(list(A),nat,size_size(list(A)),L)),K)) ).
% map_zip1
tff(fact_3902_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Y)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ht(B,fun(A,product_prod(A,B))),Y)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ).
% zip_replicate2
tff(fact_3903_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
<=> ( ( X = Y )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
& ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),minus_minus(nat,N,one_one(nat))),X) ) ) ) ).
% Cons_replicate_eq
tff(fact_3904_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
=> ? [N7: nat,Zs3: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N7))
& ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N7),Zs3)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).
% comm_append_is_replicate
tff(fact_3905_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_3906_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(fun(B,product_prod(A,B)),fun(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)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K) ).
% map_fst_mk_fst
tff(fact_3907_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A)] : linord144544945434240204of_set(B,A,F) = finite_folding_F(B,list(B),linorder_insort_key(B,A,F),nil(B)) ) ).
% sorted_key_list_of_set_def
tff(fact_3908_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(fun(B,product_prod(B,A)),fun(list(B),list(product_prod(B,A))),map(B,product_prod(B,A)),aTP_Lamp_la(A,fun(B,product_prod(B,A)),K)),L)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),L)),K) ).
% map_snd_mk_snd
tff(fact_3909_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)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_db(A,product_prod(A,A))),Xs)) ).
% Id_on_set
tff(fact_3910_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_3911_assn__basic__inequalities_I1_J,axiom,
top_top(assn) != one_one(assn) ).
% assn_basic_inequalities(1)
tff(fact_3912_assn__basic__inequalities_I5_J,axiom,
top_top(assn) != bot_bot(assn) ).
% assn_basic_inequalities(5)
tff(fact_3913_map__snd__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),aa(fun(product_prod(nat,A),A),fun(list(product_prod(nat,A)),list(A)),map(product_prod(nat,A),A),product_snd(nat,A)),enumerate(A,N,Xs)) = Xs ).
% map_snd_enumerate
tff(fact_3914_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S))
=> pp(member2(A,A3,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))) ) ).
% img_fst
tff(fact_3915_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S))
=> pp(member2(B,B2,aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S))) ) ).
% img_snd
tff(fact_3916_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_3917_map__fst__zip,axiom,
! [B: $tType,A: $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(A),aa(fun(product_prod(A,B),A),fun(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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = Xs ) ) ).
% map_fst_zip
tff(fact_3918_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),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = Ys ) ) ).
% map_snd_zip
tff(fact_3919_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs)),aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs)) = Zs ).
% zip_map_fst_snd
tff(fact_3920_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).
% map_fst_enumerate
tff(fact_3921_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_lb(product_prod(A,B),fun(product_prod(A,B),bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),L)) ) ) ).
% sorted_wrt_map_linord
tff(fact_3922_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ht(B,fun(A,product_prod(A,B))),K)),L)) = L ).
% map_fst_mk_snd
tff(fact_3923_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(fun(A,product_prod(B,A)),fun(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_3924_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_lc(product_prod(A,B),fun(product_prod(A,B),bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_3925_one__div__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).
% one_div_numeral
tff(fact_3926_one__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).
% one_mod_numeral
tff(fact_3927_norm__assertion__simps_I12_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),top_top(assn)) = top_top(assn) ).
% norm_assertion_simps(12)
tff(fact_3928_norm__assertion__simps_I11_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),top_top(assn)),X) = top_top(assn) ).
% norm_assertion_simps(11)
tff(fact_3929_norm__assertion__simps_I3_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),top_top(assn)),X) = X ).
% norm_assertion_simps(3)
tff(fact_3930_norm__assertion__simps_I4_J,axiom,
! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),X),top_top(assn)) = X ).
% norm_assertion_simps(4)
tff(fact_3931_top__set__def,axiom,
! [A: $tType] : top_top(set(A)) = collect(A,top_top(fun(A,bool))) ).
% top_set_def
tff(fact_3932_merge__true__star__ctx,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),P)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),P) ).
% merge_true_star_ctx
tff(fact_3933_fs__contract,axiom,
! [A: $tType,B: $tType,C: $tType,F: 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)),collect(product_prod(A,B),aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_ld(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),F),S))) = collect(A,aa(set(C),fun(A,bool),aTP_Lamp_le(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),F),S)) ).
% fs_contract
tff(fact_3934_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),aa(fun(product_prod(A,B),A),fun(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),aa(fun(product_prod(A,B),A),fun(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),aa(fun(product_prod(A,B),B),fun(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),aa(fun(product_prod(A,B),B),fun(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_3935_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ! [A6: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A6),X_12))
<=> ! [Z3: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),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_3936_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ? [A6: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A6),X_12))
<=> ? [Z3: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),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_3937_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(B,bool)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( pp(aa(B,bool,P,aa(product_prod(A,B),B,product_snd(A,B),X)))
=> pp(aa(B,bool,P,B2)) ) ) ).
% sndE
tff(fact_3938_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(A,bool)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),X)))
=> pp(aa(A,bool,P,A3)) ) ) ).
% fstE
tff(fact_3939_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_lf(fun(nat,bool),fun(product_prod(A,nat),bool),P)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xs),Is))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_lg(fun(nat,bool),fun(product_prod(A,nat),bool),P)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xs),aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),Is)))) ).
% nths_shift_lemma_Suc
tff(fact_3940_fn__snd__conv,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aTP_Lamp_lh(fun(B,C),fun(product_prod(A,B),C),F) = product_case_prod(A,B,C,aTP_Lamp_li(fun(B,C),fun(A,fun(B,C)),F)) ).
% fn_snd_conv
tff(fact_3941_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys) = Zs )
<=> ( ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs) = Xs )
& ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs) = Ys ) ) ) ) ).
% zip_eq_conv
tff(fact_3942_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] : aTP_Lamp_lj(fun(A,C),fun(product_prod(A,B),C),F) = product_case_prod(A,B,C,aTP_Lamp_lk(fun(A,C),fun(A,fun(B,C)),F)) ).
% fn_fst_conv
tff(fact_3943_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I: nat,Xy: product_prod(A,B)] : aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)),I),Xy) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(product_prod(A,B),A,product_fst(A,B),Xy))),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).
% update_zip
tff(fact_3944_nths__shift__lemma,axiom,
! [A: $tType,A4: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_ll(set(nat),fun(product_prod(A,nat),bool),A4)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xs),upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_lm(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A4),I)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).
% nths_shift_lemma
tff(fact_3945_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
( pp(member2(A,Y,aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S)))
=> ~ ! [X2: B] : ~ pp(member2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y),S)) ) ).
% in_snd_imageE
tff(fact_3946_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
( pp(member2(A,X,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)))
=> ~ ! [Y2: B] : ~ pp(member2(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_3947_nths__def,axiom,
! [A: $tType,Xs: list(A),A4: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),A4) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_ll(set(nat),fun(product_prod(A,nat),bool),A4)),aa(list(nat),list(product_prod(A,nat)),aa(list(A),fun(list(nat),list(product_prod(A,nat))),zip(A,nat),Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).
% nths_def
tff(fact_3948_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P: fun(product_prod(A,B),bool)] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs)))
=> pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),bool),fun(list(product_prod(A,B)),list(product_prod(A,B))),filter2(product_prod(A,B)),P),Xs)))) ) ).
% distinct_map_fst_filterI
tff(fact_3949_in__set__zip,axiom,
! [A: $tType,B: $tType,P4: product_prod(A,B),Xs: list(A),Ys: list(B)] :
( pp(member2(product_prod(A,B),P4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
<=> ? [N3: nat] :
( ( aa(nat,A,nth(A,Xs),N3) = aa(product_prod(A,B),A,product_fst(A,B),P4) )
& ( aa(nat,B,nth(B,Ys),N3) = aa(product_prod(A,B),B,product_snd(A,B),P4) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).
% in_set_zip
tff(fact_3950_entails__solve__finalize_I1_J,axiom,
! [M: list(product_prod(assn,assn)),P: assn] : fI_RESULT(M,P,one_one(assn),top_top(assn)) ).
% entails_solve_finalize(1)
tff(fact_3951_snd__image__mp,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(B,A)),B3: set(A),X: B,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A4)),B3))
=> ( pp(member2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),A4))
=> pp(member2(A,Y,B3)) ) ) ).
% snd_image_mp
tff(fact_3952_fst__image__mp,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B3: set(A),X: A,Y: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4)),B3))
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),A4))
=> pp(member2(A,X,B3)) ) ) ).
% fst_image_mp
tff(fact_3953_snd__in__Field,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R)),field2(A,R))) ).
% snd_in_Field
tff(fact_3954_fst__in__Field,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R)),field2(A,R))) ).
% fst_in_Field
tff(fact_3955_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod(nat,A),N: nat,Xs: list(A)] :
( pp(member2(product_prod(nat,A),P4,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P4)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
& ( aa(nat,A,nth(A,Xs),minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P4),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P4) ) ) ) ).
% in_set_enumerate_eq
tff(fact_3956_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xs))) ).
% sorted_enumerate
tff(fact_3957_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(C,A)),G: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),aa(fun(product_prod(A,B),fun(C,product_prod(A,B))),fun(product_prod(A,B),fun(list(C),product_prod(A,B))),foldl(product_prod(A,B),C),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_ln(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F),G))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),aa(fun(A,fun(C,A)),fun(A,fun(list(C),A)),foldl(A,C),F),A3),Xs) ).
% fst_foldl
tff(fact_3958_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z2: B] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs)))
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
=> ( pp(member2(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_3959_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(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)),aa(list(B),fun(list(A),list(product_prod(B,A))),zip(B,A),Xs),Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),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_3960_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))),Xs) ).
% map_fst_zip_take
tff(fact_3961_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list(A),Xs23: 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),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(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),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),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_3962_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),X: A,Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S))
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( aa(set(A),B,finite_folding_F(A,B,F,Z2),A4) = aa(B,B,aa(A,fun(B,B),F,X),aa(set(A),B,finite_folding_F(A,B,F,Z2),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% folding_on.remove
tff(fact_3963_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,A4: set(A),Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)),S))
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),B,finite_folding_F(A,B,F,Z2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(B,B,aa(A,fun(B,B),F,X),aa(set(A),B,finite_folding_F(A,B,F,Z2),minus_minus(set(A),A4,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_3964_top1I,axiom,
! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).
% top1I
tff(fact_3965_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list(A),Xs23: list(A),X12: A,X24: 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),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),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),X12),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs23)),Xs) ).
% mergesort_by_rel_split.simps(3)
tff(fact_3966_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A),Xs23: 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),Xs23),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),Xs23) ).
% mergesort_by_rel_split.simps(1)
tff(fact_3967_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( aa(set(A),B,finite_folding_F(A,B,F,Z2),bot_bot(set(A))) = Z2 ) ) ).
% folding_on.empty
tff(fact_3968_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A))) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_3969_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) )
=> ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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),X2),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) )
=> ! [X1: A,X23: A,Xs2: list(A)] :
( ( Xa = 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),X23),Xs2)) )
=> ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs2) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
tff(fact_3970_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),Xs23: 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),Xs23),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)),Xs23) ).
% mergesort_by_rel_split.simps(2)
tff(fact_3971_mergesort__by__rel_Osimps,axiom,
! [A: $tType,Xs: list(A),R: fun(A,fun(A,bool))] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = Xs ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),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,R),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_3972_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( Y = Xa ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),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_3973_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X12: A,X24: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R),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),X24),Xs))) = aa(product_prod(list(A),list(A)),list(A),product_case_prod(list(A),list(A),list(A),aTP_Lamp_lo(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),R)),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),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),nil(A))),Xs)) ).
% mergesort_by_rel_simps(3)
tff(fact_3974_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)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),aa(fun(product_prod(A,C),list(product_prod(A,B))),fun(list(product_prod(A,C)),list(list(product_prod(A,B)))),map(product_prod(A,C),list(product_prod(A,B))),aTP_Lamp_lq(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).
% set_relcomp
tff(fact_3975_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Ys: list(A)] : merges9089515139780605204_merge(A,R,nil(A),Ys) = Ys ).
% mergesort_by_rel_merge_simps(3)
tff(fact_3976_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Y2))
| pp(aa(A,bool,aa(A,fun(A,bool),R,Y2),X2)) )
=> ( ! [X2: A,Y2: A,Z4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R,Y2),Z4))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Z4)) ) )
=> ( sorted_wrt(A,R,merges9089515139780605204_merge(A,R,Xs,Ys))
<=> ( sorted_wrt(A,R,Xs)
& sorted_wrt(A,R,Ys) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
tff(fact_3977_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R,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_3978_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X),Y))
=> ( merges9089515139780605204_merge(A,R,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)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,R,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),R,X),Y))
=> ( merges9089515139780605204_merge(A,R,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)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R,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_3979_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A)] : merges9089515139780605204_merge(A,R,Xs,nil(A)) = Xs ).
% mergesort_by_rel_merge_simps(2)
tff(fact_3980_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),V: A,Va: list(A)] : merges9089515139780605204_merge(A,R,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ).
% mergesort_by_rel_merge.simps(3)
tff(fact_3981_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
=> ( Y = 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),X2),Xs2),Ys2)) ) ) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: A,Va2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
tff(fact_3982_trancl__Int__subset,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S4))
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R3)),S4),R3)),S4))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),S4)) ) ) ).
% trancl_Int_subset
tff(fact_3983_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [Y2: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
=> ( Y = 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),X2),Xs2),Ys2)) ) ) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: A,Va2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
tff(fact_3984_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R),Xs))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = Xs ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),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,R),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_3985_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa) = Y )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( Y = Xa ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),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)))) ) ) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa)) ) ) ) ).
% mergesort_by_rel.pelims
tff(fact_3986_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),bool))] :
( accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),A0),A1))
=> ( ! [R4: fun(A,fun(A,bool)),Xs2: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R4),Xs2))
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R4),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R4),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R4),Xs2)) ) ) )
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,A0),A1)) ) ) ).
% mergesort_by_rel.pinduct
tff(fact_3987_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: fun(A,B),Xa: list(A),Y: A] :
( ( arg_min_list(A,B,X,Xa) = Y )
=> ( accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),Xa))
=> ( ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ( ( Y = X2 )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) ) )
=> ( ! [X2: A,Y2: A,Zs3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)) )
=> ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)))),X2,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3))) )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs3)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),nil(A))) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
tff(fact_3988_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( sorted_wrt(A,X,Xa)
<=> pp(Y) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ( pp(Y)
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A))) ) )
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ( ( pp(Y)
<=> ( ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa2)) )
& sorted_wrt(A,X,Ys2) ) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2))) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
tff(fact_3989_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( sorted_wrt(A,X,Xa)
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A))) )
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)))
=> ~ ( ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa3)) )
& sorted_wrt(A,X,Ys2) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
tff(fact_3990_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ sorted_wrt(A,X,Xa)
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ~ ! [X2: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys2)))
=> ( ! [Xa4: A] :
( pp(member2(A,Xa4,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa4)) )
& sorted_wrt(A,X,Ys2) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
tff(fact_3991_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,product_prod(A,C))] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,comp(product_prod(A,C),A,B,product_fst(A,C),F)),top_top(set(B)))))
=> ( pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,comp(product_prod(A,C),C,B,product_snd(A,C),F)),top_top(set(B)))))
=> pp(aa(set(product_prod(A,C)),bool,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F),top_top(set(B))))) ) ) ).
% finite_range_prod
tff(fact_3992_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( Xs != nil(A) )
& ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),A,hd(A),Xs)) ) ) ) ).
% remdups_adj_singleton_iff
tff(fact_3993_INT__simps_I4_J,axiom,
! [G4: $tType,H2: $tType,C4: set(H2),A4: set(G4),B3: fun(H2,set(G4))] :
( ( ( C4 = bot_bot(set(H2)) )
=> ( aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),aa(fun(H2,set(G4)),fun(H2,set(G4)),aTP_Lamp_lr(set(G4),fun(fun(H2,set(G4)),fun(H2,set(G4))),A4),B3)),C4)) = top_top(set(G4)) ) )
& ( ( C4 != bot_bot(set(H2)) )
=> ( aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),aa(fun(H2,set(G4)),fun(H2,set(G4)),aTP_Lamp_lr(set(G4),fun(fun(H2,set(G4)),fun(H2,set(G4))),A4),B3)),C4)) = minus_minus(set(G4),A4,aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),B3),C4))) ) ) ) ).
% INT_simps(4)
tff(fact_3994_card__quotient__disjoint,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( inj_on(A,set(set(A)),aTP_Lamp_ls(set(product_prod(A,A)),fun(A,set(set(A))),R3),A4)
=> ( finite_card(set(A),equiv_quotient(A,A4,R3)) = finite_card(A,A4) ) ) ) ).
% card_quotient_disjoint
tff(fact_3995_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),G),Xs)) = aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),comp(B,A,C,F,G)),Xs) ).
% map_map
tff(fact_3996_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),List)) = aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),comp(B,C,A,F,G)),List) ).
% List.map.compositionality
tff(fact_3997_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: list(A)] : aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),G),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),V)) = aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),comp(B,C,A,G,F)),V) ).
% list.map_comp
tff(fact_3998_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : comp(list(B),list(C),list(A),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = aa(fun(A,C),fun(list(A),list(C)),map(A,C),comp(B,C,A,F,G)) ).
% List.map.comp
tff(fact_3999_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),aa(fun(A,C),fun(list(A),list(C)),map(A,C),G)) = aa(fun(A,B),fun(list(A),list(B)),map(A,B),comp(C,B,A,F,G)) ).
% map_comp_map
tff(fact_4000_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),list(A),remdups_adj(A),Xs) = nil(A) )
<=> ( Xs = nil(A) ) ) ).
% remdups_adj_Nil_iff
tff(fact_4001_remdups__adj__set,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),remdups_adj(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% remdups_adj_set
tff(fact_4002_remdups__adj__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),remdups_adj(A),Xs)) ).
% remdups_adj_rev
tff(fact_4003_hd__remdups__adj,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),remdups_adj(A),Xs)) = aa(list(A),A,hd(A),Xs) ).
% hd_remdups_adj
tff(fact_4004_last__remdups__adj,axiom,
! [A: $tType,Xs: list(A)] : last(A,aa(list(A),list(A),remdups_adj(A),Xs)) = last(A,Xs) ).
% last_remdups_adj
tff(fact_4005_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A4: set(A)] :
( ( aa(set(A),A,complete_Inf_Inf(A),A4) = bot_bot(A) )
<=> ! [X4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
=> ? [Xa2: A] :
( pp(member2(A,Xa2,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa2),X4)) ) ) ) ) ).
% Inf_eq_bot_iff
tff(fact_4006_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_4007_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_4008_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_4009_Inf__insert,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).
% Inf_insert
tff(fact_4010_Inf__atMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),set_ord_atMost(A,X)) = bot_bot(A) ) ).
% Inf_atMost
tff(fact_4011_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ).
% remdups_adj_Cons_alt
tff(fact_4012_INF__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F: A] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),F)),A4)) = F ) ) ) ).
% INF_const
tff(fact_4013_cINF__const,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),C3: A] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ag(A,fun(B,A),C3)),A4)) = C3 ) ) ) ).
% cINF_const
tff(fact_4014_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A4: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)) = bot_bot(A) )
<=> ! [X4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
=> ? [Xa2: B] :
( pp(member2(B,Xa2,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,Xa2)),X4)) ) ) ) ) ).
% INF_eq_bot_iff
tff(fact_4015_INT__constant,axiom,
! [B: $tType,A: $tType,A4: set(B),C3: set(A)] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),C3)),A4)) = top_top(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),C3)),A4)) = C3 ) ) ) ).
% INT_constant
tff(fact_4016_INT__insert,axiom,
! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: B,A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B3,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A4))) ).
% INT_insert
tff(fact_4017_Inf__atMostLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,X)) = bot_bot(A) ) ) ) ).
% Inf_atMostLessThan
tff(fact_4018_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set(D),A4: set(C),B3: fun(D,set(C))] :
( ( ( C4 = bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lt(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3)),C4)) = top_top(set(C)) ) )
& ( ( C4 != bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lt(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3)),C4)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A4),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C4))) ) ) ) ).
% INT_simps(2)
tff(fact_4019_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set(A),A4: fun(A,set(B)),B3: set(B)] :
( ( ( C4 = bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lu(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3)),C4)) = top_top(set(B)) ) )
& ( ( C4 != bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lu(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3)),C4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),C4))),B3) ) ) ) ).
% INT_simps(1)
tff(fact_4020_INT__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set(E4),A4: fun(E4,set(F6)),B3: set(F6)] :
( ( ( C4 = bot_bot(set(E4)) )
=> ( aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),aa(set(F6),fun(E4,set(F6)),aTP_Lamp_lv(fun(E4,set(F6)),fun(set(F6),fun(E4,set(F6))),A4),B3)),C4)) = top_top(set(F6)) ) )
& ( ( C4 != bot_bot(set(E4)) )
=> ( aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),aa(set(F6),fun(E4,set(F6)),aTP_Lamp_lv(fun(E4,set(F6)),fun(set(F6),fun(E4,set(F6))),A4),B3)),C4)) = minus_minus(set(F6),aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),A4),C4)),B3) ) ) ) ).
% INT_simps(3)
tff(fact_4021_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),remdups_adj(A),nil(A)) = nil(A) ).
% remdups_adj.simps(1)
tff(fact_4022_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: fun(A,B),Y: fun(A,B),F: fun(C,A)] :
( ( X = Y )
=> ( comp(A,B,C,X,F) = comp(A,B,C,Y,F) ) ) ).
% comp_cong_left
tff(fact_4023_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Y: fun(A,B),F: fun(B,C)] :
( ( X = Y )
=> ( comp(B,C,A,F,X) = comp(B,C,A,F,Y) ) ) ).
% comp_cong_right
tff(fact_4024_fun__comp__eq__conv,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(A,C),Fg: fun(A,B)] :
( ( comp(C,B,A,F,G) = Fg )
<=> ! [X4: A] : aa(C,B,F,aa(A,C,G,X4)) = aa(A,B,Fg,X4) ) ).
% fun_comp_eq_conv
tff(fact_4025_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),remdups_adj(A),Xs) = Xs ) ) ).
% remdups_adj_distinct
tff(fact_4026_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A)] :
( ( ( X = Y )
=> ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) )
& ( ( X != Y )
=> ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) ) ) ) ).
% remdups_adj.simps(3)
tff(fact_4027_distinct__adj__remdups__adj,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(list(A),bool,distinct_adj(A),aa(list(A),list(A),remdups_adj(A),Xs))) ).
% distinct_adj_remdups_adj
tff(fact_4028_distinct__adj__altdef,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct_adj(A),Xs))
<=> ( aa(list(A),list(A),remdups_adj(A),Xs) = Xs ) ) ).
% distinct_adj_altdef
tff(fact_4029_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: fun(bool,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(bool),set(A),image2(bool,A,A4),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(bool,A,A4,fTrue)),aa(bool,A,A4,fFalse)) ) ).
% INF_UNIV_bool_expand
tff(fact_4030_INT__bool__eq,axiom,
! [A: $tType,A4: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),A4),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(bool,set(A),A4,fTrue)),aa(bool,set(A),A4,fFalse)) ).
% INT_bool_eq
tff(fact_4031_Inf__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),U: A] :
( ! [V2: A] :
( pp(member2(A,V2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U)) )
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),U)) ) ) ) ).
% Inf_less_eq
tff(fact_4032_cInf__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2)) )
=> ( ! [Y2: A] :
( ! [X3: A] :
( pp(member2(A,X3,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),A3)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = A3 ) ) ) ) ) ).
% cInf_eq_non_empty
tff(fact_4033_cInf__greatest,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X5))) ) ) ) ).
% cInf_greatest
tff(fact_4034_cInf__lessD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z2))
=> ? [X2: A] :
( pp(member2(A,X2,X5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2)) ) ) ) ) ).
% cInf_lessD
tff(fact_4035_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),F: fun(B,A),X: A] :
( ( I5 != bot_bot(set(B)) )
=> ( ! [I3: B] :
( pp(member2(B,I3,I5))
=> ( aa(B,A,F,I3) = X ) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),I5)) = X ) ) ) ) ).
% INF_eq_const
tff(fact_4036_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_4037_Inf__finite__insert,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).
% Inf_finite_insert
tff(fact_4038_Inf__union__distrib,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),B3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ).
% Inf_union_distrib
tff(fact_4039_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X: A,Xs: list(A)] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = comp(B,B,B,aa(A,fun(B,B),F,X),aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F),Xs)) ).
% foldr.foldr_Cons
tff(fact_4040_remdups__adj_Oelims,axiom,
! [A: $tType,X: list(A),Y: list(A)] :
( ( aa(list(A),list(A),remdups_adj(A),X) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != nil(A) ) )
=> ( ! [X2: A] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ~ ( ( ( X2 = Y2 )
=> ( Y = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) )
& ( ( X2 != Y2 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ) ).
% remdups_adj.elims
tff(fact_4041_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] : aa(list(A),list(A),remdups_adj(A),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)) ).
% remdups_adj.simps(2)
tff(fact_4042_remdups__adj__length,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% remdups_adj_length
tff(fact_4043_sorted__remdups__adj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),remdups_adj(A),Xs)) ) ) ).
% sorted_remdups_adj
tff(fact_4044_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [K: B,I5: set(B),A4: fun(B,A)] :
( pp(member2(B,K,I5))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,A4,K)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A4),I5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A4),I5)) ) ) ) ).
% INF_absorb
tff(fact_4045_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A4: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lw(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4)) ) ).
% INF_inf_distrib
tff(fact_4046_Inter__insert,axiom,
! [A: $tType,A3: set(A),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),B3)) = 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)),B3)) ).
% Inter_insert
tff(fact_4047_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_4048_Inter__Un__distrib,axiom,
! [A: $tType,A4: set(set(A)),B3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),A4),B3)) = 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)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3)) ).
% Inter_Un_distrib
tff(fact_4049_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I5: set(A),A4: fun(A,set(B))] :
( pp(member2(A,K,I5))
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,K)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)) ) ) ).
% INT_absorb
tff(fact_4050_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lx(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),I5))) ).
% INT_Int_distrib
tff(fact_4051_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),C4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lx(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3)),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C4))) ).
% Int_Inter_image
tff(fact_4052_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),M2: A,F: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)))) ) ) ) ).
% cINF_greatest
tff(fact_4053_INF__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),F: fun(B,A),C3: A] :
( ( I5 != bot_bot(set(B)) )
=> ( ! [I3: B] :
( pp(member2(B,I3,I5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I3)),C3)) )
=> ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),I5)) = C3 )
<=> ! [X4: B] :
( pp(member2(B,X4,I5))
=> ( aa(B,A,F,X4) = C3 ) ) ) ) ) ) ).
% INF_eq_iff
tff(fact_4054_finite__less__Inf__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X5))
=> ( ( X5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5)))
<=> ! [X4: A] :
( pp(member2(A,X4,X5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ) ) ).
% finite_less_Inf_iff
tff(fact_4055_Inf__le__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ).
% Inf_le_Sup
tff(fact_4056_finite__Inf__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] :
( pp(member2(A,X2,A4))
=> ( pp(member2(A,Y2,A4))
=> pp(member2(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y2),A4)) ) )
=> pp(member2(A,aa(set(A),A,complete_Inf_Inf(A),A4),A4)) ) ) ) ) ).
% finite_Inf_in
tff(fact_4057_cInf__abs__ge,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3)) ) ) ) ).
% cInf_abs_ge
tff(fact_4058_less__eq__Inf__inter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3))),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)),A4),B3)))) ) ).
% less_eq_Inf_inter
tff(fact_4059_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_4060_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_4061_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),B3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),B3) = finite_fold(A,A,inf_inf(A),B3,A4) ) ) ) ).
% inf_Inf_fold_inf
tff(fact_4062_Min__Inf,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).
% Min_Inf
tff(fact_4063_cInf__eq__Min,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( pp(aa(set(A),bool,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_4064_Inf__fin__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).
% Inf_fin_Inf
tff(fact_4065_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),X5))
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X5) ) ) ) ) ).
% cInf_eq_Inf_fin
tff(fact_4066_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list(A),X: A,Y: A] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ).
% remdups_adj_append_two
tff(fact_4067_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Xs: list(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(list(B),list(B),remdups_adj(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),aa(list(A),list(A),remdups_adj(A),Xs)) ) ) ).
% remdups_adj_map_injective
tff(fact_4068_INF__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).
% INF_empty
tff(fact_4069_INF__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),C3: A] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),C3)),A4)) = top_top(A) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),C3)),A4)) = C3 ) ) ) ) ).
% INF_constant
tff(fact_4070_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),X: A,F: fun(B,A)] :
( ( I5 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ly(A,fun(fun(B,A),fun(B,A)),X),F)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),I5))) ) ) ) ).
% INF_inf_const1
tff(fact_4071_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I5: set(B),F: fun(B,A),X: A] :
( ( I5 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_lz(fun(B,A),fun(A,fun(B,A)),F),X)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),I5))),X) ) ) ) ).
% INF_inf_const2
tff(fact_4072_INF__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))) ) ).
% INF_insert
tff(fact_4073_INF__union,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [M: fun(B,A),A4: set(B),B3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M),B3))) ) ).
% INF_union
tff(fact_4074_INT__empty,axiom,
! [B: $tType,A: $tType,B3: 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),B3),bot_bot(set(B)))) = top_top(set(A)) ).
% INT_empty
tff(fact_4075_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set(D),A4: set(C),B3: fun(D,set(C))] :
( ( ( C4 = bot_bot(set(D)) )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A4),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C4))) = A4 ) )
& ( ( C4 != bot_bot(set(D)) )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A4),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C4))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lt(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3)),C4)) ) ) ) ).
% INT_extend_simps(2)
tff(fact_4076_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C4: set(A),A4: fun(A,set(B)),B3: set(B)] :
( ( ( C4 = bot_bot(set(A)) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),C4))),B3) = B3 ) )
& ( ( C4 != bot_bot(set(A)) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),C4))),B3) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lu(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3)),C4)) ) ) ) ).
% INT_extend_simps(1)
tff(fact_4077_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set(A),F: fun(B,C),A4: fun(A,set(B))] :
( ( I5 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I5))
=> inj_on(B,C,F,aa(A,set(B),A4,I3)) )
=> inj_on(B,C,F,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) ) ) ).
% inj_on_INTER
tff(fact_4078_INT__Un,axiom,
! [A: $tType,B: $tType,M: fun(B,set(A)),A4: set(B),B3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = 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),M),A4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M),B3))) ).
% INT_Un
tff(fact_4079_Inter__Un__subset,axiom,
! [A: $tType,A4: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A4),B3)))) ).
% Inter_Un_subset
tff(fact_4080_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),H: fun(B,C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( ! [X2: B,Y2: B] :
( pp(member2(B,X2,A4))
=> ( pp(member2(B,Y2,A4))
=> ( ( X2 != Y2 )
=> ( ( aa(B,C,H,X2) = aa(B,C,H,Y2) )
=> ( aa(C,A,G,aa(B,C,H,X2)) = one_one(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),comp(C,A,B,G,H)),A4) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_4081_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B12: set(set(A)),A4: set(A)] :
( ( ( B12 = bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),A4) = A4 ) )
& ( ( B12 != bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),A4) = 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_da(set(A),fun(set(A),set(A)),A4)),B12)) ) ) ) ).
% Int_Inter_eq(2)
tff(fact_4082_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B12: set(set(A)),A4: set(A)] :
( ( ( B12 = bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = A4 ) )
& ( ( B12 != bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4)),B12)) ) ) ) ).
% Int_Inter_eq(1)
tff(fact_4083_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4)))) ) ) ).
% INF_le_SUP
tff(fact_4084_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aTP_Lamp_ma(A,fun(A,fun(list(A),list(A))),X)),aa(list(A),list(A),remdups_adj(A),Xs)) ).
% remdups_adj_Cons
tff(fact_4085_cInf__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,X2,L))),E3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),minus_minus(A,aa(set(A),A,complete_Inf_Inf(A),S),L))),E3)) ) ) ) ).
% cInf_asclose
tff(fact_4086_Inf__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( ( S = bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = X ) )
& ( ( S != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S)) ) ) ) ) ) ).
% Inf_insert_finite
tff(fact_4087_Inf__fold__inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,complete_Inf_Inf(A),A4) = finite_fold(A,A,inf_inf(A),top_top(A),A4) ) ) ) ).
% Inf_fold_inf
tff(fact_4088_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xs)),I) != aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xs)),aa(nat,nat,suc,I)) ) ) ).
% remdups_adj_adjacent
tff(fact_4089_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N = zero_zero(nat) )
=> ( aa(list(A),list(A),remdups_adj(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(list(A),list(A),remdups_adj(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).
% remdups_adj_replicate
tff(fact_4090_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( aa(list(A),list(A),remdups_adj(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
=> ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),X) ) ) ).
% remdups_adj_singleton
tff(fact_4091_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list(A),X: A,Xs_2: list(A)] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2)))) ).
% remdups_adj_append
tff(fact_4092_INT__extend__simps_I3_J,axiom,
! [F6: $tType,E4: $tType,C4: set(E4),A4: fun(E4,set(F6)),B3: set(F6)] :
( ( ( C4 = bot_bot(set(E4)) )
=> ( minus_minus(set(F6),aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),A4),C4)),B3) = minus_minus(set(F6),top_top(set(F6)),B3) ) )
& ( ( C4 != bot_bot(set(E4)) )
=> ( minus_minus(set(F6),aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),A4),C4)),B3) = aa(set(set(F6)),set(F6),complete_Inf_Inf(set(F6)),aa(set(E4),set(set(F6)),image2(E4,set(F6),aa(set(F6),fun(E4,set(F6)),aTP_Lamp_lv(fun(E4,set(F6)),fun(set(F6),fun(E4,set(F6))),A4),B3)),C4)) ) ) ) ).
% INT_extend_simps(3)
tff(fact_4093_INT__extend__simps_I4_J,axiom,
! [G4: $tType,H2: $tType,C4: set(H2),A4: set(G4),B3: fun(H2,set(G4))] :
( ( ( C4 = bot_bot(set(H2)) )
=> ( minus_minus(set(G4),A4,aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),B3),C4))) = A4 ) )
& ( ( C4 != bot_bot(set(H2)) )
=> ( minus_minus(set(G4),A4,aa(set(set(G4)),set(G4),complete_Sup_Sup(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),B3),C4))) = aa(set(set(G4)),set(G4),complete_Inf_Inf(set(G4)),aa(set(H2),set(set(G4)),image2(H2,set(G4),aa(fun(H2,set(G4)),fun(H2,set(G4)),aTP_Lamp_lr(set(G4),fun(fun(H2,set(G4)),fun(H2,set(G4))),A4),B3)),C4)) ) ) ) ).
% INT_extend_simps(4)
tff(fact_4094_remdups__adj__append_H,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( ( Xs = nil(A) )
| ( Ys = nil(A) )
| ( last(A,Xs) != aa(list(A),A,hd(A),Ys) ) )
=> ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),Xs)),aa(list(A),list(A),remdups_adj(A),Ys)) ) ) ).
% remdups_adj_append'
tff(fact_4095_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xs)))) ) ).
% remdups_adj_length_ge1
tff(fact_4096_quotient__empty,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R3) = bot_bot(set(set(A))) ).
% quotient_empty
tff(fact_4097_quotient__is__empty,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( ( equiv_quotient(A,A4,R3) = bot_bot(set(set(A))) )
<=> ( A4 = bot_bot(set(A)) ) ) ).
% quotient_is_empty
tff(fact_4098_quotient__is__empty2,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( ( bot_bot(set(set(A))) = equiv_quotient(A,A4,R3) )
<=> ( A4 = bot_bot(set(A)) ) ) ).
% quotient_is_empty2
tff(fact_4099_quotient__diff1,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),A3: A] :
( inj_on(A,set(set(A)),aTP_Lamp_ls(set(product_prod(A,A)),fun(A,set(set(A))),R3),A4)
=> ( pp(member2(A,A3,A4))
=> ( equiv_quotient(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),R3) = minus_minus(set(set(A)),equiv_quotient(A,A4,R3),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),R3)) ) ) ) ).
% quotient_diff1
tff(fact_4100_size__list__map,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) = aa(list(B),nat,size_list(B,comp(A,nat,B,F,G)),Xs) ).
% size_list_map
tff(fact_4101_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A4: set(B),C3: multiset(A)] :
( ( A4 != bot_bot(set(B)) )
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_ac(multiset(A),fun(B,multiset(A)),C3)),A4)) = C3 ) ) ).
% subset_mset.cINF_const
tff(fact_4102_length__filter__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),comp(A,bool,B,P,F)),Xs)) ).
% length_filter_map
tff(fact_4103_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : comp(list(B),nat,list(A),size_list(B,F),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = size_list(A,comp(B,nat,A,F,G)) ).
% list.size_gen_o_map
tff(fact_4104_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : comp(C,set(B),A,aTP_Lamp_mb(C,set(B)),F) = comp(set(D),set(B),A,image2(D,B,G),aTP_Lamp_mc(A,set(D))) ).
% empty_natural
tff(fact_4105_UNIV__bool,axiom,
top_top(set(bool)) = aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert2(bool),fFalse),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert2(bool),fTrue),bot_bot(set(bool)))) ).
% UNIV_bool
tff(fact_4106_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] : comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),Y),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X)) = comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),X),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),Y)) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4107_filter__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),comp(A,bool,B,P,F)),Xs)) ).
% filter_map
tff(fact_4108_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),comp(A,bool,B,P,F)),Xs)) ).
% takeWhile_map
tff(fact_4109_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C),A3: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Xs)),A3) = aa(A,A,aa(list(C),fun(A,A),aa(fun(C,fun(A,A)),fun(list(C),fun(A,A)),foldr(C,A),comp(B,fun(A,A),C,G,F)),Xs),A3) ).
% foldr_map
tff(fact_4110_rotate__add,axiom,
! [A: $tType,M2: nat,N: nat] : aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = comp(list(A),list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M2),aa(nat,fun(list(A),list(A)),rotate(A),N)) ).
% rotate_add
tff(fact_4111_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A),X: B,Y: B] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( pp(member2(B,X,S))
=> ( pp(member2(B,Y,S))
=> ( comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),Y),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),X)) = comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),X),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F),Y)) ) ) ) ) ).
% folding_insort_key.insort_key_commute
tff(fact_4112_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X3: list(A)] : aa(list(A),set(B),comp(list(B),set(B),list(A),set2(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F)),X3) = aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),X3)) ).
% set_oo_map_alt
tff(fact_4113_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),Xs: list(A),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),bool),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),comp(A,bool,product_prod(A,B),P,product_fst(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) ).
% zip_takeWhile_fst
tff(fact_4114_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list(A),P: fun(B,bool),Ys: list(B)] : aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),P),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),bool),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),comp(B,bool,product_prod(A,B),P,product_snd(A,B))),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys)) ).
% zip_takeWhile_snd
tff(fact_4115_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B3: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member2(set(B),X2,B3))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [A13: set(B)] :
( pp(member2(set(B),A13,B3))
=> ! [A23: set(B)] :
( pp(member2(set(B),A23,B3))
=> ( ( A13 != A23 )
=> ! [X2: B] :
( pp(member2(B,X2,A13))
=> ( pp(member2(B,X2,A23))
=> ( aa(B,A,G,X2) = one_one(A) ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A),groups7121269368397514597t_prod(B,A)),G),B3) ) ) ) ) ).
% prod.Union_comp
tff(fact_4116_remove__rev__def,axiom,
! [A: $tType,X: A] : remove_rev(A,X) = aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),comp(bool,bool,A,fNot,aa(A,fun(A,bool),fequal(A),X))) ).
% remove_rev_def
tff(fact_4117_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),A4) ) ).
% prod.eq_fold
tff(fact_4118_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ( comp(list(A),list(A),list(A),remdups(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A))) = mergesort_remdups(A) ) ) ).
% remdup_sort_mergesort_remdups
tff(fact_4119_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [C4: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C4))
=> ( ( X2 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X2),Xa4) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C4)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A),groups7311177749621191930dd_sum(B,A)),G),C4) ) ) ) ) ).
% sum.Union_disjoint
tff(fact_4120_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C4: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C4))
=> ( ( X2 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X2),Xa4) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C4)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A),groups7121269368397514597t_prod(B,A)),G),C4) ) ) ) ) ).
% prod.Union_disjoint
tff(fact_4121_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),B3: A,F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))) = finite_fold(B,A,comp(A,fun(A,A),B,inf_inf(A),F),B3,A4) ) ) ) ).
% inf_INF_fold_inf
tff(fact_4122_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: 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,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_atMost_Suc_shift
tff(fact_4123_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: 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,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_lessThan_Suc_shift
tff(fact_4124_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4)) = finite_fold(B,A,comp(A,fun(A,A),B,sup_sup(A),F),bot_bot(A),A4) ) ) ) ).
% SUP_fold_sup
tff(fact_4125_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)) = finite_fold(B,A,comp(A,fun(A,A),B,inf_inf(A),F),top_top(A),A4) ) ) ) ).
% INF_fold_inf
tff(fact_4126_remdups__adj_Opelims,axiom,
! [A: $tType,X: list(A),Y: list(A)] :
( ( aa(list(A),list(A),remdups_adj(A),X) = Y )
=> ( accp(list(A),remdups_adj_rel(A),X)
=> ( ( ( X = nil(A) )
=> ( ( Y = nil(A) )
=> ~ accp(list(A),remdups_adj_rel(A),nil(A)) ) )
=> ( ! [X2: A] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( ( ( ( X2 = Y2 )
=> ( Y = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) )
& ( ( X2 != Y2 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) )
=> ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ).
% remdups_adj.pelims
tff(fact_4127_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),remdups_adj(A),Xs) = Ys )
<=> ? [F5: fun(nat,nat)] :
( order_mono(nat,nat,F5)
& ( aa(set(nat),set(nat),image2(nat,nat,F5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),aa(nat,nat,F5,I2)) ) )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) )
<=> ( aa(nat,nat,F5,I2) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat))) ) ) ) ) ) ).
% remdups_adj_altdef
tff(fact_4128_range__prod,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,comp(product_prod(A,B),A,C,product_fst(A,B),F)),top_top(set(C))),aTP_Lamp_md(fun(C,product_prod(A,B)),fun(A,set(B)),F)))) ).
% range_prod
tff(fact_4129_lists__empty,axiom,
! [A: $tType] : aa(set(A),set(list(A)),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_4130_Cons__in__lists__iff,axiom,
! [A: $tType,X: A,Xs: list(A),A4: set(A)] :
( pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(set(A),set(list(A)),lists(A),A4)))
<=> ( pp(member2(A,X,A4))
& pp(member2(list(A),Xs,aa(set(A),set(list(A)),lists(A),A4))) ) ) ).
% Cons_in_lists_iff
tff(fact_4131_in__listsI,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(A,X2,A4)) )
=> pp(member2(list(A),Xs,aa(set(A),set(list(A)),lists(A),A4))) ) ).
% in_listsI
tff(fact_4132_lists__Int__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(set(A),set(list(A)),lists(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),A4)),aa(set(A),set(list(A)),lists(A),B3)) ).
% lists_Int_eq
tff(fact_4133_append__in__lists__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),A4: set(A)] :
( pp(member2(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),aa(set(A),set(list(A)),lists(A),A4)))
<=> ( pp(member2(list(A),Xs,aa(set(A),set(list(A)),lists(A),A4)))
& pp(member2(list(A),Ys,aa(set(A),set(list(A)),lists(A),A4))) ) ) ).
% append_in_lists_conv
tff(fact_4134_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A4: set(A),X5: set(B)] : minus_minus(set(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),X5)),product_Sigma(A,B,A4,aTP_Lamp_mf(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_UNIV_cancel
tff(fact_4135_lists__UNIV,axiom,
! [A: $tType] : aa(set(A),set(list(A)),lists(A),top_top(set(A))) = top_top(set(list(A))) ).
% lists_UNIV
tff(fact_4136_set__product,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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),product(A,B),Xs),Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_mg(list(B),fun(A,set(B)),Ys)) ).
% set_product
tff(fact_4137_pairself__image__cart,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B3: 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,F)),product_Sigma(B,B,A4,aTP_Lamp_mh(set(B),fun(B,set(B)),B3))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),A4),aa(set(B),fun(A,set(A)),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(A,set(A))),F),B3)) ).
% pairself_image_cart
tff(fact_4138_lists_ONil,axiom,
! [A: $tType,A4: set(A)] : pp(member2(list(A),nil(A),aa(set(A),set(list(A)),lists(A),A4))) ).
% lists.Nil
tff(fact_4139_mono__add,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A3)) ) ).
% mono_add
tff(fact_4140_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,M2)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)) ) ) ) ).
% min_of_mono
tff(fact_4141_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,M2)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),M2),N)) ) ) ) ).
% max_of_mono
tff(fact_4142_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).
% mono_strict_invE
tff(fact_4143_monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))) ) ) ) ).
% monoD
tff(fact_4144_monoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))) ) ) ) ).
% monoE
tff(fact_4145_monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y2))) )
=> order_mono(A,B,F) ) ) ).
% monoI
tff(fact_4146_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_mono(A,B,F)
<=> ! [X4: A,Y3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y3))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,F,Y3))) ) ) ) ).
% mono_def
tff(fact_4147_lists__IntI,axiom,
! [A: $tType,L: list(A),A4: set(A),B3: set(A)] :
( pp(member2(list(A),L,aa(set(A),set(list(A)),lists(A),A4)))
=> ( pp(member2(list(A),L,aa(set(A),set(list(A)),lists(A),B3)))
=> pp(member2(list(A),L,aa(set(A),set(list(A)),lists(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))) ) ) ).
% lists_IntI
tff(fact_4148_listsE,axiom,
! [A: $tType,X: A,L: list(A),A4: set(A)] :
( pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),aa(set(A),set(list(A)),lists(A),A4)))
=> ~ ( pp(member2(A,X,A4))
=> ~ pp(member2(list(A),L,aa(set(A),set(list(A)),lists(A),A4))) ) ) ).
% listsE
tff(fact_4149_lists_OCons,axiom,
! [A: $tType,A3: A,A4: set(A),L: list(A)] :
( pp(member2(A,A3,A4))
=> ( pp(member2(list(A),L,aa(set(A),set(list(A)),lists(A),A4)))
=> pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),L),aa(set(A),set(list(A)),lists(A),A4))) ) ) ).
% lists.Cons
tff(fact_4150_listrel__subset,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))))
=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),product_Sigma(list(A),list(A),aa(set(A),set(list(A)),lists(A),A4),aTP_Lamp_mk(set(A),fun(list(A),set(list(A))),A4)))) ) ).
% listrel_subset
tff(fact_4151_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_ml(fun(A,A),fun(nat,A),Q)) ) ) ).
% mono_funpow
tff(fact_4152_in__lists__conv__set,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( pp(member2(list(A),Xs,aa(set(A),set(list(A)),lists(A),A4)))
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(A,X4,A4)) ) ) ).
% in_lists_conv_set
tff(fact_4153_in__listsD,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( pp(member2(list(A),Xs,aa(set(A),set(list(A)),lists(A),A4)))
=> ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(A,X3,A4)) ) ) ).
% in_listsD
tff(fact_4154_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mono_invE
tff(fact_4155_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf(A)
& semilattice_inf(B) )
=> ! [F: fun(A,B),A4: A,B3: A] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,A4)),aa(A,B,F,B3)))) ) ) ).
% mono_inf
tff(fact_4156_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup(A)
& semilattice_sup(B) )
=> ! [F: fun(A,B),A4: A,B3: A] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,A4)),aa(A,B,F,B3))),aa(A,B,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3)))) ) ) ).
% mono_sup
tff(fact_4157_lists_Osimps,axiom,
! [A: $tType,A3: list(A),A4: set(A)] :
( pp(member2(list(A),A3,aa(set(A),set(list(A)),lists(A),A4)))
<=> ( ( A3 = nil(A) )
| ? [A6: A,L4: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),L4) )
& pp(member2(A,A6,A4))
& pp(member2(list(A),L4,aa(set(A),set(list(A)),lists(A),A4))) ) ) ) ).
% lists.simps
tff(fact_4158_lists_Ocases,axiom,
! [A: $tType,A3: list(A),A4: set(A)] :
( pp(member2(list(A),A3,aa(set(A),set(list(A)),lists(A),A4)))
=> ( ( A3 != nil(A) )
=> ~ ! [A5: A,L2: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),L2) )
=> ( pp(member2(A,A5,A4))
=> ~ pp(member2(list(A),L2,aa(set(A),set(list(A)),lists(A),A4))) ) ) ) ) ).
% lists.cases
tff(fact_4159_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list(A),F: fun(B,A),Q: set(B)] :
( pp(member2(list(A),X,aa(set(A),set(list(A)),lists(A),aa(set(B),set(A),image2(B,A,F),Q))))
=> ~ ! [Xo: list(B)] :
( pp(member2(list(B),Xo,aa(set(B),set(list(B)),lists(B),Q)))
=> ( X != aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xo) ) ) ) ).
% lists_image_witness
tff(fact_4160_in__prod__fst__sndI,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A4: set(A),B3: set(B)] :
( pp(member2(A,aa(product_prod(A,B),A,product_fst(A,B),X),A4))
=> ( pp(member2(B,aa(product_prod(A,B),B,product_snd(A,B),X),B3))
=> pp(member2(product_prod(A,B),X,product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))) ) ) ).
% in_prod_fst_sndI
tff(fact_4161_R__subset__Field,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,field2(A,R),aTP_Lamp_mm(set(product_prod(A,A)),fun(A,set(A)),R)))) ).
% R_subset_Field
tff(fact_4162_listrel__refl__on,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( refl_on(A,A4,R3)
=> refl_on(list(A),aa(set(A),set(list(A)),lists(A),A4),listrel(A,A,R3)) ) ).
% listrel_refl_on
tff(fact_4163_lists__eq__set,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(list(A)),lists(A),A4) = collect(list(A),aTP_Lamp_mn(set(A),fun(list(A),bool),A4)) ).
% lists_eq_set
tff(fact_4164_Rings_Omono__mult,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> order_mono(A,A,aa(A,fun(A,A),times_times(A),A3)) ) ) ).
% Rings.mono_mult
tff(fact_4165_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [F: fun(A,A),P4: A,K: nat] :
( order_mono(A,A,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,P4)),P4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,compow(fun(A,A),K,F),bot_bot(A))),P4)) ) ) ) ).
% Kleene_iter_lpfp
tff(fact_4166_lists__mono,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(A),set(list(A)),lists(A),A4)),aa(set(A),set(list(A)),lists(A),B3))) ) ).
% lists_mono
tff(fact_4167_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))))
<=> ( ( A4 = bot_bot(set(A)) )
| ( B3 = bot_bot(set(B)) )
| ( pp(aa(set(A),bool,finite_finite2(A),A4))
& pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ) ).
% finite_cartesian_product_iff
tff(fact_4168_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))))
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ).
% finite_cartesian_productD2
tff(fact_4169_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))))
=> ( ( B3 != bot_bot(set(B)) )
=> pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ).
% finite_cartesian_productD1
tff(fact_4170_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: fun(A,set(B))] :
( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_mo(set(A),fun(fun(A,set(B)),fun(A,bool)),A4),B3))))
=> ( ! [A5: A] :
( pp(member2(A,A5,A4))
=> pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,A5))) )
=> pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,B3))) ) ) ).
% finite_SigmaI2
tff(fact_4171_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E5: set(product_prod(A,A)),U2: set(A)] :
( pp(member2(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))),E5),product_Sigma(A,A,U2,aTP_Lamp_mj(set(A),fun(A,set(A)),U2))))))
=> pp(member2(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,E5))) ) ).
% Restr_trancl_mono
tff(fact_4172_set__zip__cart,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L3: list(B)] :
( pp(member2(product_prod(A,B),X,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L),L3))))
=> pp(member2(product_prod(A,B),X,product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_mg(list(B),fun(A,set(B)),L3)))) ) ).
% set_zip_cart
tff(fact_4173_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),S: set(A)] :
( order_mono(A,B,F)
=> ( ? [X3: A] :
( pp(member2(A,X3,S))
& ! [Xa4: A] :
( pp(member2(A,Xa4,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa4)) ) )
=> ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_mp(fun(A,B),fun(set(A),fun(B,bool)),F),S)) = aa(A,B,F,ord_Least(A,aTP_Lamp_mq(set(A),fun(A,bool),S))) ) ) ) ) ).
% Least_mono
tff(fact_4174_lists__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] : aa(set(A),set(list(A)),lists(A),aa(set(B),set(A),image2(B,A,F),A4)) = aa(set(list(B)),set(list(A)),image2(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),aa(set(B),set(list(B)),lists(B),A4)) ).
% lists_image
tff(fact_4175_funpow__decreasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [M2: nat,N: nat,F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( order_mono(A,A,F)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,compow(fun(A,A),M2,F),bot_bot(A))),aa(A,A,compow(fun(A,A),N,F),bot_bot(A)))) ) ) ) ).
% funpow_decreasing
tff(fact_4176_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_mono(A,B,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798349783984er_Max(A,A4)) = lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F),A4)) ) ) ) ) ) ).
% mono_Max_commute
tff(fact_4177_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_mono(A,B,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798350308766er_Min(A,A4)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F),A4)) ) ) ) ) ) ).
% mono_Min_commute
tff(fact_4178_lfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),K: nat] :
( order_mono(A,A,F)
=> ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F),bot_bot(A)) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) )
=> ( complete_lattice_lfp(A,F) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) ) ) ) ) ).
% lfp_Kleene_iter
tff(fact_4179_lists__of__len__fin2,axiom,
! [A: $tType,P: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),P))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),P)),collect(list(A),aTP_Lamp_mr(nat,fun(list(A),bool),N))))) ) ).
% lists_of_len_fin2
tff(fact_4180_lists__of__len__fin1,axiom,
! [A: $tType,P: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),P))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),aa(set(A),set(list(A)),lists(A),P)),collect(list(A),aTP_Lamp_ms(nat,fun(list(A),bool),N))))) ) ).
% lists_of_len_fin1
tff(fact_4181_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A4: 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_me(set(B),fun(A,set(B)),A4))) = finite_card(B,A4) ).
% card_cartesian_product_singleton
tff(fact_4182_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_mt(set(A),fun(nat,fun(list(A),bool)),A4),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_dn(list(A),fun(A,list(A))))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),A4),N)),aTP_Lamp_mu(set(A),fun(list(A),set(A)),A4))) ).
% lists_length_Suc_eq
tff(fact_4183_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set(B),B3: set(A)] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_an(set(A),fun(B,set(A)),B3))) = bot_bot(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_an(set(A),fun(B,set(A)),B3))) = B3 ) ) ) ).
% snd_image_times
tff(fact_4184_fst__image__times,axiom,
! [B: $tType,A: $tType,B3: set(B),A4: set(A)] :
( ( ( B3 = bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))) = bot_bot(set(A)) ) )
& ( ( B3 != bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))) = A4 ) ) ) ).
% fst_image_times
tff(fact_4185_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] : product_Sigma(A,B,A4,aTP_Lamp_mv(A,set(B))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty2
tff(fact_4186_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)) = bot_bot(set(product_prod(A,B))) )
<=> ( ( A4 = bot_bot(set(A)) )
| ( B3 = bot_bot(set(B)) ) ) ) ).
% Times_empty
tff(fact_4187_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B3: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B3) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty1
tff(fact_4188_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,bool))] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),complete_lattice_lfp(set(product_prod(A,B)),F)))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
=> ( ! [A5: A,B4: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,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)),F)),collect(product_prod(A,B),product_case_prod(A,B,bool,P))))))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B4)) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) ) ) ) ).
% lfp_induct2
tff(fact_4189_mono__Int,axiom,
! [B: $tType,A: $tType,F: fun(set(A),set(B)),A4: set(A),B3: set(A)] :
( order_mono(set(A),set(B),F)
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F,A4)),aa(set(A),set(B),F,B3)))) ) ).
% mono_Int
tff(fact_4190_mono__Un,axiom,
! [B: $tType,A: $tType,F: fun(set(A),set(B)),A4: set(A),B3: set(A)] :
( order_mono(set(A),set(B),F)
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F,A4)),aa(set(A),set(B),F,B3))),aa(set(A),set(B),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))) ) ).
% mono_Un
tff(fact_4191_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: fun(A,fun(A,bool))] : order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_kc(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).
% ord.lexordp.mono
tff(fact_4192_finite_Omono,axiom,
! [A: $tType] : order_mono(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_kt(fun(set(A),bool),fun(set(A),bool))) ).
% finite.mono
tff(fact_4193_lexordp_Omono,axiom,
! [A: $tType] :
( ord(A)
=> order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_ka(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ).
% lexordp.mono
tff(fact_4194_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B),C4: set(A),D2: set(B)] :
( ( product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)) = product_Sigma(A,B,C4,aTP_Lamp_me(set(B),fun(A,set(B)),D2)) )
<=> ( ( ( A4 = C4 )
& ( B3 = D2 ) )
| ( ( ( A4 = bot_bot(set(A)) )
| ( B3 = bot_bot(set(B)) ) )
& ( ( C4 = bot_bot(set(A)) )
| ( D2 = bot_bot(set(B)) ) ) ) ) ) ).
% times_eq_iff
tff(fact_4195_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set(A),J4: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I5),J4),C4) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J4,C4)) ).
% Sigma_Int_distrib1
tff(fact_4196_Sigma__empty__iff,axiom,
! [A: $tType,B: $tType,I5: set(A),X5: fun(A,set(B))] :
( ( product_Sigma(A,B,I5,X5) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] :
( pp(member2(A,X4,I5))
=> ( aa(A,set(B),X5,X4) = bot_bot(set(B)) ) ) ) ).
% Sigma_empty_iff
tff(fact_4197_Times__Int__distrib1,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3),aTP_Lamp_me(set(B),fun(A,set(B)),C4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B3,aTP_Lamp_me(set(B),fun(A,set(B)),C4))) ).
% Times_Int_distrib1
tff(fact_4198_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B)),B3: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_mw(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(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))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,A4)),product_Sigma(A,B,I5,B3)) ).
% Sigma_Int_distrib2
tff(fact_4199_Times__Int__Times,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B),C4: set(A),D2: 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,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))),product_Sigma(A,B,C4,aTP_Lamp_me(set(B),fun(A,set(B)),D2))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4),aa(set(B),fun(A,set(B)),aTP_Lamp_mx(set(B),fun(set(B),fun(A,set(B))),B3),D2)) ).
% Times_Int_Times
tff(fact_4200_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),C4: set(B),B3: set(A),D2: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B3,aTP_Lamp_me(set(B),fun(A,set(B)),D2))))
<=> ( ( A4 = bot_bot(set(A)) )
| ( C4 = bot_bot(set(B)) )
| ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),D2)) ) ) ) ).
% times_subset_iff
tff(fact_4201_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: 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,A4,B3)) = collect(A,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_mo(set(A),fun(fun(A,set(B)),fun(A,bool)),A4),B3)) ).
% fst_image_Sigma
tff(fact_4202_Refl__Field__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R3),R3)
=> ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),field2(A,R3)),A4) ) ) ).
% Refl_Field_Restr
tff(fact_4203_Refl__Field__Restr2,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R3),R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R3)))
=> ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).
% Refl_Field_Restr2
tff(fact_4204_Total__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( total_on(A,field2(A,R3),R3)
=> total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Total_Restr
tff(fact_4205_total__on__imp__Total__Restr,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( total_on(A,A4,R3)
=> total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% total_on_imp_Total_Restr
tff(fact_4206_mono__compose,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Q: fun(A,fun(B,C)),F: 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_my(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F)) ) ) ).
% mono_compose
tff(fact_4207_Restr__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A),R3: set(product_prod(A,A))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))) = 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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))) ) ) ).
% Restr_subset
tff(fact_4208_Restr__Field,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,field2(A,R3),aTP_Lamp_mm(set(product_prod(A,A)),fun(A,set(A)),R3))) = R3 ).
% Restr_Field
tff(fact_4209_Field__Restr__subset,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))))),A4)) ).
% Field_Restr_subset
tff(fact_4210_Refl__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R3),R3)
=> refl_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Refl_Restr
tff(fact_4211_def__lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: A,F: fun(A,A),P: A] :
( ( A4 = complete_lattice_lfp(A,F) )
=> ( order_mono(A,A,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),P))),P))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),P)) ) ) ) ) ).
% def_lfp_induct
tff(fact_4212_lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),P: A] :
( order_mono(A,A,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F)),P))),P))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F)),P)) ) ) ) ).
% lfp_induct
tff(fact_4213_def__lfp__induct__set,axiom,
! [A: $tType,A4: set(A),F: fun(set(A),set(A)),A3: A,P: fun(A,bool)] :
( ( A4 = complete_lattice_lfp(set(A),F) )
=> ( order_mono(set(A),set(A),F)
=> ( pp(member2(A,A3,A4))
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),collect(A,P)))))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ) ) ).
% def_lfp_induct_set
tff(fact_4214_lfp__induct__set,axiom,
! [A: $tType,A3: A,F: fun(set(A),set(A)),P: fun(A,bool)] :
( pp(member2(A,A3,complete_lattice_lfp(set(A),F)))
=> ( order_mono(set(A),set(A),F)
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F)),collect(A,P)))))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% lfp_induct_set
tff(fact_4215_partition__filter__conv,axiom,
! [A: $tType,F: fun(A,bool),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),F),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(fun(A,bool),fun(list(A),list(A)),filter2(A),F),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),comp(bool,bool,A,fNot,F)),Xs)) ).
% partition_filter_conv
tff(fact_4216_partition__rev__filter__conv,axiom,
! [A: $tType,P: fun(A,bool),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),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),comp(bool,bool,A,fNot,P)),Xs))),No2)) ).
% partition_rev_filter_conv
tff(fact_4217_rtrancl__last__visit__node,axiom,
! [A: $tType,S4: A,S5: A,R: set(product_prod(A,A)),Sh: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S4),S5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( ( ( S4 != Sh )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S4),S5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mz(A,fun(A,set(A)),Sh)))))) )
| ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S4),Sh),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mz(A,fun(A,set(A)),Sh)))))) ) ) ) ).
% rtrancl_last_visit_node
tff(fact_4218_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),A4: 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_na(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),A4) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_nb(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),A4)) ).
% image_split_eq_Sigma
tff(fact_4219_vimageI,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A3: B,B2: A,B3: set(A)] :
( ( aa(B,A,F,A3) = B2 )
=> ( pp(member2(A,B2,B3))
=> pp(member2(B,A3,aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),B3))) ) ) ).
% vimageI
tff(fact_4220_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F: fun(A,B),B3: set(B)] :
( pp(member2(A,A3,aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)))
<=> pp(member2(B,aa(A,B,F,A3),B3)) ) ).
% vimage_eq
tff(fact_4221_vimage__ident,axiom,
! [A: $tType,Y5: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_ab(A,A)),Y5) = Y5 ).
% vimage_ident
tff(fact_4222_vimage__Collect__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,B),P: fun(B,bool)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),collect(B,P)) = collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_nc(fun(A,B),fun(fun(B,bool),fun(A,bool)),F),P)) ).
% vimage_Collect_eq
tff(fact_4223_vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),bot_bot(set(B))) = bot_bot(set(A)) ).
% vimage_empty
tff(fact_4224_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),top_top(set(B))) = top_top(set(A)) ).
% vimage_UNIV
tff(fact_4225_vimage__Int,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).
% vimage_Int
tff(fact_4226_vimage__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).
% vimage_Un
tff(fact_4227_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : pp(member2(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),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))) ).
% listrel_rtrancl_refl
tff(fact_4228_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A4: set(B)] :
( ( pp(member2(B,C3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_ck(B,fun(A,B),C3)),A4) = top_top(set(A)) ) )
& ( ~ pp(member2(B,C3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_ck(B,fun(A,B),C3)),A4) = bot_bot(set(A)) ) ) ) ).
% vimage_const
tff(fact_4229_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(A)] : aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))) ).
% image_vimage_eq
tff(fact_4230_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A4: set(B),D3: B,B3: set(A)] :
( ( pp(member2(B,C3,A4))
=> ( ( pp(member2(B,D3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_nd(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B3)),A4) = top_top(set(A)) ) )
& ( ~ pp(member2(B,D3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_nd(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B3)),A4) = B3 ) ) ) )
& ( ~ pp(member2(B,C3,A4))
=> ( ( pp(member2(B,D3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_nd(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B3)),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B3) ) )
& ( ~ pp(member2(B,D3,A4))
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_nd(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B3)),A4) = bot_bot(set(A)) ) ) ) ) ) ).
% vimage_if
tff(fact_4231_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,B),G: fun(A,B),Y: set(B)] :
( ! [W3: A] :
( pp(member2(A,W3,S))
=> ( aa(A,B,F,W3) = aa(A,B,G,W3) ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),G),Y)),S) ) ) ).
% vimage_inter_cong
tff(fact_4232_vimage__Diff,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),minus_minus(set(B),A4,B3)) = minus_minus(set(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).
% vimage_Diff
tff(fact_4233_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R: set(product_prod(A,A))] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh),R))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R))) ) ) ) ) ).
% converse_rtranclE'
tff(fact_4234_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3))),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)))) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_4235_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F: fun(A,B),A4: set(B)] :
( pp(member2(A,A3,aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)))
=> pp(member2(B,aa(A,B,F,A3),A4)) ) ).
% vimageD
tff(fact_4236_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F: fun(A,B),B3: set(B)] :
( pp(member2(A,A3,aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)))
=> pp(member2(B,aa(A,B,F,A3),B3)) ) ).
% vimageE
tff(fact_4237_vimageI2,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A3: B,A4: set(A)] :
( pp(member2(A,aa(B,A,F,A3),A4))
=> pp(member2(B,A3,aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4))) ) ).
% vimageI2
tff(fact_4238_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: fun(B,bool),F: fun(A,B),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(B,bool,P,aa(A,B,F,X2)))
<=> pp(aa(A,bool,Q,X2)) )
=> ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),collect(B,P)) = collect(A,Q) ) ) ).
% vimage_Collect
tff(fact_4239_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)) ).
% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_4240_vimage__Compl,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),uminus_uminus(set(B)),A4)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)) ).
% vimage_Compl
tff(fact_4241_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),S3))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),S3))) ) ).
% rtrancl_mono_rightI
tff(fact_4242_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set(product_prod(A,A)),V3: set(product_prod(A,A)),X: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V3))
=> ( pp(member2(product_prod(A,A),X,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),U2)))
=> pp(member2(product_prod(A,A),X,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),V3))) ) ) ).
% rtrancl_mono_mp
tff(fact_4243_r__le__rtrancl,axiom,
! [A: $tType,S: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),S))) ).
% r_le_rtrancl
tff(fact_4244_subset__vimage__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),B3: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)))
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(member2(B,aa(A,B,F,X4),B3)) ) ) ).
% subset_vimage_iff
tff(fact_4245_vimage__mono,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(A),F: fun(B,A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),B3))) ) ).
% vimage_mono
tff(fact_4246_vimage__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3) = collect(A,aa(set(B),fun(A,bool),aTP_Lamp_ne(fun(A,B),fun(set(B),fun(A,bool)),F),B3)) ).
% vimage_def
tff(fact_4247_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod(A,A),R: set(product_prod(A,A)),R3: product_prod(A,A)] :
( pp(member2(product_prod(A,A),X,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> pp(member2(product_prod(A,A),X,aa(set(product_prod(A,A)),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)),R3),R)))) ) ).
% in_rtrancl_insert
tff(fact_4248_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),A4)),B3))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),B3))) ) ).
% image_subset_iff_subset_vimage
tff(fact_4249_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4))),A4)) ).
% image_vimage_subset
tff(fact_4250_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F: fun(A,B),B2: B] :
( pp(member2(A,A3,aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B))))))
<=> ( aa(A,B,F,A3) = B2 ) ) ).
% vimage_singleton_eq
tff(fact_4251_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))
=> ( pp(member2(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),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3))))
=> pp(member2(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)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_4252_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)),aa(set(product_prod(A,A)),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)),X),R)))) ).
% rtrancl_sub_insert_rtrancl
tff(fact_4253_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Zs: list(A)] :
( pp(member2(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),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3))))
=> ( pp(member2(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),Zs),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3))))
=> pp(member2(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),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))) ) ) ).
% listrel_rtrancl_trans
tff(fact_4254_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,product_prod(B,C)),A4: set(B),B3: set(C)] : aa(set(product_prod(B,C)),set(A),aa(fun(A,product_prod(B,C)),fun(set(product_prod(B,C)),set(A)),vimage(A,product_prod(B,C)),F),product_Sigma(B,C,A4,aTP_Lamp_nf(set(C),fun(B,set(C)),B3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),comp(product_prod(B,C),B,A,product_fst(B,C),F)),A4)),aa(set(C),set(A),aa(fun(A,C),fun(set(C),set(A)),vimage(A,C),comp(product_prod(B,C),C,A,product_snd(B,C),F)),B3)) ).
% vimage_Times
tff(fact_4255_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A4: set(B),F: fun(B,set(A))] :
( ( pp(member2(B,X,A4))
=> ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A4,F)) = aa(B,set(A),F,X) ) )
& ( ~ pp(member2(B,X,A4))
=> ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A4,F)) = bot_bot(set(A)) ) ) ) ).
% Pair_vimage_Sigma
tff(fact_4256_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A4) = bot_bot(set(B)) )
<=> ( A4 = bot_bot(set(A)) ) ) ) ).
% surj_vimage_empty
tff(fact_4257_vimage__insert,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A3: B,B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3)) ).
% vimage_insert
tff(fact_4258_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F2: set(A),H: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( inj_on(B,A,H,A4)
=> pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),F2)),A4))) ) ) ).
% finite_vimage_IntI
tff(fact_4259_inv__image__partition,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( ! [Y2: A] :
( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Ys)))
=> ~ pp(aa(A,bool,P,Y2)) )
=> ( aa(set(product_prod(list(A),list(A))),set(list(A)),aa(fun(list(A),product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(list(A))),vimage(list(A),product_prod(list(A),list(A))),aa(fun(A,bool),fun(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)))))) = aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys) ) ) ) ).
% inv_image_partition
tff(fact_4260_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E5: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
( pp(member2(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))),E5),U2))))
=> ( ~ pp(member2(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,E5)))
=> ? [X2: A,Y2: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),X2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E5),U2))))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2),U2))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),W),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E5),U2)))) ) ) ) ).
% trancl_union_outside
tff(fact_4261_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),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,if(product_prod(list(A),list(A)),aa(A,bool,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_4262_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool),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_4263_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),X: A] :
( pp(member2(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),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3))))
=> pp(member2(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),X),Ys)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)))) ) ).
% rtrancl_listrel1_ConsI1
tff(fact_4264_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E5: set(product_prod(A,A))] :
( pp(member2(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)),E5))))
=> ( ~ pp(member2(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,E5)))
=> ~ ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))) ) ) ) ).
% trancl_over_edgeE
tff(fact_4265_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3))))
=> ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).
% rtrancl_listrel1_eq_len
tff(fact_4266_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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,R3)))
=> pp(member2(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),listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))) ) ).
% listrel_reflcl_if_listrel1
tff(fact_4267_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),listrel(A,A,R3)))
=> pp(member2(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),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)))) ) ).
% rtrancl_listrel1_if_listrel
tff(fact_4268_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E5: set(product_prod(A,A)),U2: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E5),product_Sigma(A,A,U2,aTP_Lamp_mj(set(A),fun(A,set(A)),U2))))))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5))) ) ).
% Restr_rtrancl_mono
tff(fact_4269_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite2(B),A4))
=> ? [X2: A] :
( pp(member2(A,X2,aa(set(B),set(A),image2(B,A,F),A4)))
& ~ pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))) ) ) ) ).
% inf_img_fin_dom
tff(fact_4270_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite2(B),A4))
=> ~ ! [Y2: A] :
( pp(member2(A,Y2,aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),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_4271_partition_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)) ).
% partition.simps(1)
tff(fact_4272_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F2: set(A),H: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),F2))
=> ( ! [Y2: A] :
( pp(member2(A,Y2,F2))
=> pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A4))) )
=> pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),H),F2)),A4))) ) ) ).
% finite_finite_vimage_IntI
tff(fact_4273_partition__P,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
=> ( ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Yes2)))
=> pp(aa(A,bool,P,X3)) )
& ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),No2)))
=> ~ pp(aa(A,bool,P,X3)) ) ) ) ).
% partition_P
tff(fact_4274_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E5: set(product_prod(A,A)),F: fun(A,B)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)))
=> pp(member2(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,F,A3)),aa(A,B,F,B2)),aa(set(product_prod(B,B)),set(product_prod(B,B)),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,F)),E5)))) ) ).
% rtrancl_mapI
tff(fact_4275_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R3)))) ).
% listrel_subset_rtrancl_listrel1
tff(fact_4276_vimage__eq__UN,axiom,
! [A: $tType,B: $tType,F: fun(A,B),B3: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ng(fun(A,B),fun(B,set(A)),F)),B3)) ).
% vimage_eq_UN
tff(fact_4277_partition_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),product_case_prod(list(A),list(A),product_prod(list(A),list(A)),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_nh(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),Xs)) ).
% partition.simps(2)
tff(fact_4278_partition__filter1,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ).
% partition_filter1
tff(fact_4279_rtrancl__last__touch,axiom,
! [A: $tType,Q2: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( pp(member2(A,Q2,S))
=> ~ ! [Qt: A] :
( pp(member2(A,Qt,S))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),R,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).
% rtrancl_last_touch
tff(fact_4280_rtrancl__last__visit_H,axiom,
! [A: $tType,Q2: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),R,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),S))))))
=> ~ ! [Qt: A] :
( pp(member2(A,Qt,S))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),R,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).
% rtrancl_last_visit'
tff(fact_4281_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite2(B),A4))
=> ? [X2: A] :
( pp(member2(A,X2,aa(set(B),set(A),image2(B,A,F),A4)))
& ~ pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))),A4))) ) ) ) ).
% inf_img_fin_dom'
tff(fact_4282_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite2(B),A4))
=> ~ ! [Y2: A] :
( pp(member2(A,Y2,aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))),A4))) ) ) ) ).
% inf_img_fin_domE'
tff(fact_4283_partition__rev_Oelims,axiom,
! [A: $tType,X: fun(A,bool),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) )
=> ! [X2: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X2),aa(list(A),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),X2),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),X2),No))),Xs2) ) ) ) ) ) ).
% partition_rev.elims
tff(fact_4284_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F: fun(A,B),D2: set(A),A4: set(B)] :
( inj_on(A,B,F,D2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),A4)),D2))),finite_card(B,A4))) ) ) ).
% card_vimage_inj_on_le
tff(fact_4285_partition__set,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes2)),aa(list(A),set(A),set2(A),No2)) = aa(list(A),set(A),set2(A),Xs) ) ) ).
% partition_set
tff(fact_4286_rtrancl__last__visit,axiom,
! [A: $tType,Q2: A,Q4: A,R: set(product_prod(A,A)),S: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),R,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),S))))))
=> ~ ! [Qt: A] :
( pp(member2(A,Qt,S))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),Qt),transitive_trancl(A,R)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),R,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).
% rtrancl_last_visit
tff(fact_4287_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),M2: A,X5: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),M2),bot_bot(set(A))),aTP_Lamp_mj(set(A),fun(A,set(A)),X5))))))
=> ( ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R3)))
=> ~ ! [X2: A] :
( pp(member2(A,X2,X5))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3))) ) ) ) ) ).
% trancl_multi_insert2
tff(fact_4288_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),X5: set(A),M2: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),product_Sigma(A,A,X5,aTP_Lamp_ni(A,fun(A,set(A)),M2))))))
=> ( ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R3)))
=> ~ ! [X2: A] :
( pp(member2(A,X2,X5))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3))) ) ) ) ) ).
% trancl_multi_insert
tff(fact_4289_partition__filter2,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(fun(A,bool),fun(list(A),product_prod(list(A),list(A))),partition(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),comp(bool,bool,A,fNot,P)),Xs) ).
% partition_filter2
tff(fact_4290_partition__rev_Opelims,axiom,
! [A: $tType,X: fun(A,bool),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),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) )
=> ~ accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( 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) )
=> ! [X2: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X2),aa(list(A),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),X2),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),X2),No))),Xs2) )
=> ~ accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)))) ) ) ) ) ) ) ).
% partition_rev.pelims
tff(fact_4291_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R,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),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_nj(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R),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_4292_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),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 ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != aa(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_nj(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X2)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) ) ) ) ) ).
% quicksort_by_rel.elims
tff(fact_4293_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),A3: B] :
( inj_on(A,B,F,A4)
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nk(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F),A4),A3))),bot_bot(set(A))))) ) ).
% inj_on_vimage_singleton
tff(fact_4294_set__quicksort__by__rel,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R,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_4295_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R,Sl2),nil(A)) = Sl2 ).
% quicksort_by_rel.simps(1)
tff(fact_4296_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list(A),R: fun(A,fun(A,bool)),Xs: list(A)] :
( ( Sl2 != nil(A) )
=> ( aa(list(A),list(A),quicksort_by_rel(A,R,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,R,nil(A)),Xs)),Sl2) ) ) ).
% quicksort_by_rel_remove_acc_guared
tff(fact_4297_quicksort__by__rel__remove__acc,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R,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,R,nil(A)),Xs)),Sl2) ).
% quicksort_by_rel_remove_acc
tff(fact_4298_sorted__wrt__quicksort__by__rel,axiom,
! [X6: $tType,R: fun(X6,fun(X6,bool)),Xs: list(X6)] :
( ! [X2: X6,Y2: X6] :
( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Y2))
| pp(aa(X6,bool,aa(X6,fun(X6,bool),R,Y2),X2)) )
=> ( ! [X2: X6,Y2: X6,Z4: X6] :
( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Y2))
=> ( pp(aa(X6,bool,aa(X6,fun(X6,bool),R,Y2),Z4))
=> pp(aa(X6,bool,aa(X6,fun(X6,bool),R,X2),Z4)) ) )
=> sorted_wrt(X6,R,aa(list(X6),list(X6),quicksort_by_rel(X6,R,nil(X6)),Xs)) ) ) ).
% sorted_wrt_quicksort_by_rel
tff(fact_4299_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_4300_the__elem__def,axiom,
! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_nl(set(A),fun(A,bool),X5)) ).
% the_elem_def
tff(fact_4301_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).
% sort_quicksort_by_rel
tff(fact_4302_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: B] :
( inj_on(A,B,F,top_top(set(A)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,bool),aTP_Lamp_nm(fun(A,B),fun(B,fun(A,bool)),F),A3))),bot_bot(set(A))))) ) ).
% inj_vimage_singleton
tff(fact_4303_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R),aa(list(A),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,R,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),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_nj(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R),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_4304_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa),Xb) = Y )
=> ( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = aa(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_nj(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X2)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)))) ) ) ) ) ) ).
% quicksort_by_rel.pelims
tff(fact_4305_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)))] :
( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22)))
=> ( ! [R4: fun(A,fun(A,bool)),Sl: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R4),Sl),nil(A))) )
=> ( ! [R4: fun(A,fun(A,bool)),Sl: list(A),X2: A,Xs2: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),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),X2),Xs2))))
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y4: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R4),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y4) = Xa3 )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R4),Sl),Y4)) ) )
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y4: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R4),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y4) = Xa3 )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),quicksort_by_rel(A,R4,Sl),Y4))),Xb2)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R4),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2))) ) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,A0),A1),A22)) ) ) ) ).
% quicksort_by_rel.pinduct
tff(fact_4306_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A)] :
( accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R),aa(list(A),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,R,Sl2),nil(A)) = Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
tff(fact_4307_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),minus_minus(set(product_prod(A,A)),E5,product_Sigma(A,A,top_top(set(A)),aTP_Lamp_mj(set(A),fun(A,set(A)),R))))))
=> ( ~ pp(member2(A,U,R))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),rel_restrict(A,E5,R)))) ) ) ).
% rtrancl_restrictI
tff(fact_4308_Preorder__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_preorder_on(A,field2(A,R3),R3)
=> order_preorder_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Preorder_Restr
tff(fact_4309_sum__mult__sum__if__inj,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [F: fun(A,B),G: fun(C,B),A4: set(A),B3: set(C)] :
( inj_on(product_prod(A,C),B,product_case_prod(A,C,B,aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_gh(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F),G)),product_Sigma(A,C,A4,aTP_Lamp_nn(set(C),fun(A,set(C)),B3)))
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A4)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B3)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),collect(B,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_no(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F),G),A4),B3))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_4310_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),A4: set(A),Z2: B,Y: B,A3: A] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),S))
=> ( finite_fold_graph(A,B,F,Z2,A4,Y)
=> ( pp(member2(A,A3,A4))
=> ? [Y6: B] :
( ( Y = aa(B,B,aa(A,fun(B,B),F,A3),Y6) )
& finite_fold_graph(A,B,F,Z2,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),Y6) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_4311_list_Omap__id0,axiom,
! [A: $tType] : aa(fun(A,A),fun(list(A),list(A)),map(A,A),id(A)) = id(list(A)) ).
% list.map_id0
tff(fact_4312_rel__restrict__empty,axiom,
! [A: $tType,R: set(product_prod(A,A))] : rel_restrict(A,R,bot_bot(set(A))) = R ).
% rel_restrict_empty
tff(fact_4313_rotate0,axiom,
! [A: $tType] : aa(nat,fun(list(A),list(A)),rotate(A),zero_zero(nat)) = id(list(A)) ).
% rotate0
tff(fact_4314_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B] : finite_fold_graph(A,B,F,Z2,bot_bot(set(A)),Z2) ).
% fold_graph.emptyI
tff(fact_4315_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,X: B] :
( finite_fold_graph(A,B,F,Z2,bot_bot(set(A)),X)
=> ( X = Z2 ) ) ).
% empty_fold_graphE
tff(fact_4316_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A4: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,A4,R)))
=> ~ pp(member2(A,Y,R)) ) ).
% rel_restrict_notR(2)
tff(fact_4317_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A4: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,A4,R)))
=> ~ pp(member2(A,X,R)) ) ).
% rel_restrict_notR(1)
tff(fact_4318_rel__restrictI,axiom,
! [A: $tType,X: A,R: set(A),Y: A,E5: set(product_prod(A,A))] :
( ~ pp(member2(A,X,R))
=> ( ~ pp(member2(A,Y,R))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E5))
=> pp(member2(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,E5,R))) ) ) ) ).
% rel_restrictI
tff(fact_4319_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,E5,R)))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E5)) ) ).
% rel_restrict_lift
tff(fact_4320_List_Omap_Oidentity,axiom,
! [A: $tType] : aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ab(A,A)) = id(list(A)) ).
% List.map.identity
tff(fact_4321_list_Omap__id,axiom,
! [A: $tType,T4: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),id(A)),T4) = T4 ).
% list.map_id
tff(fact_4322_rel__restrict__mono,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B3: set(product_prod(A,A)),R: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A4),B3))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A4,R)),rel_restrict(A,B3,R))) ) ).
% rel_restrict_mono
tff(fact_4323_rel__restrict__sub,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R,A4)),R)) ).
% rel_restrict_sub
tff(fact_4324_rel__restrict__union,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B3: set(A)] : rel_restrict(A,R,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = rel_restrict(A,rel_restrict(A,R,A4),B3) ).
% rel_restrict_union
tff(fact_4325_finite__rel__restrict,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),rel_restrict(A,R,A4))) ) ).
% finite_rel_restrict
tff(fact_4326_foldr__Nil,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B))] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F),nil(A)) = id(B) ).
% foldr_Nil
tff(fact_4327_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F,Z2,A1,A22)
=> ( ( ( A1 = bot_bot(set(A)) )
=> ( A22 != Z2 ) )
=> ~ ! [X2: A,A8: set(A)] :
( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),A8) )
=> ! [Y2: B] :
( ( A22 = aa(B,B,aa(A,fun(B,B),F,X2),Y2) )
=> ( ~ pp(member2(A,X2,A8))
=> ~ finite_fold_graph(A,B,F,Z2,A8,Y2) ) ) ) ) ) ).
% fold_graph.cases
tff(fact_4328_fold__graph_Osimps,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F,Z2,A1,A22)
<=> ( ( ( A1 = bot_bot(set(A)) )
& ( A22 = Z2 ) )
| ? [X4: A,A9: set(A),Y3: B] :
( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),A9) )
& ( A22 = aa(B,B,aa(A,fun(B,B),F,X4),Y3) )
& ~ pp(member2(A,X4,A9))
& finite_fold_graph(A,B,F,Z2,A9,Y3) ) ) ) ).
% fold_graph.simps
tff(fact_4329_foldr__filter,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_np(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F),P)),Xs) ).
% foldr_filter
tff(fact_4330_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A4: set(product_prod(A,A)),R: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,rel_restrict(A,A4,R))))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),rel_restrict(A,transitive_trancl(A,A4),R))) ) ).
% rel_restrict_trancl_mem
tff(fact_4331_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,E5,R))))
=> ~ pp(member2(A,V,R)) ) ).
% rel_restrict_trancl_notR(1)
tff(fact_4332_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,E5,R))))
=> ~ pp(member2(A,W,R)) ) ).
% rel_restrict_trancl_notR(2)
tff(fact_4333_rel__restrict__mono2,axiom,
! [A: $tType,R: set(A),S: set(A),A4: set(product_prod(A,A))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R),S))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A4,S)),rel_restrict(A,A4,R))) ) ).
% rel_restrict_mono2
tff(fact_4334_rel__restrict__trancl__sub,axiom,
! [A: $tType,A4: set(product_prod(A,A)),R: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,A4,R))),rel_restrict(A,transitive_trancl(A,A4),R))) ).
% rel_restrict_trancl_sub
tff(fact_4335_rel__restrict__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R,A4) = collect(product_prod(A,A),product_case_prod(A,A,bool,aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_nq(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),R),A4))) ).
% rel_restrict_def
tff(fact_4336_rel__restrict__Int__empty,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),field2(A,R)) = bot_bot(set(A)) )
=> ( rel_restrict(A,R,A4) = R ) ) ).
% rel_restrict_Int_empty
tff(fact_4337_Field__rel__restrict,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,rel_restrict(A,R,A4))),minus_minus(set(A),field2(A,R),A4))) ).
% Field_rel_restrict
tff(fact_4338_rel__restrict__compl,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: 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,R,A4)),rel_restrict(A,R,aa(set(A),set(A),uminus_uminus(set(A)),A4))) = bot_bot(set(product_prod(A,A))) ).
% rel_restrict_compl
tff(fact_4339_homo__rel__restrict__mono,axiom,
! [A: $tType,R: set(product_prod(A,A)),B3: set(A),A4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R,A4)),product_Sigma(A,A,minus_minus(set(A),B3,A4),aa(set(A),fun(A,set(A)),aTP_Lamp_nr(set(A),fun(set(A),fun(A,set(A))),B3),A4)))) ) ).
% homo_rel_restrict_mono
tff(fact_4340_rel__restrict__alt__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R,A4) = 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))),R),product_Sigma(A,A,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_ns(set(A),fun(A,set(A)),A4))) ).
% rel_restrict_alt_def
tff(fact_4341_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A4: set(A),R: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),R)),transitive_trancl(A,product_Sigma(A,A,minus_minus(set(A),A4,R),aa(set(A),fun(A,set(A)),aTP_Lamp_nr(set(A),fun(set(A),fun(A,set(A))),A4),R))))) ).
% rel_restrict_Sigma_sub
tff(fact_4342_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_4343_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),comp(A,fun(A,A),B,inf_inf(A),F)),Xs),top_top(A)) ) ).
% INF_set_fold
tff(fact_4344_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),R: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P,R)) = same_fst(A,B,P,aTP_Lamp_nt(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R)) ).
% same_fst_trancl
tff(fact_4345_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),comp(A,fun(A,A),B,sup_sup(A),F)),Xs),bot_bot(A)) ) ).
% SUP_set_fold
tff(fact_4346_fold__append,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = comp(A,A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),Ys),aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),Xs)) ).
% fold_append
tff(fact_4347_fold__replicate,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),N: nat,X: B] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = compow(fun(A,A),N,aa(B,fun(A,A),F,X)) ).
% fold_replicate
tff(fact_4348_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,A)),S4: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),nil(B)),S4) = S4 ).
% fold_simps(1)
tff(fact_4349_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,A)),X: B,Xs: list(B),S4: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),S4) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),Xs),aa(A,A,aa(B,fun(A,A),F,X),S4)) ).
% fold_simps(2)
tff(fact_4350_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,bool),P: fun(B,bool),S4: B,F: fun(A,fun(B,B))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,Q,X2)) )
=> ( pp(aa(B,bool,P,S4))
=> ( ! [X2: A,S6: B] :
( pp(aa(A,bool,Q,X2))
=> ( pp(aa(B,bool,P,S6))
=> pp(aa(B,bool,P,aa(B,B,aa(A,fun(B,B),F,X2),S6))) ) )
=> pp(aa(B,bool,P,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs),S4))) ) ) ) ).
% fold_invariant
tff(fact_4351_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,Xs: list(B),Ys: list(B),F: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
( ( A3 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),Xs)))
=> ( aa(B,fun(A,A),F,X2) = aa(B,fun(A,A),G,X2) ) )
=> ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),Xs),A3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),Ys),B2) ) ) ) ) ).
% List.fold_cong
tff(fact_4352_foldl__conv__fold,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,A)),S4: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F),S4),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aTP_Lamp_de(fun(A,fun(B,A)),fun(B,fun(A,A)),F)),Xs),S4) ).
% foldl_conv_fold
tff(fact_4353_fold__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X: A,Xs: list(A)] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = comp(B,B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs),aa(A,fun(B,B),F,X)) ).
% fold_Cons
tff(fact_4354_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,C,B,H,aa(A,fun(B,B),G,X2)) = comp(C,C,B,aa(A,fun(C,C),F,X2),H) ) )
=> ( comp(B,C,B,H,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xs)) = comp(C,C,B,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),fold(A,C),F),Xs),H) ) ) ).
% fold_commute
tff(fact_4355_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C)),S4: B] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,C,B,H,aa(A,fun(B,B),G,X2)) = comp(C,C,B,aa(A,fun(C,C),F,X2),H) ) )
=> ( aa(B,C,H,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xs),S4)) = aa(C,C,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),fold(A,C),F),Xs),aa(B,C,H,S4)) ) ) ).
% fold_commute_apply
tff(fact_4356_fold__Nil,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B))] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),nil(A)) = id(B) ).
% fold_Nil
tff(fact_4357_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,fun(B,B),F,X2) = id(B) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs) = id(B) ) ) ).
% fold_id
tff(fact_4358_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Xs)) = aa(list(C),fun(A,A),aa(fun(C,fun(A,A)),fun(list(C),fun(A,A)),fold(C,A),comp(B,fun(A,A),C,G,F)),Xs) ).
% fold_map
tff(fact_4359_foldr__conv__fold,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F),Xs) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),aa(list(B),list(B),rev(B),Xs)) ).
% foldr_conv_fold
tff(fact_4360_fold__filter,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_np(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F),P)),Xs) ).
% fold_filter
tff(fact_4361_union__set__fold,axiom,
! [A: $tType,Xs: list(A),A4: 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)),A4) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),insert2(A)),Xs),A4) ).
% union_set_fold
tff(fact_4362_rev__conv__fold,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xs),nil(A)) ).
% rev_conv_fold
tff(fact_4363_fold__Cons__rev,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xs) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)) ).
% fold_Cons_rev
tff(fact_4364_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,B,B,aa(A,fun(B,B),F,Y2),aa(A,fun(B,B),F,X2)) = comp(B,B,B,aa(A,fun(B,B),F,X2),aa(A,fun(B,B),F,Y2)) ) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs) ) ) ).
% fold_rev
tff(fact_4365_List_Ounion__def,axiom,
! [A: $tType] : union(A) = aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),insert(A)) ).
% List.union_def
tff(fact_4366_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,B,B,aa(A,fun(B,B),F,Y2),aa(A,fun(B,B),F,X2)) = comp(B,B,B,aa(A,fun(B,B),F,X2),aa(A,fun(B,B),F,Y2)) ) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F),Xs) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs) ) ) ).
% foldr_fold
tff(fact_4367_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B)),X: A] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,B,B,aa(A,fun(B,B),F,X2),aa(A,fun(B,B),F,Y2)) = comp(B,B,B,aa(A,fun(B,B),F,Y2),aa(A,fun(B,B),F,X2)) ) ) )
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs) = comp(B,B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)),aa(A,fun(B,B),F,X)) ) ) ) ).
% fold_remove1_split
tff(fact_4368_minus__set__fold,axiom,
! [A: $tType,A4: set(A),Xs: list(A)] : minus_minus(set(A),A4,aa(list(A),set(A),set2(A),Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xs),A4) ).
% minus_set_fold
tff(fact_4369_sort__conv__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_az(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),linorder_insort_key(A,A,aTP_Lamp_az(A,A))),Xs),nil(A)) ) ).
% sort_conv_fold
tff(fact_4370_inter__coset__fold,axiom,
! [A: $tType,A4: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),coset(A,Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xs),A4) ).
% inter_coset_fold
tff(fact_4371_fold__append__concat__rev,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),fold(list(A),list(A)),append(A)),Xss2) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xss2))) ).
% fold_append_concat_rev
tff(fact_4372_Sup__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xs),bot_bot(A)) ) ).
% Sup_set_fold
tff(fact_4373_Inf__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xs),top_top(A)) ) ).
% Inf_set_fold
tff(fact_4374_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Xs: list(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xs),X) ) ).
% Inf_fin.set_eq_fold
tff(fact_4375_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Xs: list(A)] : lattic5882676163264333800up_fin(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xs),X) ) ).
% Sup_fin.set_eq_fold
tff(fact_4376_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] : lattic643756798349783984er_Max(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_max(A)),Xs),X) ) ).
% Max.set_eq_fold
tff(fact_4377_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] : lattic643756798350308766er_Min(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_min(A)),Xs),X) ) ).
% Min.set_eq_fold
tff(fact_4378_semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A,Xs: list(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),F),Xs),X) ) ) ).
% semilattice_set.set_eq_fold
tff(fact_4379_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] :
( inj_on(B,A,F,aa(list(B),set(B),set2(B),Xs))
=> ( aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(fun(B,fun(list(B),list(B))),fun(list(B),fun(list(B),list(B))),fold(B,list(B)),linorder_insort_key(B,A,F)),Xs),nil(B)) ) ) ) ).
% sort_key_conv_fold
tff(fact_4380_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Xs: list(A),Y: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
=> ( finite_fold(A,B,F,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),aa(list(A),list(A),remdups(A),Xs)),Y) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_4381_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_4382_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Xs: list(A),Y: B] :
( finite673082921795544331dem_on(A,B,S,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
=> ( finite_fold(A,B,F,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F),Xs),Y) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
tff(fact_4383_list_Oin__rel,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A3),B2))
<=> ? [Z3: list(product_prod(A,B))] :
( pp(member2(list(product_prod(A,B)),Z3,collect(list(product_prod(A,B)),aTP_Lamp_nu(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R))))
& ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Z3) = A3 )
& ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Z3) = B2 ) ) ) ).
% list.in_rel
tff(fact_4384_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Ys: list(B),Xs: list(B),F: fun(B,A)] :
( ( mset(B,Ys) = mset(B,Xs) )
=> ( ! [K2: B] :
( pp(member2(B,K2,aa(list(B),set(B),set2(B),Ys)))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(B,fun(B,bool),aTP_Lamp_nv(fun(B,A),fun(B,fun(B,bool)),F),K2)),Ys) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(B,fun(B,bool),aTP_Lamp_nv(fun(B,A),fun(B,fun(B,bool)),F),K2)),Xs) ) )
=> ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys))
=> ( aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs) = Ys ) ) ) ) ) ).
% properties_for_sort_key
tff(fact_4385_list__all2__Nil2,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),nil(B)))
<=> ( Xs = nil(A) ) ) ).
% list_all2_Nil2
tff(fact_4386_list__all2__Nil,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),nil(A)),Ys))
<=> ( Ys = nil(B) ) ) ).
% list_all2_Nil
tff(fact_4387_mset__zero__iff,axiom,
! [A: $tType,X: list(A)] :
( ( mset(A,X) = zero_zero(multiset(A)) )
<=> ( X = nil(A) ) ) ).
% mset_zero_iff
tff(fact_4388_mset__zero__iff__right,axiom,
! [A: $tType,X: list(A)] :
( ( zero_zero(multiset(A)) = mset(A,X) )
<=> ( X = nil(A) ) ) ).
% mset_zero_iff_right
tff(fact_4389_list__all2__rev,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),rev(A),Xs)),aa(list(B),list(B),rev(B),Ys)))
<=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys)) ) ).
% list_all2_rev
tff(fact_4390_mergesort__by__rel__permutes,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A)] : mset(A,aa(list(A),list(A),mergesort_by_rel(A,R),Xs)) = mset(A,Xs) ).
% mergesort_by_rel_permutes
tff(fact_4391_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] : mset(A,merges9089515139780605204_merge(A,R,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_4392_quicksort__permutes,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : mset(A,aa(list(A),list(A),linorder_quicksort(A),Xs)) = mset(A,Xs) ) ).
% quicksort_permutes
tff(fact_4393_quicksort__by__rel__permutes,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : mset(A,aa(list(A),list(A),quicksort_by_rel(A,R,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_4394_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list(A),Xs23: 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),Xs23),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),Xs23),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,Xs23)) ).
% mset_mergesort_by_rel_split
tff(fact_4395_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X: list(A),Ya: list(A),Y: list(B),Xa: list(B),R: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),Ya)))
=> ( pp(member2(B,Yb,aa(list(B),set(B),set2(B),Xa)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z4),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z4),Yb)) ) ) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
<=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra),Ya),Xa)) ) ) ) ) ).
% list.rel_cong
tff(fact_4396_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Ra: fun(A,fun(B,bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
=> ( ! [Z4: A,Yb: B] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),X)))
=> ( pp(member2(B,Yb,aa(list(B),set(B),set2(B),Y)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z4),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z4),Yb)) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra),X),Y)) ) ) ).
% list.rel_mono_strong
tff(fact_4397_list_Orel__refl__strong,axiom,
! [A: $tType,X: list(A),Ra: fun(A,fun(A,bool))] :
( ! [Z4: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z4),Z4)) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),Ra),X),X)) ) ).
% list.rel_refl_strong
tff(fact_4398_list__all2__same,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Xs))
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X4),X4)) ) ) ).
% list_all2_same
tff(fact_4399_list_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,X22: list(A),Y21: B,Y22: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22)))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X22),Y22)) ) ) ).
% list.rel_inject(2)
tff(fact_4400_list_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,Y21: B,X22: list(A),Y22: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X22),Y22))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22))) ) ) ).
% list.rel_intros(2)
tff(fact_4401_list__all2__Cons,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Y: B,Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),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)))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys)) ) ) ).
% list_all2_Cons
tff(fact_4402_list__all2__Cons1,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
<=> ? [Z3: B,Zs2: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
& pp(aa(B,bool,aa(A,fun(B,bool),P,X),Z3))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Zs2)) ) ) ).
% list_all2_Cons1
tff(fact_4403_list__all2__Cons2,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Y: B,Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)))
<=> ? [Z3: A,Zs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs2) )
& pp(aa(B,bool,aa(A,fun(B,bool),P,Z3),Y))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Zs2),Ys)) ) ) ).
% list_all2_Cons2
tff(fact_4404_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R),Ra))
=> pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra))) ) ).
% list.rel_mono
tff(fact_4405_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: list(A),B2: list(B),C3: list(A),D3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A3),B2))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),C3),D3))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A3),C3)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B2),D3))) ) ) ).
% list_all2_appendI
tff(fact_4406_list__all2__dropI,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys))) ) ).
% list_all2_dropI
tff(fact_4407_list__all2__update__cong,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),X: A,Y: B,I: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),Y))) ) ) ).
% list_all2_update_cong
tff(fact_4408_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).
% list_all2_lengthD
tff(fact_4409_list__all2__antisym,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Q,Y2),X2))
=> ( X2 = Y2 ) ) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Ys))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),Q),Ys),Xs))
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
tff(fact_4410_list__all2__trans,axiom,
! [B: $tType,A: $tType,C: $tType,P1: fun(A,fun(B,bool)),P22: fun(B,fun(C,bool)),P32: fun(A,fun(C,bool)),As2: list(A),Bs: list(B),Cs2: list(C)] :
( ! [A5: A,B4: B,C2: C] :
( pp(aa(B,bool,aa(A,fun(B,bool),P1,A5),B4))
=> ( pp(aa(C,bool,aa(B,fun(C,bool),P22,B4),C2))
=> pp(aa(C,bool,aa(A,fun(C,bool),P32,A5),C2)) ) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P1),As2),Bs))
=> ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(fun(B,fun(C,bool)),fun(list(B),fun(list(C),bool)),list_all2(B,C),P22),Bs),Cs2))
=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),P32),As2),Cs2)) ) ) ) ).
% list_all2_trans
tff(fact_4411_list__all2__refl,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,X2),X2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Xs)) ) ).
% list_all2_refl
tff(fact_4412_list__all2__mono,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Q: fun(A,fun(B,bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( ! [Xs2: A,Ys2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,Xs2),Ys2))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,Xs2),Ys2)) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Q),Xs),Ys)) ) ) ).
% list_all2_mono
tff(fact_4413_list__all2__eq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = Ys )
<=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),fequal(A)),Xs),Ys)) ) ).
% list_all2_eq
tff(fact_4414_list_Orel__refl,axiom,
! [B: $tType,Ra: fun(B,fun(B,bool)),X: list(B)] :
( ! [X2: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra,X2),X2))
=> pp(aa(list(B),bool,aa(list(B),fun(list(B),bool),aa(fun(B,fun(B,bool)),fun(list(B),fun(list(B),bool)),list_all2(B,B),Ra),X),X)) ) ).
% list.rel_refl
tff(fact_4415_list_Orel__eq,axiom,
! [A: $tType] : aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),fequal(A)) = fequal(list(A)) ).
% list.rel_eq
tff(fact_4416_list__all2__rev1,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),rev(A),Xs)),Ys))
<=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),rev(B),Ys))) ) ).
% list_all2_rev1
tff(fact_4417_list__all2__takeI,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys))) ) ).
% list_all2_takeI
tff(fact_4418_mset_Osimps_I1_J,axiom,
! [A: $tType] : mset(A,nil(A)) = zero_zero(multiset(A)) ).
% mset.simps(1)
tff(fact_4419_list_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),nil(A)),nil(B))) ).
% list.ctr_transfer(1)
tff(fact_4420_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As2: list(A),F: fun(C,B),Bs: list(C)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),As2),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Bs)))
<=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_nw(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F)),As2),Bs)) ) ).
% list_all2_map2
tff(fact_4421_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F: fun(C,A),As2: list(C),Bs: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F),As2)),Bs))
<=> pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),aa(fun(C,fun(B,bool)),fun(list(C),fun(list(B),bool)),list_all2(C,B),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_nx(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F)),As2),Bs)) ) ).
% list_all2_map1
tff(fact_4422_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: list(A),Y: list(B)] :
( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),aa(fun(C,fun(B,bool)),fun(list(C),fun(list(B),bool)),list_all2(C,B),Sb),aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),I),X)),Y))
<=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ny(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y)) ) ).
% list.rel_map(1)
tff(fact_4423_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G: fun(B,C),Y: list(B)] :
( pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),Sa),X),aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),G),Y)))
<=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_nz(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y)) ) ).
% list.rel_map(2)
tff(fact_4424_list_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: A,Y22: list(A)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y22)),nil(B))) ).
% list.rel_distinct(2)
tff(fact_4425_list_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: B,Y22: list(B)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22))) ).
% list.rel_distinct(1)
tff(fact_4426_list_Orel__cases,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A3),B2))
=> ( ( ( A3 = nil(A) )
=> ( B2 != nil(B) ) )
=> ~ ! [X1: A,X23: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),X23) )
=> ! [Y1: B,Y23: list(B)] :
( ( B2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y1),Y23) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,X1),Y1))
=> ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X23),Y23)) ) ) ) ) ) ).
% list.rel_cases
tff(fact_4427_list_Orel__induct,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Q: fun(list(A),fun(list(B),bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
=> ( ! [A21: A,A222: list(A),B21: B,B222: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,A21),B21))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,A222),B222))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A21),A222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B21),B222))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,X),Y)) ) ) ) ).
% list.rel_induct
tff(fact_4428_List_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),R: fun(list(A),fun(list(B),bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,nil(A)),nil(B)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys2: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),Y2))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs2),Ys2))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs2),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs),Ys)) ) ) ) ).
% List.list_all2_induct
tff(fact_4429_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),L: list(A),L3: list(B),Q: fun(list(A),fun(list(B),bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),L),L3))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
=> ( ! [X2: A,X8: B,Ls2: list(A),Ls4: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),X8))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Ls2),Ls4))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,Ls2),Ls4))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ls2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X8),Ls4))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,L),L3)) ) ) ) ).
% Misc.list_all2_induct
tff(fact_4430_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(A,fun(B,bool)),Us2: list(A),Vs: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs)))
<=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Us2),Vs)) ) ) ) ).
% list_all2_append
tff(fact_4431_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(A),Zs: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs))
<=> ? [Us: list(B),Vs2: list(B)] :
( ( Zs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs2) )
& ( aa(list(B),nat,size_size(list(B)),Us) = aa(list(A),nat,size_size(list(A)),Xs) )
& ( aa(list(B),nat,size_size(list(B)),Vs2) = aa(list(A),nat,size_size(list(A)),Ys) )
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Us))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Ys),Vs2)) ) ) ).
% list_all2_append1
tff(fact_4432_list__all2__append2,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Zs: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)))
<=> ? [Us: list(A),Vs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs2) )
& ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(B),nat,size_size(list(B)),Ys) )
& ( aa(list(A),nat,size_size(list(A)),Vs2) = aa(list(B),nat,size_size(list(B)),Zs) )
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Us),Ys))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Vs2),Zs)) ) ) ).
% list_all2_append2
tff(fact_4433_list__all2__conv__all__nth,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2))) ) ) ) ).
% list_all2_conv_all_nth
tff(fact_4434_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ! [N7: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),aa(list(A),nat,size_size(list(A)),A3)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),N7)),aa(nat,B,nth(B,B2),N7))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A3),B2)) ) ) ).
% list_all2_all_nthI
tff(fact_4435_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P4: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),aa(list(B),nat,size_size(list(B)),Ys)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P4)),aa(nat,B,nth(B,Ys),P4))) ) ) ).
% list_all2_nthD2
tff(fact_4436_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P4: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P4)),aa(nat,B,nth(B,Ys),P4))) ) ) ).
% list_all2_nthD
tff(fact_4437_list_Orel__sel,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A3),B2))
<=> ( ( ( A3 = nil(A) )
<=> ( B2 = nil(B) ) )
& ( ( A3 != nil(A) )
=> ( ( B2 != nil(B) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,aa(list(A),A,hd(A),A3)),aa(list(B),B,hd(B),B2)))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),tl(A),A3)),aa(list(B),list(B),tl(B),B2))) ) ) ) ) ) ).
% list.rel_sel
tff(fact_4438_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_4439_product__lists__set,axiom,
! [A: $tType,Xss2: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss2)) = collect(list(A),aTP_Lamp_ob(list(list(A)),fun(list(A),bool),Xss2)) ).
% product_lists_set
tff(fact_4440_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
( ! [X2: product_prod(A,B)] :
( pp(member2(product_prod(A,B),X2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),A3),B2))))
=> pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X2)) )
=> ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A3),B2)) ) ) ).
% list_all2I
tff(fact_4441_Linear__order__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_679001287576687338der_on(A,field2(A,R3),R3)
=> order_679001287576687338der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Linear_order_Restr
tff(fact_4442_list__all2__iff,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X4: product_prod(A,B)] :
( pp(member2(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))))
=> pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X4)) ) ) ) ).
% list_all2_iff
tff(fact_4443_sort__key__inj__key__eq,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),Ys: list(B),F: fun(B,A)] :
( ( mset(B,Xs) = mset(B,Ys) )
=> ( inj_on(B,A,F,aa(list(B),set(B),set2(B),Xs))
=> ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys))
=> ( aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F),Xs) = Ys ) ) ) ) ) ).
% sort_key_inj_key_eq
tff(fact_4444_linear__order__on__Restr,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A] :
( order_679001287576687338der_on(A,A4,R3)
=> order_679001287576687338der_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),order_above(A,R3,X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_above(A,R3,X),aa(A,fun(A,set(A)),aTP_Lamp_oc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X)))) ) ).
% linear_order_on_Restr
tff(fact_4445_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),J: nat,X: A,Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( mset(product_prod(A,B),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),Xs)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),J),Ys)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),J),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),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),Ys))) ) ) ) ).
% mset_zip_take_Cons_drop_twice
tff(fact_4446_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),X)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),X)),Xs))) ) ) ) ).
% takeWhile_neq_rev
tff(fact_4447_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Sk: fun(fun(B,A),fun(list(B),list(B)))] :
( linord3483353639454293061rt_key(B,A,Sk)
<=> ! [F5: fun(B,A),Xs3: list(B),K4: A] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_od(fun(B,A),fun(A,fun(B,bool)),F5),K4)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F5),Xs3)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_od(fun(B,A),fun(A,fun(B,bool)),F5),K4)),Xs3) ) ) ).
% stable_sort_key_def
tff(fact_4448_predicate2I,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
( ! [X2: A,Y2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),Y2))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X2),Y2)) )
=> pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q)) ) ).
% predicate2I
tff(fact_4449_dropWhile__idem,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) ).
% dropWhile_idem
tff(fact_4450_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = nil(A) )
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X4)) ) ) ).
% dropWhile_eq_Nil_conv
tff(fact_4451_dropWhile__append2,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Ys) ) ) ).
% dropWhile_append2
tff(fact_4452_dropWhile__append1,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),Ys) ) ) ) ).
% dropWhile_append1
tff(fact_4453_dropWhile__replicate,axiom,
! [A: $tType,P: fun(A,bool),X: A,N: nat] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) ) ) ).
% dropWhile_replicate
tff(fact_4454_takeWhile__dropWhile__id,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = Xs ).
% takeWhile_dropWhile_id
tff(fact_4455_mset__single__iff__right,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( add_mset(A,X,zero_zero(multiset(A))) = mset(A,Xs) )
<=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).
% mset_single_iff_right
tff(fact_4456_mset__single__iff,axiom,
! [A: $tType,Xs: list(A),X: A] :
( ( mset(A,Xs) = add_mset(A,X,zero_zero(multiset(A))) )
<=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).
% mset_single_iff
tff(fact_4457_union__diff__assoc,axiom,
! [A: $tType,C4: multiset(A),B3: multiset(A),A4: multiset(A)] :
( ( minus_minus(multiset(A),C4,B3) = 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)),A4),B3),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),minus_minus(multiset(A),B3,C4)) ) ) ).
% union_diff_assoc
tff(fact_4458_mset__single__cases,axiom,
! [A: $tType,S4: A,C3: multiset(A),R6: A,C9: multiset(A)] :
( ( add_mset(A,S4,C3) = add_mset(A,R6,C9) )
=> ( ( ( S4 = R6 )
=> ( C3 != C9 ) )
=> ~ ( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A))))) )
=> ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R6,zero_zero(multiset(A)))),minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A))))) )
=> ( minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A)))) != minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases
tff(fact_4459_mset__single__cases_H,axiom,
! [A: $tType,S4: A,C3: multiset(A),R6: A,C9: multiset(A)] :
( ( add_mset(A,S4,C3) = add_mset(A,R6,C9) )
=> ( ( ( S4 = R6 )
=> ( C3 != C9 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),Cc) )
=> ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R6,zero_zero(multiset(A)))),Cc) )
=> ( ( minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A)))) = Cc )
=> ( minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases'
tff(fact_4460_mset__single__cases2,axiom,
! [A: $tType,S4: A,C3: multiset(A),R6: A,C9: multiset(A)] :
( ( add_mset(A,S4,C3) = add_mset(A,R6,C9) )
=> ( ( ( S4 = R6 )
=> ( C3 != C9 ) )
=> ~ ( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A))))),add_mset(A,S4,zero_zero(multiset(A)))) )
=> ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A))))),add_mset(A,R6,zero_zero(multiset(A)))) )
=> ( minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A)))) != minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases2
tff(fact_4461_mset__single__cases2_H,axiom,
! [A: $tType,S4: A,C3: multiset(A),R6: A,C9: multiset(A)] :
( ( add_mset(A,S4,C3) = add_mset(A,R6,C9) )
=> ( ( ( S4 = R6 )
=> ( C3 != C9 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C9 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,S4,zero_zero(multiset(A)))) )
=> ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,R6,zero_zero(multiset(A)))) )
=> ( ( minus_minus(multiset(A),C9,add_mset(A,S4,zero_zero(multiset(A)))) = Cc )
=> ( minus_minus(multiset(A),C3,add_mset(A,R6,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases2'
tff(fact_4462_mset__unplusm__dist__cases,axiom,
! [A: $tType,S4: A,A4: multiset(A),B3: multiset(A),C4: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),C4) )
=> ( ( ( B3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),minus_minus(multiset(A),B3,add_mset(A,S4,zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B3,add_mset(A,S4,zero_zero(multiset(A))))),C4) ) )
=> ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),minus_minus(multiset(A),C4,add_mset(A,S4,zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),minus_minus(multiset(A),C4,add_mset(A,S4,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases
tff(fact_4463_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B3: multiset(A),C4: multiset(A),S4: A,A4: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),A4) )
=> ( ( ( B3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),minus_minus(multiset(A),B3,add_mset(A,S4,zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B3,add_mset(A,S4,zero_zero(multiset(A))))),C4) ) )
=> ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,S4,zero_zero(multiset(A)))),minus_minus(multiset(A),C4,add_mset(A,S4,zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),minus_minus(multiset(A),C4,add_mset(A,S4,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases2
tff(fact_4464_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),nil(A)) = nil(A) ).
% dropWhile.simps(1)
tff(fact_4465_mset__union__2__elem,axiom,
! [A: $tType,A3: A,B2: A,C3: A,M: multiset(A)] :
( ( add_mset(A,A3,add_mset(A,B2,zero_zero(multiset(A)))) = add_mset(A,C3,M) )
=> ( ( ( add_mset(A,A3,zero_zero(multiset(A))) = M )
& ( B2 = C3 ) )
| ( ( A3 = C3 )
& ( add_mset(A,B2,zero_zero(multiset(A))) = M ) ) ) ) ).
% mset_union_2_elem
tff(fact_4466_mset__distrib,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),M: multiset(A),N4: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M),N4) )
=> ~ ! [Am: multiset(A),An: multiset(A)] :
( ( A4 = 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)] :
( ( B3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
=> ( ( M = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
=> ( N4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).
% mset_distrib
tff(fact_4467_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,Q: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
=> ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).
% rev_predicate2D
tff(fact_4468_predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),X: A,Y: B] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).
% predicate2D
tff(fact_4469_distinct__dropWhile,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> pp(aa(list(A),bool,distinct(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))) ) ).
% distinct_dropWhile
tff(fact_4470_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ) ).
% dropWhile.simps(2)
tff(fact_4471_set__dropWhileD,axiom,
! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% set_dropWhileD
tff(fact_4472_dropWhile__cong,axiom,
! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
( ( L = K )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),L)))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) ) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),L) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),Q),K) ) ) ) ).
% dropWhile_cong
tff(fact_4473_mset_Osimps_I2_J,axiom,
! [A: $tType,A3: A,X: list(A)] : mset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X)) = add_mset(A,A3,mset(A,X)) ).
% mset.simps(2)
tff(fact_4474_dropWhile__append3,axiom,
! [A: $tType,P: fun(A,bool),Y: A,Xs: list(A),Ys: list(A)] :
( ~ pp(aa(A,bool,P,Y))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),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),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).
% dropWhile_append3
tff(fact_4475_length__dropWhile__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_dropWhile_le
tff(fact_4476_sorted__dropWhile,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) ) ) ).
% sorted_dropWhile
tff(fact_4477_hd__dropWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) != nil(A) )
=> ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)))) ) ).
% hd_dropWhile
tff(fact_4478_dropWhile__eq__self__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = Xs )
<=> ( ( Xs = nil(A) )
| ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).
% dropWhile_eq_self_iff
tff(fact_4479_dropWhile__last,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ~ pp(aa(A,bool,P,X))
=> ( last(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = last(A,Xs) ) ) ) ).
% dropWhile_last
tff(fact_4480_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),dropWhile(B),comp(A,bool,B,P,F)),Xs)) ).
% dropWhile_map
tff(fact_4481_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_ad(A,fun(A,bool),X)),Xs))) ).
% remdups_adj_Cons'
tff(fact_4482_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Y: A,Ys: list(A)] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
<=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) )
& ~ pp(aa(A,bool,P,Y)) ) ) ).
% dropWhile_eq_Cons_conv
tff(fact_4483_takeWhile__eq__filter,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
=> ~ pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).
% takeWhile_eq_filter
tff(fact_4484_dropWhile__eq__drop,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),Xs) ).
% dropWhile_eq_drop
tff(fact_4485_dropWhile__append,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X2)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Ys) ) )
& ( ~ ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X3)) )
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),Ys) ) ) ) ).
% dropWhile_append
tff(fact_4486_length__dropWhile__takeWhile,axiom,
! [A: $tType,X: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_dropWhile_takeWhile
tff(fact_4487_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_ad(A,fun(A,bool),Y)),Ys))) ).
% remdups_adj_append_dropWhile
tff(fact_4488_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != nil(A) )
=> ( aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),remdups_adj(A),Xs)),aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_oe(list(A),fun(A,bool),Xs)),Ys))) ) ) ).
% remdups_adj_append''
tff(fact_4489_tl__remdups__adj,axiom,
! [A: $tType,Ys: list(A)] :
( ( Ys != nil(A) )
=> ( aa(list(A),list(A),tl(A),aa(list(A),list(A),remdups_adj(A),Ys)) = aa(list(A),list(A),remdups_adj(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_of(list(A),fun(A,bool),Ys)),aa(list(A),list(A),tl(A),Ys))) ) ) ).
% tl_remdups_adj
tff(fact_4490_stable__sort__key__sort__key,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> linord3483353639454293061rt_key(A,B,linorder_sort_key(A,B)) ) ).
% stable_sort_key_sort_key
tff(fact_4491_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)))) ) ) ).
% dropWhile_nth
tff(fact_4492_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list(A),X: A] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),X)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),X)),Xs))) ) ) ) ).
% dropWhile_neq_rev
tff(fact_4493_sorted__list__of__multiset__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : linord6283353356039996273ltiset(A,add_mset(A,X,zero_zero(multiset(A)))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ).
% sorted_list_of_multiset_singleton
tff(fact_4494_shuffles_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),X),Xa) = Y )
=> ( 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)))) )
=> ~ 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)))) )
=> ~ 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))) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ! [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),X2)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs2),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)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys2))) )
=> ~ 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),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_4495_map__rec,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(B),list(A),aa(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A)),aa(list(A),fun(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A))),rec_list(list(A),B),nil(A)),aTP_Lamp_og(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F)),Xs) ).
% map_rec
tff(fact_4496_set__rec,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),aa(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A)),aa(set(A),fun(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A))),rec_list(set(A),A),bot_bot(set(A))),aTP_Lamp_oh(A,fun(list(A),fun(set(A),set(A))))),Xs) ).
% set_rec
tff(fact_4497_sorted__list__of__multiset__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord6283353356039996273ltiset(A,zero_zero(multiset(A))) = nil(A) ) ) ).
% sorted_list_of_multiset_empty
tff(fact_4498_shuffles_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
=> ( ! [Ys2: list(A)] :
( 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)),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys2)) )
=> ( ! [Xs2: list(A)] :
( 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)),Xs2),nil(A)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),nil(A))) )
=> ( ! [X2: A,Xs2: list(A),Y2: A,Ys2: list(A)] :
( 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),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2)))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2))) ) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).
% shuffles.pinduct
tff(fact_4499_list_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: fun(A,fun(list(A),fun(C,C))),X21: A,X22: list(A)] : aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(C,C,aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),F22,X21),X22),aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),X22)) ).
% list.simps(7)
tff(fact_4500_list_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: fun(A,fun(list(A),fun(C,C)))] : aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),nil(A)) = F1 ).
% list.simps(6)
tff(fact_4501_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X: A,Xs: list(A)] :
( ( F = aa(fun(A,fun(list(A),fun(B,B))),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),fun(B,B))),fun(list(A),B)),rec_list(B,A),F1),F22) )
=> ( aa(list(A),B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X),Xs),aa(list(A),B,F,Xs)) ) ) ).
% rec_list_Cons_imp
tff(fact_4502_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
( ( F = aa(fun(A,fun(list(A),fun(B,B))),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),fun(B,B))),fun(list(A),B)),rec_list(B,A),F1),F22) )
=> ( aa(list(A),B,F,nil(A)) = F1 ) ) ).
% rec_list_Nil_imp
tff(fact_4503_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( 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))
=> ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(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.psimps(1)
tff(fact_4504_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( 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)))
=> ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(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.psimps(2)
tff(fact_4505_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] :
( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
=> ( aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))) ) ) ).
% shuffles.psimps(3)
tff(fact_4506_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,fun(list(B),fun(C,C))),F: fun(A,B)] : comp(list(B),C,list(A),aa(fun(B,fun(list(B),fun(C,C))),fun(list(B),C),aa(C,fun(fun(B,fun(list(B),fun(C,C))),fun(list(B),C)),rec_list(C,B),G),Ga),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F)) = aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),G),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_oi(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F)) ).
% list.rec_o_map
tff(fact_4507_revg_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X,Xa) = Y )
=> ( 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 )
=> ~ 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,As: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( ( Y = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) )
=> ~ 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),As)),Xa)) ) ) ) ) ) ).
% revg.pelims
tff(fact_4508_merge_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X,Xa) = Y )
=> ( 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 )
=> ~ 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)) ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ~ 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),V2),Va2)),nil(A))) ) ) )
=> ~ ! [X1: A,L1: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X23))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X23))
=> ( ( ( X1 = X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,L22)) ) )
& ( ( X1 != X23 )
=> ( Y = 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),X1),L1),L22)) ) ) ) ) )
=> ~ 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),X1),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) ) ) ) ) ) ) ) ).
% merge.pelims
tff(fact_4509_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 )
=> ( 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) )
=> ~ 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)) )
=> ! [L2: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))) )
=> ( ( Y = L2 )
=> ~ 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)),L2),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)) )
=> ~ 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) )
=> ! [L2: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),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)),L2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) )
=> ~ 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)),L2),nil(list(A))))) ) ) )
=> ~ ! [L1: list(A),L22: list(A),Ls2: 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),Ls2)) )
=> ( ( 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),Ls2) )
=> ~ 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),Ls2)))) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
tff(fact_4510_splice_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
( accp(product_prod(list(A),list(A)),splice_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)),A0),A1))
=> ( ! [Ys2: list(A)] :
( accp(product_prod(list(A),list(A)),splice_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)),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys2)) )
=> ( ! [X2: A,Xs2: list(A),Ys2: list(A)] :
( accp(product_prod(list(A),list(A)),splice_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),X2),Xs2)),Ys2))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys2),Xs2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys2)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).
% splice.pinduct
tff(fact_4511_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( 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_4512_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A)),L: list(A)] :
( 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_4513_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc22: list(list(A))] :
( 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_4514_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( 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_4515_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)),bool))] :
( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1))
=> ( ( accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),nil(list(A)))) )
=> ( ! [L2: list(A)] :
( 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)),L2),nil(list(A)))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( 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))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),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)))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),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)),L2: list(A)] :
( 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)),L2),nil(list(A)))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),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)),L2),nil(list(A))))) ) )
=> ( ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] :
( 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),Ls2))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),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)),Ls2))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),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),Ls2)))) ) )
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,A0),A1)) ) ) ) ) ) ) ) ).
% merge_list.pinduct
tff(fact_4516_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: list(list(A))] :
( 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),Ls))))
=> ( 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),Ls))) = 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),Ls) ) ) ) ).
% merge_list.psimps(5)
tff(fact_4517_splice_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),X),Xa) = Y )
=> ( accp(product_prod(list(A),list(A)),splice_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 )
=> ~ accp(product_prod(list(A),list(A)),splice_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)) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa),Xs2)) )
=> ~ accp(product_prod(list(A),list(A)),splice_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),X2),Xs2)),Xa)) ) ) ) ) ) ).
% splice.pelims
tff(fact_4518_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( accp(product_prod(list(A),list(A)),splice_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))
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ) ) ).
% splice.psimps(1)
tff(fact_4519_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = 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)),splice(A),Ys),Xs)) ) ) ).
% splice.psimps(2)
tff(fact_4520_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M2: list(product_prod(A,B))] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),M2)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),M2)) = map_of(A,B,M2) ) ) ).
% map_of_rev_distinct
tff(fact_4521_split__Nil__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys) = nil(A) )
<=> ( ( Xs = nil(A) )
& ( Ys = nil(A) ) ) ) ).
% split_Nil_iff
tff(fact_4522_splice__Nil2,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),nil(A)) = Xs ).
% splice_Nil2
tff(fact_4523_splice__in__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : pp(member2(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys),aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys))) ).
% splice_in_shuffles
tff(fact_4524_length__splice,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).
% length_splice
tff(fact_4525_splice__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M2),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),X) ).
% splice_replicate
tff(fact_4526_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = 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)),splice(A),Ys),Xs)) ).
% splice.simps(2)
tff(fact_4527_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ).
% splice.simps(1)
tff(fact_4528_splice_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),X),Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa),Xs2)) ) ) ) ) ).
% splice.elims
tff(fact_4529_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4530_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B)),M2: fun(A,option(B))] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),L)))
=> ( ( map_of(A,B,L) = M2 )
<=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M2) ) ) ) ).
% map_of_map_to_set
tff(fact_4531_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
( pp(aa(list(A),bool,distinct(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4532_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] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))))
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4533_map__to__set__empty,axiom,
! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_oj(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% map_to_set_empty
tff(fact_4534_map__to__set__ran,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A))] : ran(B,A,M2) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M2)) ).
% map_to_set_ran
tff(fact_4535_map__to__set__empty__iff_I1_J,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B))] :
( ( map_to_set(A,B,M2) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] : aa(A,option(B),M2,X4) = none(B) ) ).
% map_to_set_empty_iff(1)
tff(fact_4536_map__to__set__empty__iff_I2_J,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B))] :
( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M2) )
<=> ! [X4: A] : aa(A,option(B),M2,X4) = none(B) ) ).
% map_to_set_empty_iff(2)
tff(fact_4537_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),X: B,P: fun(product_prod(B,A),bool)] :
( ( aa(B,option(A),map_of(B,A,Xs),X) = none(A) )
=> ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(fun(product_prod(B,A),bool),fun(list(product_prod(B,A)),list(product_prod(B,A))),filter2(product_prod(B,A)),P),Xs)),X) = none(A) ) ) ).
% map_of_None_filterD
tff(fact_4538_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M2)) ).
% inj_on_fst_map_to_set
tff(fact_4539_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)))) ).
% ran_map_of
tff(fact_4540_ran__empty,axiom,
! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_ok(B,option(A))) = bot_bot(set(A)) ).
% ran_empty
tff(fact_4541_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : rel_of(A,B,aTP_Lamp_oj(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).
% rel_of_empty
tff(fact_4542_inf__None__2,axiom,
! [A: $tType] :
( inf(A)
=> ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),none(A)) = none(A) ) ).
% inf_None_2
tff(fact_4543_inf__None__1,axiom,
! [A: $tType] :
( inf(A)
=> ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),none(A)),Y) = none(A) ) ).
% inf_None_1
tff(fact_4544_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] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))))
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4545_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] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))))
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4546_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),K: A,V: B] : map_to_set(A,B,fun_upd(A,option(B),M2,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,M2),collect(product_prod(A,B),aTP_Lamp_ol(A,fun(product_prod(A,B),bool),K)))) ).
% map_to_set_upd
tff(fact_4547_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))) ).
% set_to_map_ran
tff(fact_4548_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
( ! [X4: 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),X4),Y3))
<=> ( V = none(product_prod(A,B)) ) ) ).
% not_Some_eq2
tff(fact_4549_set__to__map__empty,axiom,
! [A: $tType,B: $tType,X3: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X3) = none(B) ).
% set_to_map_empty
tff(fact_4550_inf__Some,axiom,
! [A: $tType] :
( inf(A)
=> ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) ) ).
% inf_Some
tff(fact_4551_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),K: A,V: B] :
( ( aa(A,option(B),M2,K) = none(B) )
=> ( ran(A,B,aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_om(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),M2),K),V)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),V),ran(A,B,M2)) ) ) ).
% map_update_eta_repair(2)
tff(fact_4552_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_on(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).
% ran_nth_set_encoding_conv
tff(fact_4553_le__some__optE,axiom,
! [A: $tType] :
( preorder(A)
=> ! [M2: A,X: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),M2)),X))
=> ~ ! [M5: A] :
( ( X = aa(A,option(A),some(A),M5) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),M5)) ) ) ) ).
% le_some_optE
tff(fact_4554_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) )
<=> pp(member2(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_4555_Some__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4)) ) ) ) ).
% Some_Sup
tff(fact_4556_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M2)) = M2 ).
% map_to_set_inverse
tff(fact_4557_map__to__set__def,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : map_to_set(A,B,M2) = collect(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_oo(fun(A,option(B)),fun(A,fun(B,bool)),M2))) ).
% map_to_set_def
tff(fact_4558_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [A4: set(A),F: fun(A,B)] :
( ( A4 != 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,F),A4))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_op(fun(A,B),fun(A,option(B)),F)),A4)) ) ) ) ).
% Some_SUP
tff(fact_4559_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod(A,B),S: set(product_prod(A,B))] :
( ~ pp(member2(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_4560_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S),X4) = none(B)
<=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(1)
tff(fact_4561_rel__of__def,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),P: fun(product_prod(A,B),bool)] : rel_of(A,B,M2,P) = collect(product_prod(A,B),product_case_prod(A,B,bool,aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_oq(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),M2),P))) ).
% rel_of_def
tff(fact_4562_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ( aTP_Lamp_oj(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_4563_map__of__zip__map,axiom,
! [A: $tType,B: $tType,Xs: list(A),F: fun(A,B),X3: A] :
( ( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs))),X3) = aa(B,option(B),some(B),aa(A,B,F,X3)) ) )
& ( ~ pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,option(B),map_of(A,B,aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Xs),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs))),X3) = none(B) ) ) ) ).
% map_of_zip_map
tff(fact_4564_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)),Zs3: 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)),Zs3)) )
& ( aa(B,option(A),map_of(B,A,Ys2),K) = none(A) ) ) ) ).
% map_of_Some_split
tff(fact_4565_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_4566_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),bool)] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ( ~ pp(aa(product_prod(B,A),bool,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)))
=> ( pp(aa(list(B),bool,distinct(B),aa(list(product_prod(B,A)),list(B),aa(fun(product_prod(B,A),B),fun(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,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(fun(product_prod(B,A),bool),fun(list(product_prod(B,A)),list(product_prod(B,A))),filter2(product_prod(B,A)),P),Xs)),K) = none(A) ) ) ) ) ).
% map_of_Some_filter_not_in
tff(fact_4567_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] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))))
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_4568_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M2: fun(A,option(B)),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( map_upds(A,B,M2,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) = fun_upd(A,option(B),map_upds(A,B,M2,Xs,Ys),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).
% map_upds_append1
tff(fact_4569_extract__def,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A),none(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_or(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),comp(bool,bool,A,fNot,P)),Xs)) ).
% extract_def
tff(fact_4570_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_os(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F),P),Xs) ).
% map_filter_map_filter
tff(fact_4571_extract__Some__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
<=> ( ( Xs = 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),Zs)) )
& pp(aa(A,bool,P,Y))
& ~ ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Ys)))
& pp(aa(A,bool,P,X4)) ) ) ) ).
% extract_Some_iff
tff(fact_4572_map__upds__Nil1,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),Bs: list(B)] : map_upds(A,B,M2,nil(A),Bs) = M2 ).
% map_upds_Nil1
tff(fact_4573_map__upds__Nil2,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),As2: list(A)] : map_upds(A,B,M2,As2,nil(B)) = M2 ).
% map_upds_Nil2
tff(fact_4574_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),A3: A,As2: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M2,A3,aa(B,option(B),some(B),B2)),As2,Bs) ).
% map_upds_Cons
tff(fact_4575_extract__Nil__code,axiom,
! [A: $tType,P: fun(A,bool)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).
% extract_Nil_code
tff(fact_4576_extract__None__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
<=> ~ ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) ) ) ).
% extract_None_iff
tff(fact_4577_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,option(A))] : map_filter(B,A,F,nil(B)) = nil(A) ).
% map_filter_simps(2)
tff(fact_4578_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)) = collect(A,aa(list(B),fun(A,bool),aTP_Lamp_ot(fun(B,option(A)),fun(list(B),fun(A,bool)),G),Xs)) ).
% set_map_filter
tff(fact_4579_extract__SomeE,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
=> ( ( Xs = 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),Zs)) )
& pp(aa(A,bool,P,Y))
& ~ ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Ys)))
& pp(aa(A,bool,P,X3)) ) ) ) ).
% extract_SomeE
tff(fact_4580_extract__Cons__code,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X),Xs))) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))),aTP_Lamp_ov(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P,Xs)) ) ) ) ).
% extract_Cons_code
tff(fact_4581_find__dropWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(list(A),option(A),aa(fun(A,fun(list(A),option(A))),fun(list(A),option(A)),aa(option(A),fun(fun(A,fun(list(A),option(A))),fun(list(A),option(A))),case_list(option(A),A),none(A)),aTP_Lamp_ow(A,fun(list(A),option(A)))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),comp(bool,bool,A,fNot,P)),Xs)) ).
% find_dropWhile
tff(fact_4582_sorted__find__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ? [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X3)) )
=> ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(A,option(A),some(A),lattic643756798350308766er_Min(A,collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ox(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).
% sorted_find_Min
tff(fact_4583_find__cong,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Ys)))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) ) )
=> ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),Q),Ys) ) ) ) ).
% find_cong
tff(fact_4584_find__SomeD_I1_J,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
( ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(A,option(A),some(A),X) )
=> pp(aa(A,bool,P,X)) ) ).
% find_SomeD(1)
tff(fact_4585_find_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X))
=> ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) ) ) ) ).
% find.simps(2)
tff(fact_4586_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: fun(A,bool)] : aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),Uu),nil(A)) = none(A) ).
% find.simps(1)
tff(fact_4587_find__SomeD_I2_J,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
( ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(A,option(A),some(A),X) )
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% find_SomeD(2)
tff(fact_4588_find__None__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = none(A) )
<=> ~ ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) ) ) ).
% find_None_iff
tff(fact_4589_find__None__iff2,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( none(A) = aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) )
<=> ~ ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) ) ) ).
% find_None_iff2
tff(fact_4590_find__Some__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
( ( aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) = aa(A,option(A),some(A),X) )
<=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2)))
& ( X = aa(nat,A,nth(A,Xs),I2) )
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I2))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J2))) ) ) ) ).
% find_Some_iff
tff(fact_4591_find__Some__iff2,axiom,
! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
( ( aa(A,option(A),some(A),X) = aa(list(A),option(A),aa(fun(A,bool),fun(list(A),option(A)),find(A),P),Xs) )
<=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2)))
& ( X = aa(nat,A,nth(A,Xs),I2) )
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I2))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J2))) ) ) ) ).
% find_Some_iff2
tff(fact_4592_inf__option__def,axiom,
! [A: $tType] :
( inf(A)
=> ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),Y) = case_option(option(A),A,none(A),aTP_Lamp_oz(option(A),fun(A,option(A)),Y),X) ) ).
% inf_option_def
tff(fact_4593_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Y: B] :
( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(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_oj(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_4594_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M2,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,M2,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_4595_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D2: set(A),M2: fun(A,option(B))] :
( ( pp(member2(A,X,D2))
=> ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),X,none(B)) = restrict_map(A,B,M2,minus_minus(set(A),D2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ) )
& ( ~ pp(member2(A,X,D2))
=> ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),X,none(B)) = restrict_map(A,B,M2,D2) ) ) ) ).
% fun_upd_None_restrict
tff(fact_4596_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B))] : restrict_map(A,B,F,top_top(set(A))) = F ).
% restrict_map_UNIV
tff(fact_4597_restrict__restrict,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),A4: set(A),B3: set(A)] : restrict_map(A,B,restrict_map(A,B,M2,A4),B3) = restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).
% restrict_restrict
tff(fact_4598_restrict__map__to__empty,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,M2,bot_bot(set(A))),X3) = none(B) ).
% restrict_map_to_empty
tff(fact_4599_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D2: set(A),M2: fun(A,option(B)),Y: option(B)] :
( pp(member2(A,X,D2))
=> ( fun_upd(A,option(B),restrict_map(A,B,M2,D2),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,minus_minus(set(A),D2,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_4600_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D2: set(A),M2: fun(A,option(B)),Y: option(B)] :
( ( pp(member2(A,X,D2))
=> ( restrict_map(A,B,fun_upd(A,option(B),M2,X,Y),D2) = fun_upd(A,option(B),restrict_map(A,B,M2,minus_minus(set(A),D2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ) )
& ( ~ pp(member2(A,X,D2))
=> ( restrict_map(A,B,fun_upd(A,option(B),M2,X,Y),D2) = restrict_map(A,B,M2,D2) ) ) ) ).
% restrict_fun_upd
tff(fact_4601_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M2: fun(A,option(B)),X5: set(A)] : pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M2,X5)),M2)) ) ).
% le_map_restrict
tff(fact_4602_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M2: fun(A,option(B)),M6: fun(A,option(B)),N: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M2),M6))
=> pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M2,N)),map_add(A,B,M6,N))) ) ) ).
% map_add_first_le
tff(fact_4603_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),R: set(A),M6: fun(A,option(B)),R7: set(A)] :
( ( restrict_map(A,B,M2,R) = M6 )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R7),R))
=> ( restrict_map(A,B,M2,R7) = restrict_map(A,B,M6,R7) ) ) ) ).
% restrict_map_subset_eq
tff(fact_4604_map__add__left__None,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),K: B,G: fun(B,option(A))] :
( ( aa(B,option(A),F,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F,G),K) = aa(B,option(A),G,K) ) ) ).
% map_add_left_None
tff(fact_4605_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: fun(B,option(A)),K: B,F: fun(B,option(A))] :
( ( aa(B,option(A),G,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F,G),K) = aa(B,option(A),F,K) ) ) ).
% map_add_find_left
tff(fact_4606_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A)),A4: set(B),K: B,V: A] :
( ( aa(B,option(A),restrict_map(B,A,M2,A4),K) = aa(A,option(A),some(A),V) )
<=> ( ( aa(B,option(A),M2,K) = aa(A,option(A),some(A),V) )
& pp(member2(B,K,A4)) ) ) ).
% restrict_map_eq(2)
tff(fact_4607_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A)),X: B,S4: set(B)] :
( ( aa(B,option(A),M2,X) = none(A) )
=> ( restrict_map(B,A,M2,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),S4))) = restrict_map(B,A,M2,aa(set(B),set(B),uminus_uminus(set(B)),S4)) ) ) ).
% map_restrict_insert_none_simp
tff(fact_4608_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),S: set(A),K: A,V: B] : fun_upd(A,option(B),restrict_map(A,B,F,S),K,aa(B,option(B),some(B),V)) = restrict_map(A,B,fun_upd(A,option(B),F,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_4609_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),D2: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M2,D2),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,minus_minus(set(A),D2,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_4610_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss2: list(list(product_prod(A,B)))] : map_of(A,B,aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),Xss2)) = aa(fun(A,option(B)),fun(A,option(B)),aa(list(list(product_prod(A,B))),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),fun(list(list(product_prod(A,B))),fun(fun(A,option(B)),fun(A,option(B)))),foldr(list(product_prod(A,B)),fun(A,option(B))),aTP_Lamp_pa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))))),Xss2),aTP_Lamp_oj(A,option(B))) ).
% map_of_concat
tff(fact_4611_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : restrict_map(A,B,F,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),F,X,none(B)) ).
% restrict_complement_singleton_eq
tff(fact_4612_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A] : fun_upd(A,option(B),M2,X,none(B)) = restrict_map(A,B,M2,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_4613_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: option(bool)] :
( ( subset_eq_mset_impl(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa),nil(A)))) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( Y != case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),product_case_prod(list(A),product_prod(A,list(A)),option(bool),aTP_Lamp_pc(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X2),Xa)) ) ) ) ) ).
% subset_eq_mset_impl.elims
tff(fact_4614_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: option(bool)] :
( ( subset_eq_mset_impl(A,X,Xa) = Y )
=> ( accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa),nil(A)))) )
=> ~ accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ~ ! [X2: A,Xs2: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> ( ( Y = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),product_case_prod(list(A),product_prod(A,list(A)),option(bool),aTP_Lamp_pc(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X2),Xa)) )
=> ~ accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xa)) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
tff(fact_4615_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),product_case_prod(list(A),product_prod(A,list(A)),option(bool),aTP_Lamp_pc(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs)),extract(A,aa(A,fun(A,bool),fequal(A),X),Ys)) ).
% subset_eq_mset_impl.simps(2)
tff(fact_4616_subset__eq__mset__impl_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : subset_eq_mset_impl(A,nil(A),Ys) = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Ys),nil(A)))) ).
% subset_eq_mset_impl.simps(1)
tff(fact_4617_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_pd(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Inf_fin.eq_fold'
tff(fact_4618_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: fun(B,option(A)),X: B,Y: A,Z2: A] :
( ( aa(B,option(A),M2,X) = aa(A,option(A),some(A),Y) )
=> ( inj_on(B,option(A),M2,dom(B,A,M2))
=> ( ~ pp(member2(A,Z2,ran(B,A,M2)))
=> ( ran(B,A,fun_upd(B,option(A),M2,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,M2),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_4619_Eps__Opt__eq__None,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( eps_Opt(A,P) = none(A) )
<=> ~ ? [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).
% Eps_Opt_eq_None
tff(fact_4620_restrict__map__self,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : restrict_map(A,B,M2,dom(A,B,M2)) = M2 ).
% restrict_map_self
tff(fact_4621_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] : eps_Opt(A,aa(A,fun(A,bool),fequal(A),X)) = aa(A,option(A),some(A),X) ).
% some_opt_sym_eq_trivial
tff(fact_4622_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_ad(A,fun(A,bool),X)) = aa(A,option(A),some(A),X) ).
% some_opt_eq_trivial
tff(fact_4623_some__opt__false__trivial,axiom,
! [A: $tType] : eps_Opt(A,aTP_Lamp_bo(A,bool)) = none(A) ).
% some_opt_false_trivial
tff(fact_4624_dom__eq__empty__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B))] :
( ( dom(A,B,F) = bot_bot(set(A)) )
<=> ! [X4: A] : aa(A,option(B),F,X4) = none(B) ) ).
% dom_eq_empty_conv
tff(fact_4625_dom__restrict,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),A4: set(A)] : dom(A,B,restrict_map(A,B,M2,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M2)),A4) ).
% dom_restrict
tff(fact_4626_dom__empty,axiom,
! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_oj(A,option(B))) = bot_bot(set(A)) ).
% dom_empty
tff(fact_4627_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: 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_pe(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K),V),M2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),dom(A,B,M2)) ).
% map_update_eta_repair(1)
tff(fact_4628_dom__const_H,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : dom(A,B,aTP_Lamp_pf(fun(A,B),fun(A,option(B)),F)) = top_top(set(A)) ).
% dom_const'
tff(fact_4629_restrict__map__inv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F))),X3) = none(B) ).
% restrict_map_inv
tff(fact_4630_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option(B),F: fun(A,option(B)),X: A] :
( ( ( Y = none(B) )
=> ( dom(A,B,fun_upd(A,option(B),F,X,Y)) = minus_minus(set(A),dom(A,B,F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) )
& ( ( Y != none(B) )
=> ( dom(A,B,fun_upd(A,option(B),F,X,Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),dom(A,B,F)) ) ) ) ).
% dom_fun_upd
tff(fact_4631_ran__map__add,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,M1,M22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M1)),ran(A,B,M22)) ) ) ).
% ran_map_add
tff(fact_4632_ran__add,axiom,
! [B: $tType,A: $tType,F: 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,F)),dom(A,B,G)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,F,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F)),ran(A,B,G)) ) ) ).
% ran_add
tff(fact_4633_nempty__dom,axiom,
! [B: $tType,A: $tType,E3: fun(A,option(B))] :
( ~ ! [X3: A] : aa(A,option(B),E3,X3) = none(B)
=> ~ ! [M4: A] : ~ pp(member2(A,M4,dom(A,B,E3))) ) ).
% nempty_dom
tff(fact_4634_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D2: set(A),M2: fun(A,option(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D2),dom(A,B,M2)))
=> ( inj_on(A,option(B),M2,D2)
=> inj_on(A,B,comp(option(B),B,A,the2(B),M2),D2) ) ) ).
% inj_on_map_the
tff(fact_4635_ran__is__image,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(B),set(A),image2(B,A,comp(option(A),A,B,the2(A),M)),dom(B,A,M)) ).
% ran_is_image
tff(fact_4636_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] :
( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
=> pp(aa(set(B),bool,finite_finite2(B),ran(A,B,M))) ) ).
% map_dom_ran_finite
tff(fact_4637_dom__if,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),F: fun(A,option(B)),G: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_pg(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F),G)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F)),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,G)),collect(A,aTP_Lamp_bi(fun(A,bool),fun(A,bool),P)))) ).
% dom_if
tff(fact_4638_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M2: fun(A,option(B)),M6: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M2),M6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,M2)),dom(A,B,M6))) ) ) ).
% le_map_dom_mono
tff(fact_4639_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: fun(A,bool),X: A] :
( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
=> pp(aa(A,bool,P,X)) ) ).
% Eps_Opt_eq_Some_implies
tff(fact_4640_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: fun(A,bool),X: A] :
( ! [X8: A] :
( pp(aa(A,bool,P,X))
=> ( pp(aa(A,bool,P,X8))
=> ( X8 = X ) ) )
=> ( ( eps_Opt(A,P) = aa(A,option(A),some(A),X) )
<=> pp(aa(A,bool,P,X)) ) ) ).
% Eps_Opt_eq_Some
tff(fact_4641_map__add__comm,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
=> ( map_add(A,B,M1,M22) = map_add(A,B,M22,M1) ) ) ).
% map_add_comm
tff(fact_4642_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A4: fun(A,option(B)),B3: fun(A,option(B)),C4: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,A4)),dom(A,B,B3)) = bot_bot(set(A)) )
=> ( map_add(A,B,A4,map_add(A,B,B3,C4)) = map_add(A,B,B3,map_add(A,B,A4,C4)) ) ) ).
% map_add_left_comm
tff(fact_4643_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A)),A4: set(B),K: B] :
( ( aa(B,option(A),restrict_map(B,A,M2,A4),K) = none(A) )
<=> ~ pp(member2(B,K,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M2)),A4))) ) ).
% restrict_map_eq(1)
tff(fact_4644_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: A] :
( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
=> ( ( finite_card(A,dom(A,B,M)) = finite_card(B,ran(A,B,M)) )
=> ( pp(member2(A,X,dom(A,B,M)))
=> ( ( aa(A,option(B),M,X) = aa(A,option(B),M,Y) )
<=> ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
tff(fact_4645_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] :
( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),map_to_set(A,B,M2)))
<=> pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M2))) ) ).
% finite_map_to_set
tff(fact_4646_map__to__set__dom,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : dom(A,B,M2) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M2)) ).
% map_to_set_dom
tff(fact_4647_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_4648_card__map__to__set,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : finite_card(product_prod(A,B),map_to_set(A,B,M2)) = finite_card(A,dom(A,B,M2)) ).
% card_map_to_set
tff(fact_4649_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_ph(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs),aa(B,option(A),F,X)) ).
% map_filter_simps(1)
tff(fact_4650_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M2: fun(A,option(B)),M6: fun(A,option(B)),N: fun(A,option(B)),N9: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M2),M6))
=> ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),N),N9))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M6)),dom(A,B,N9)) = bot_bot(set(A)) )
=> pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M2,N)),map_add(A,B,M6,N9))) ) ) ) ) ).
% map_add_distinct_le
tff(fact_4651_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] :
( ( dom(A,B,F) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
<=> ? [V4: B] : F = fun_upd(A,option(B),aTP_Lamp_oj(A,option(B)),X,aa(B,option(B),some(B),V4)) ) ).
% dom_eq_singleton_conv
tff(fact_4652_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,bool),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,bool)),S),K)) ).
% set_to_map_def
tff(fact_4653_map__filter__def,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F,Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),comp(option(B),B,A,the2(B),F)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_pj(fun(A,option(B)),fun(A,bool),F)),Xs)) ).
% map_filter_def
tff(fact_4654_dom__override__on,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B)),A4: set(A)] : dom(A,B,override_on(A,option(B),F,G,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),minus_minus(set(A),dom(A,B,F),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_pk(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4)))),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_pl(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4))) ).
% dom_override_on
tff(fact_4655_graph__map__add,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
=> ( graph(A,B,map_add(A,B,M1,M22)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M1)),graph(A,B,M22)) ) ) ).
% graph_map_add
tff(fact_4656_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S4: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S4),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))))
& ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S4)) ) )
<=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
& ( S4 = minus_minus(set(A),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4)),dom(A,B,A4)) ) ) ) ).
% eq_f_restr_conv
tff(fact_4657_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] : override_on(A,B,F,G,bot_bot(set(A))) = F ).
% override_on_emptyset
tff(fact_4658_map__leI,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
( ! [X2: A,V2: B] :
( ( aa(A,option(B),M1,X2) = aa(B,option(B),some(B),V2) )
=> ( aa(A,option(B),M22,X2) = aa(B,option(B),some(B),V2) ) )
=> map_le(A,B,M1,M22) ) ).
% map_leI
tff(fact_4659_map__leD,axiom,
! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),K: A,V: B] :
( map_le(A,B,M1,M22)
=> ( ( aa(A,option(B),M1,K) = aa(B,option(B),some(B),V) )
=> ( aa(A,option(B),M22,K) = aa(B,option(B),some(B),V) ) ) ) ).
% map_leD
tff(fact_4660_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S4: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S4),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))))
=> ( ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S4)) )
<=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
& ( S4 = minus_minus(set(A),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4)),dom(A,B,A4)) ) ) ) ) ).
% eq_f_restr_ss_eq
tff(fact_4661_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),K5: set(A),V: B] : map_le(A,B,M2,map_mmupd(A,B,M2,minus_minus(set(A),K5,dom(A,B,M2)),V)) ).
% le_map_mmupd_not_dom
tff(fact_4662_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K5: set(A),K6: set(A),M2: fun(A,option(B)),V: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K5),K6))
=> map_le(A,B,map_mmupd(A,B,M2,minus_minus(set(A),K5,dom(A,B,M2)),V),map_mmupd(A,B,M2,minus_minus(set(A),K6,dom(A,B,M2)),V)) ) ).
% map_mmupd_update_less
tff(fact_4663_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_4664_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K5: set(A),M2: fun(A,option(B)),V: B] :
( ~ pp(member2(A,K,K5))
=> ( aa(A,option(B),map_mmupd(A,B,M2,K5,V),K) = aa(A,option(B),M2,K) ) ) ).
% mmupd_notin_upd
tff(fact_4665_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),V: B] : map_mmupd(A,B,M2,bot_bot(set(A)),V) = M2 ).
% map_mmupd_empty
tff(fact_4666_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K5: set(A),M2: fun(A,option(B)),V: B] :
( pp(member2(A,K,K5))
=> ( aa(A,option(B),map_mmupd(A,B,M2,K5,V),K) = aa(B,option(B),some(B),V) ) ) ).
% mmupd_in_upd
tff(fact_4667_the__dflt__None__empty,axiom,
! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).
% the_dflt_None_empty
tff(fact_4668_dom__mmupd,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),K5: set(A),V: B] : dom(A,B,map_mmupd(A,B,M2,K5,V)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M2)),K5) ).
% dom_mmupd
tff(fact_4669_map__mmupd__def,axiom,
! [A: $tType,B: $tType,K: B,K5: set(B),M2: fun(B,option(A)),V: A] :
( ( pp(member2(B,K,K5))
=> ( aa(B,option(A),map_mmupd(B,A,M2,K5,V),K) = aa(A,option(A),some(A),V) ) )
& ( ~ pp(member2(B,K,K5))
=> ( aa(B,option(A),map_mmupd(B,A,M2,K5,V),K) = aa(B,option(A),M2,K) ) ) ) ).
% map_mmupd_def
tff(fact_4670_map__mmupdE,axiom,
! [B: $tType,A: $tType,M2: fun(B,option(A)),K5: set(B),V: A,K: B,X: A] :
( ( aa(B,option(A),map_mmupd(B,A,M2,K5,V),K) = aa(A,option(A),some(A),X) )
=> ( ( ~ pp(member2(B,K,K5))
=> ( aa(B,option(A),M2,K) != aa(A,option(A),some(A),X) ) )
=> ~ ( pp(member2(B,K,K5))
=> ( X != V ) ) ) ) ).
% map_mmupdE
tff(fact_4671_dflt__None__set__def,axiom,
! [A: $tType,S: set(A)] :
( ( ( S = bot_bot(set(A)) )
=> ( dflt_None_set(A,S) = none(set(A)) ) )
& ( ( S != bot_bot(set(A)) )
=> ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ) ).
% dflt_None_set_def
tff(fact_4672_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_4673_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
( order_preorder_on(A,field2(A,R3),R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( pp(member2(A,B2,field2(A,R3)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
<=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R3)) ) ) ) ) ).
% subset_Image1_Image1_iff
tff(fact_4674_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A3: A,M: multiset(A)] :
( ~ pp(member2(A,A3,set_mset(A,minus_minus(multiset(A),M,add_mset(A,A3,zero_zero(multiset(A)))))))
=> ( set_mset(A,minus_minus(multiset(A),M,add_mset(A,A3,zero_zero(multiset(A))))) = minus_minus(set(A),set_mset(A,M),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ).
% at_most_one_mset_mset_diff
tff(fact_4675_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(B,A))] : image(B,A,R,bot_bot(set(B))) = bot_bot(set(A)) ).
% Image_empty2
tff(fact_4676_set__mset__eq__empty__iff,axiom,
! [A: $tType,M: multiset(A)] :
( ( set_mset(A,M) = bot_bot(set(A)) )
<=> ( M = zero_zero(multiset(A)) ) ) ).
% set_mset_eq_empty_iff
tff(fact_4677_set__mset__empty,axiom,
! [A: $tType] : set_mset(A,zero_zero(multiset(A))) = bot_bot(set(A)) ).
% set_mset_empty
tff(fact_4678_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_4679_Image__Id__on,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : image(A,A,id_on(A,A4),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) ).
% Image_Id_on
tff(fact_4680_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A3: B] :
( pp(member2(A,B2,image(B,A,R3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))))
<=> pp(member2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2),R3)) ) ).
% Image_singleton_iff
tff(fact_4681_mset__diff__cancel1elem,axiom,
! [A: $tType,A3: A,B3: multiset(A)] :
( ~ pp(member2(A,A3,set_mset(A,B3)))
=> ( minus_minus(multiset(A),add_mset(A,A3,zero_zero(multiset(A))),B3) = add_mset(A,A3,zero_zero(multiset(A))) ) ) ).
% mset_diff_cancel1elem
tff(fact_4682_listrel__Nil,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(B,A))] : image(list(B),list(A),listrel(B,A,R3),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_4683_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E5: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U)),E5) = image(B,A,E5,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_4684_mset__right__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset(A),B3: multiset(A)] :
( pp(member2(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3))))
=> ( ~ pp(member2(A,A3,set_mset(A,B3)))
=> pp(member2(A,A3,set_mset(A,A4))) ) ) ).
% mset_right_cancel_union
tff(fact_4685_mset__left__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset(A),B3: multiset(A)] :
( pp(member2(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3))))
=> ( ~ pp(member2(A,A3,set_mset(A,A4)))
=> pp(member2(A,A3,set_mset(A,B3))) ) ) ).
% mset_left_cancel_union
tff(fact_4686_mset__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset(A),B3: multiset(A)] :
( pp(member2(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3))))
=> ( ~ pp(member2(A,A3,set_mset(A,A4)))
=> pp(member2(A,A3,set_mset(A,B3))) ) ) ).
% mset_un_cases
tff(fact_4687_ex__Melem__conv,axiom,
! [A: $tType,A4: multiset(A)] :
( ? [X4: A] : pp(member2(A,X4,set_mset(A,A4)))
<=> ( A4 != zero_zero(multiset(A)) ) ) ).
% ex_Melem_conv
tff(fact_4688_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,A4)),image(B,A,R,B3)))) ).
% Image_Int_subset
tff(fact_4689_rtrancl__image__advance,axiom,
! [A: $tType,Q2: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
( pp(member2(A,Q2,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0)))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),X),R))
=> pp(member2(A,X,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0))) ) ) ).
% rtrancl_image_advance
tff(fact_4690_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q2: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
( pp(member2(A,Q2,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0)))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),X),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> pp(member2(A,X,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),Q0))) ) ) ).
% rtrancl_image_advance_rtrancl
tff(fact_4691_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B3: set(B),C4: set(product_prod(B,A))] :
( pp(aa(set(A),bool,finite_finite2(A),image(B,A,A4,B3)))
=> ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),C4),A4))
=> pp(aa(set(A),bool,finite_finite2(A),image(B,A,C4,B3))) ) ) ).
% finite_Image_subset
tff(fact_4692_rtrancl__image__unfold__right,axiom,
! [A: $tType,E5: set(product_prod(A,A)),V3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E5,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),V3))),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),V3))) ).
% rtrancl_image_unfold_right
tff(fact_4693_rtrancl__reachable__induct,axiom,
! [A: $tType,I5: set(A),INV: set(A),E5: set(product_prod(A,A))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I5),INV))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E5,INV)),INV))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),I5)),INV)) ) ) ).
% rtrancl_reachable_induct
tff(fact_4694_trancl__Image__unfold__right,axiom,
! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : image(A,A,transitive_trancl(A,E5),S) = image(A,A,E5,image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),S)) ).
% trancl_Image_unfold_right
tff(fact_4695_trancl__Image__unfold__left,axiom,
! [A: $tType,E5: set(product_prod(A,A)),S: set(A)] : image(A,A,transitive_trancl(A,E5),S) = image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),image(A,A,E5,S)) ).
% trancl_Image_unfold_left
tff(fact_4696_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R: set(product_prod(A,A)),V: A] :
( ( image(A,A,R,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,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),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_4697_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R: set(product_prod(A,A)),V: A] :
( ( image(A,A,R,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,R),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_4698_reachable__mono,axiom,
! [A: $tType,R: set(product_prod(A,A)),R7: set(product_prod(A,A)),X5: set(A),X9: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),R7))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),X9))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),X5)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R7),X9))) ) ) ).
% reachable_mono
tff(fact_4699_quotientI,axiom,
! [A: $tType,X: A,A4: set(A),R3: set(product_prod(A,A))] :
( pp(member2(A,X,A4))
=> pp(member2(set(A),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),equiv_quotient(A,A4,R3))) ) ).
% quotientI
tff(fact_4700_quotientE,axiom,
! [A: $tType,X5: set(A),A4: set(A),R3: set(product_prod(A,A))] :
( pp(member2(set(A),X5,equiv_quotient(A,A4,R3)))
=> ~ ! [X2: A] :
( ( X5 = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))) )
=> ~ pp(member2(A,X2,A4)) ) ) ).
% quotientE
tff(fact_4701_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(B,A,A4,B3)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A4))) ).
% Image_subset_snd_image
tff(fact_4702_trancl__image__by__rtrancl,axiom,
! [A: $tType,E5: 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,E5),Vi)),Vi) = image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),Vi) ).
% trancl_image_by_rtrancl
tff(fact_4703_mset__left__cancel__elem,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset(A)] :
( pp(member2(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,B2,zero_zero(multiset(A)))),A4))))
=> ( ( A3 != B2 )
=> pp(member2(A,A3,set_mset(A,A4))) ) ) ).
% mset_left_cancel_elem
tff(fact_4704_mset__right__cancel__elem,axiom,
! [A: $tType,A3: A,A4: multiset(A),B2: A] :
( pp(member2(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),add_mset(A,B2,zero_zero(multiset(A)))))))
=> ( ( A3 != B2 )
=> pp(member2(A,A3,set_mset(A,A4))) ) ) ).
% mset_right_cancel_elem
tff(fact_4705_Image__singleton,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: B] : image(B,A,R3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))) = collect(A,aa(B,fun(A,bool),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,bool)),R3),A3)) ).
% Image_singleton
tff(fact_4706_Image__fold,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R))
=> ( image(A,B,R,S) = finite_fold(product_prod(A,B),set(B),product_case_prod(A,B,fun(set(B),set(B)),aTP_Lamp_pm(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R) ) ) ).
% Image_fold
tff(fact_4707_listrel__Cons,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B,Xs: list(B)] : image(list(B),list(A),listrel(B,A,R3),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))))) = aa(set(list(A)),set(list(A)),aa(set(A),fun(set(list(A)),set(list(A))),set_Cons(A),image(B,A,R3,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,R3),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_4708_rtrancl__apply__insert,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,S: set(A)] : image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),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,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ).
% rtrancl_apply_insert
tff(fact_4709_rtrancl__Image__in__Field,axiom,
! [A: $tType,R: set(product_prod(A,A)),V3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R),V3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R)),V3))) ).
% rtrancl_Image_in_Field
tff(fact_4710_set__mset__single,axiom,
! [A: $tType,B2: A] : set_mset(A,add_mset(A,B2,zero_zero(multiset(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) ).
% set_mset_single
tff(fact_4711_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E5,R)),R))
=> ( ~ pp(member2(A,V,R))
=> ~ ( ~ pp(member2(A,U,R))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),rel_restrict(A,E5,R)))) ) ) ) ) ).
% E_closed_restr_reach_cases
tff(fact_4712_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E5: set(product_prod(A,A)),R: set(A)] :
( pp(member2(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,E5)))
=> ( ~ pp(member2(A,X,R))
=> ( ~ pp(member2(A,Y,R))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E5,R)),R))
=> pp(member2(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,E5,R)))) ) ) ) ) ).
% rel_restrict_tranclI
tff(fact_4713_mset__contains__eq,axiom,
! [A: $tType,M2: A,M: multiset(A)] :
( pp(member2(A,M2,set_mset(A,M)))
<=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,M2,zero_zero(multiset(A)))),minus_minus(multiset(A),M,add_mset(A,M2,zero_zero(multiset(A))))) = M ) ) ).
% mset_contains_eq
tff(fact_4714_mset__union__diff__comm,axiom,
! [A: $tType,T4: A,S: multiset(A),T: multiset(A)] :
( pp(member2(A,T4,set_mset(A,S)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T),minus_minus(multiset(A),S,add_mset(A,T4,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)),T),S),add_mset(A,T4,zero_zero(multiset(A)))) ) ) ).
% mset_union_diff_comm
tff(fact_4715_diff__union__single__conv2,axiom,
! [A: $tType,A3: A,J4: multiset(A),I5: multiset(A)] :
( pp(member2(A,A3,set_mset(A,J4)))
=> ( minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),J4),I5),add_mset(A,A3,zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),J4,add_mset(A,A3,zero_zero(multiset(A))))),I5) ) ) ).
% diff_union_single_conv2
tff(fact_4716_mset__un__single__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset(A),B3: multiset(A),C4: multiset(A)] :
( ( add_mset(A,A3,A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),C4) )
=> ( ( pp(member2(A,A3,set_mset(A,B3)))
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),minus_minus(multiset(A),B3,add_mset(A,A3,zero_zero(multiset(A))))),C4) ) )
=> ~ ( pp(member2(A,A3,set_mset(A,C4)))
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B3),minus_minus(multiset(A),C4,add_mset(A,A3,zero_zero(multiset(A))))) ) ) ) ) ).
% mset_un_single_un_cases
tff(fact_4717_Image__eq__UN,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),B3: set(B)] : image(B,A,R3,B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_pn(set(product_prod(B,A)),fun(B,set(A)),R3)),B3)) ).
% Image_eq_UN
tff(fact_4718_Sigma__Image,axiom,
! [A: $tType,B: $tType,A4: set(B),B3: fun(B,set(A)),X5: set(B)] : image(B,A,product_Sigma(B,A,A4,B3),X5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X5),A4))) ).
% Sigma_Image
tff(fact_4719_finite__reachable__advance,axiom,
! [A: $tType,E5: set(product_prod(A,A)),V0: A,V: A] :
( pp(aa(set(A),bool,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V0),bot_bot(set(A))))))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5)))
=> pp(aa(set(A),bool,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A)))))) ) ) ).
% finite_reachable_advance
tff(fact_4720_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E5: set(product_prod(A,A))] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),V),bot_bot(set(A))))),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),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_4721_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E5: set(product_prod(A,A))] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,E5),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,E5),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_4722_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_4723_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] : the_default(A,X,none(A)) = X ).
% the_default.simps(2)
tff(fact_4724_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E5: set(product_prod(A,A)),S: set(A)] :
( pp(member2(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,E5)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E5,S)),S))
=> ( pp(member2(A,U,S))
=> pp(member2(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))),E5),product_Sigma(A,A,S,aTP_Lamp_mj(set(A),fun(A,set(A)),S)))))) ) ) ) ).
% trancl_restrict_reachable
tff(fact_4725_singleton__quotient,axiom,
! [A: $tType,X: A,R3: 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))),R3) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),image(A,A,R3,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_4726_quotient__def,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] : equiv_quotient(A,A4,R3) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_po(set(product_prod(A,A)),fun(A,set(set(A))),R3)),A4)) ).
% quotient_def
tff(fact_4727_sum__wcount__Int,axiom,
! [A: $tType,A4: set(A),F: fun(A,nat),N4: multiset(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F,N4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),set_mset(A,N4))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F,N4)),A4) ) ) ).
% sum_wcount_Int
tff(fact_4728_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N = zero_zero(nat) )
=> ( set_mset(A,replicate_mset(A,N,X)) = bot_bot(set(A)) ) )
& ( ( N != zero_zero(nat) )
=> ( set_mset(A,replicate_mset(A,N,X)) = 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_4729_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
( refl_on(A,field2(A,R3),R3)
=> ( antisym(A,R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( pp(member2(A,B2,field2(A,R3)))
=> ( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
<=> ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
tff(fact_4730_antisym__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( antisym(A,R3)
=> antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% antisym_Restr
tff(fact_4731_size__diff__se,axiom,
! [A: $tType,T4: A,S: multiset(A)] :
( pp(member2(A,T4,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,T4,zero_zero(multiset(A)))))),one_one(nat)) ) ) ).
% size_diff_se
tff(fact_4732_sorted__list__of__multiset__def,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: multiset(A)] : linord6283353356039996273ltiset(A,M) = fold_mset(A,list(A),linorder_insort_key(A,A,aTP_Lamp_az(A,A)),nil(A),M) ) ).
% sorted_list_of_multiset_def
tff(fact_4733_flat__lub__def,axiom,
! [A: $tType,A4: set(A),B2: A] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
=> ( partial_flat_lub(A,B2,A4) = B2 ) )
& ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
=> ( partial_flat_lub(A,B2,A4) = the(A,aa(A,fun(A,bool),aTP_Lamp_pp(set(A),fun(A,fun(A,bool)),A4),B2)) ) ) ) ).
% flat_lub_def
tff(fact_4734_mset__size__le1__cases,axiom,
! [A: $tType,M: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M)),aa(nat,nat,suc,zero_zero(nat))))
=> ( ( M != zero_zero(multiset(A)) )
=> ~ ! [M4: A] : M != add_mset(A,M4,zero_zero(multiset(A))) ) ) ).
% mset_size_le1_cases
tff(fact_4735_mset__size1elem,axiom,
! [A: $tType,P: multiset(A),Q2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),one_one(nat)))
=> ( pp(member2(A,Q2,set_mset(A,P)))
=> ( P = add_mset(A,Q2,zero_zero(multiset(A))) ) ) ) ).
% mset_size1elem
tff(fact_4736_mset__size2elem,axiom,
! [A: $tType,P: multiset(A),Q2: A,Q4: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q2,zero_zero(multiset(A)))),add_mset(A,Q4,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,Q2,zero_zero(multiset(A)))),add_mset(A,Q4,zero_zero(multiset(A)))) ) ) ) ).
% mset_size2elem
tff(fact_4737_mset__set__Union,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(aa(set(A),bool,finite_finite2(A),B3))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
=> ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A4)),mset_set(A,B3)) ) ) ) ) ).
% mset_set_Union
tff(fact_4738_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
( order_7125193373082350890der_on(A,field2(A,R3),R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( pp(member2(A,B2,field2(A,R3)))
=> ( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
<=> ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
tff(fact_4739_mset__set_Oempty,axiom,
! [A: $tType] : mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ).
% mset_set.empty
tff(fact_4740_mset__le__addE,axiom,
! [A: $tType,Xs: multiset(A),Ys: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xs),Ys))
=> ~ ! [Zs3: multiset(A)] : Ys != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs3) ) ).
% mset_le_addE
tff(fact_4741_mset__le__distrib,axiom,
! [A: $tType,X5: multiset(A),A4: multiset(A),B3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)))
=> ~ ! [Xa5: multiset(A),Xb3: multiset(A)] :
( ( X5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa5),Xb3) )
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xa5),A4))
=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xb3),B3)) ) ) ) ).
% mset_le_distrib
tff(fact_4742_mset__union__subset,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),C4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)),C4))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),C4))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),C4)) ) ) ).
% mset_union_subset
tff(fact_4743_mset__le__decr__left1,axiom,
! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_decr_left1
tff(fact_4744_mset__le__decr__left2,axiom,
! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_decr_left2
tff(fact_4745_mset__le__incr__right1,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3))) ) ).
% mset_le_incr_right1
tff(fact_4746_mset__le__incr__right2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B2))) ) ).
% mset_le_incr_right2
tff(fact_4747_mset__le__subtract,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),C4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B3))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),minus_minus(multiset(A),A4,C4)),minus_minus(multiset(A),B3,C4))) ) ).
% mset_le_subtract
tff(fact_4748_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C3: A,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,C3,A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_add_mset_decr_left1
tff(fact_4749_mset__le__add__mset,axiom,
! [A: $tType,X: A,B3: multiset(A),C4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,B3)),C4))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),C4))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),C4)) ) ) ).
% mset_le_add_mset
tff(fact_4750_mset__le__single__cases,axiom,
! [A: $tType,M: multiset(A),A3: A] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),add_mset(A,A3,zero_zero(multiset(A)))))
=> ( ( M != zero_zero(multiset(A)) )
=> ( M = add_mset(A,A3,zero_zero(multiset(A))) ) ) ) ).
% mset_le_single_cases
tff(fact_4751_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C3: A,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,C3,A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,C3,zero_zero(multiset(A)))),B2)) ) ).
% mset_le_add_mset_decr_left2
tff(fact_4752_mset__le__subtract__left,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),X5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)),X5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),minus_minus(multiset(A),X5,A4)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),X5)) ) ) ).
% mset_le_subtract_left
tff(fact_4753_mset__le__subtract__right,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A),X5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)),X5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),minus_minus(multiset(A),X5,B3)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),X5)) ) ) ).
% mset_le_subtract_right
tff(fact_4754_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_4755_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B3: multiset(A),X5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,B3)),X5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),minus_minus(multiset(A),X5,B3)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),X5)) ) ) ).
% mset_le_subtract_add_mset_right
tff(fact_4756_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B3: multiset(A),X5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,B3)),X5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),minus_minus(multiset(A),X5,add_mset(A,X,zero_zero(multiset(A))))))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),add_mset(A,X,zero_zero(multiset(A)))),X5)) ) ) ).
% mset_le_subtract_add_mset_left
tff(fact_4757_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A4: set(B),M2: multiset(A),F: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M2),aa(B,multiset(A),F,X2))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M2),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)))) ) ) ).
% subset_mset.cINF_greatest
tff(fact_4758_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A4: set(B),F: fun(B,multiset(A)),M: multiset(A)] :
( ( A4 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,X2)),M)) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))),M)) ) ) ).
% subset_mset.cSUP_least
tff(fact_4759_mset__set__empty__iff,axiom,
! [A: $tType,A4: set(A)] :
( ( mset_set(A,A4) = zero_zero(multiset(A)) )
<=> ( ( A4 = bot_bot(set(A)) )
| ~ pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ).
% mset_set_empty_iff
tff(fact_4760_infinite__set__mset__mset__set,axiom,
! [A: $tType,A4: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( set_mset(A,mset_set(A,A4)) = bot_bot(set(A)) ) ) ).
% infinite_set_mset_mset_set
tff(fact_4761_mset__le__mono__add__single,axiom,
! [A: $tType,A3: A,Ys: multiset(A),B2: A,Ws2: multiset(A)] :
( pp(member2(A,A3,set_mset(A,Ys)))
=> ( pp(member2(A,B2,set_mset(A,Ws2)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B2,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys),Ws2))) ) ) ).
% mset_le_mono_add_single
tff(fact_4762_mset__union__subset__s,axiom,
! [A: $tType,A3: A,B3: multiset(A),C4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),B3)),C4))
=> ( pp(member2(A,A3,set_mset(A,C4)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B3),C4)) ) ) ).
% mset_union_subset_s
tff(fact_4763_mset__2dist2__cases,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset(A),B3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B2,zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B3)))
=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B2,zero_zero(multiset(A))))),A4))
=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A3,zero_zero(multiset(A)))),add_mset(A,B2,zero_zero(multiset(A))))),B3))
=> ( ( pp(member2(A,A3,set_mset(A,A4)))
=> ~ pp(member2(A,B2,set_mset(A,B3))) )
=> ~ ( pp(member2(A,A3,set_mset(A,B3)))
=> ~ pp(member2(A,B2,set_mset(A,A4))) ) ) ) ) ) ).
% mset_2dist2_cases
tff(fact_4764_Partial__order__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_7125193373082350890der_on(A,field2(A,R3),R3)
=> order_7125193373082350890der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Partial_order_Restr
tff(fact_4765_mset__set_Oinsert__remove,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( mset_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = add_mset(A,X,mset_set(A,minus_minus(set(A),A4,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_4766_mset__set_Oremove,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( mset_set(A,A4) = add_mset(A,X,mset_set(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).
% mset_set.remove
tff(fact_4767_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_bot(A)
=> ordering_top(A,aTP_Lamp_pq(A,fun(A,bool)),aTP_Lamp_pr(A,fun(A,bool)),bot_bot(A)) ) ).
% bot.ordering_top_axioms
tff(fact_4768_prod__mset_Oremove,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,A4: multiset(A)] :
( pp(member2(A,X,set_mset(A,A4)))
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),minus_minus(multiset(A),A4,add_mset(A,X,zero_zero(multiset(A)))))) ) ) ) ).
% prod_mset.remove
tff(fact_4769_proj__def,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B] : equiv_proj(B,A,R3,X) = image(B,A,R3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))) ).
% proj_def
tff(fact_4770_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_4771_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,N4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),add_mset(A,X,N4)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N4)) ) ).
% prod_mset.add_mset
tff(fact_4772_prod__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: multiset(A),N4: 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)),M),N4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N4)) ) ).
% prod_mset.union
tff(fact_4773_prod__mset__Un,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(A),B3: 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)),A4),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),B3)) ) ).
% prod_mset_Un
tff(fact_4774_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),Top)) ) ).
% ordering_top.extremum
tff(fact_4775_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A3)) ) ).
% ordering_top.extremum_strict
tff(fact_4776_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
<=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
tff(fact_4777_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( ( A3 != Top )
<=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),Top)) ) ) ).
% ordering_top.not_eq_extremum
tff(fact_4778_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
tff(fact_4779_prod__mset_Oneutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(A)] :
( ! [X2: A] :
( pp(member2(A,X2,set_mset(A,A4)))
=> ( X2 = one_one(A) ) )
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4) = one_one(A) ) ) ) ).
% prod_mset.neutral
tff(fact_4780_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: multiset(A)] :
( pp(dvd_dvd(A,aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4),one_one(A)))
<=> ! [X4: A] :
( pp(member2(A,X4,set_mset(A,A4)))
=> pp(dvd_dvd(A,X4,one_one(A))) ) ) ) ).
% is_unit_prod_mset_iff
tff(fact_4781_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M) = fold_mset(A,A,times_times(A),one_one(A),M) ) ).
% prod_mset.eq_fold
tff(fact_4782_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_4783_proj__iff,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A4))
=> ( ( equiv_proj(A,A,R3,X) = equiv_proj(A,A,R3,Y) )
<=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)) ) ) ) ).
% proj_iff
tff(fact_4784_cSUP__UNION,axiom,
! [D: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [A4: set(C),B3: fun(C,set(D)),F: fun(D,B)] :
( ( A4 != bot_bot(set(C)) )
=> ( ! [X2: C] :
( pp(member2(C,X2,A4))
=> ( aa(C,set(D),B3,X2) != bot_bot(set(D)) ) )
=> ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ps(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B3),F)),A4)))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B3),A4)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_pt(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B3),F)),A4)) ) ) ) ) ) ).
% cSUP_UNION
tff(fact_4785_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,A4)
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A4)))) ) ) ) ) ).
% mono_cSup
tff(fact_4786_bdd__above__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit941137186595557371_above(A,bot_bot(set(A))) ) ).
% bdd_above_empty
tff(fact_4787_equiv__listrel,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A))] :
( equiv_equiv(A,A4,R3)
=> equiv_equiv(list(A),aa(set(A),set(list(A)),lists(A),A4),listrel(A,A,R3)) ) ).
% equiv_listrel
tff(fact_4788_bdd__above__Int1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: set(A),B3: set(A)] :
( condit941137186595557371_above(A,A4)
=> condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% bdd_above_Int1
tff(fact_4789_bdd__above__Int2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: set(A),A4: set(A)] :
( condit941137186595557371_above(A,B3)
=> condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% bdd_above_Int2
tff(fact_4790_in__quotient__imp__non__empty,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X5: set(A)] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(set(A),X5,equiv_quotient(A,A4,R3)))
=> ( X5 != bot_bot(set(A)) ) ) ) ).
% in_quotient_imp_non_empty
tff(fact_4791_cSup__le__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,S)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3))
<=> ! [X4: A] :
( pp(member2(A,X4,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A3)) ) ) ) ) ) ).
% cSup_le_iff
tff(fact_4792_cSup__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B3: set(A),A4: set(A)] :
( ( B3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A4)
=> ( ! [B4: A] :
( pp(member2(A,B4,B3))
=> ? [X3: A] :
( pp(member2(A,X3,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),X3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B3)),aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ) ).
% cSup_mono
tff(fact_4793_less__cSup__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,X5)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5)))
<=> ? [X4: A] :
( pp(member2(A,X4,X5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X4)) ) ) ) ) ) ).
% less_cSup_iff
tff(fact_4794_equiv__class__self,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),A3: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(A,A3,A4))
=> pp(member2(A,A3,image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).
% equiv_class_self
tff(fact_4795_quotient__disj,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X5: set(A),Y5: set(A)] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(set(A),X5,equiv_quotient(A,A4,R3)))
=> ( pp(member2(set(A),Y5,equiv_quotient(A,A4,R3)))
=> ( ( X5 = Y5 )
| ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),Y5) = bot_bot(set(A)) ) ) ) ) ) ).
% quotient_disj
tff(fact_4796_cSUP__le__iff,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),U: A] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),U))
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),U)) ) ) ) ) ) ).
% cSUP_le_iff
tff(fact_4797_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),G: fun(C,A),B3: set(C),F: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,G),B3))
=> ( ! [N7: B] :
( pp(member2(B,N7,A4))
=> ? [X3: C] :
( pp(member2(C,X3,B3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N7)),aa(C,A,G,X3))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3)))) ) ) ) ) ).
% cSUP_mono
tff(fact_4798_cSup__subset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).
% cSup_subset_mono
tff(fact_4799_cSup__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: 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),A3),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).
% cSup_insert
tff(fact_4800_cSup__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( condit941137186595557371_above(A,X5)
=> ( ( ( X5 = bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = A3 ) )
& ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ) ).
% cSup_insert_If
tff(fact_4801_cSup__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A4)
=> ( ( B3 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,B3)
=> ( 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)),A4),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ) ) ).
% cSup_union_distrib
tff(fact_4802_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(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))
<=> ( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
& pp(member2(A,X,A4))
& pp(member2(A,Y,A4)) ) ) ) ).
% equiv_class_eq_iff
tff(fact_4803_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(A,X,A4))
=> ( pp(member2(A,Y,A4))
=> ( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
<=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)) ) ) ) ) ).
% eq_equiv_class_iff
tff(fact_4804_equiv__class__eq,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3))
=> ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ) ).
% equiv_class_eq
tff(fact_4805_eq__equiv__class,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,A4: set(A)] :
( ( image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
=> ( equiv_equiv(A,A4,R3)
=> ( pp(member2(A,B2,A4))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3)) ) ) ) ).
% eq_equiv_class
tff(fact_4806_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(A,X,A4))
=> ( pp(member2(A,Y,A4))
=> ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R3) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))),R3) )
<=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)) ) ) ) ) ).
% eq_equiv_class_iff2
tff(fact_4807_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S))
=> ( equiv_equiv(A,A4,R)
=> ( equiv_equiv(A,A4,S)
=> ( image(A,A,S,image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq2
tff(fact_4808_refines__equiv__class__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S))
=> ( equiv_equiv(A,A4,R)
=> ( equiv_equiv(A,A4,S)
=> ( image(A,A,R,image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq
tff(fact_4809_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( condit6923001295902523014norder(A)
=> ! [A4: set(B),F: fun(B,A),A3: A] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))))
<=> ? [X4: B] :
( pp(member2(B,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X4))) ) ) ) ) ) ).
% less_cSUP_iff
tff(fact_4810_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),G: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A4))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_pu(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4)) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_4811_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B),F: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),B3))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ) ).
% cSUP_subset_mono
tff(fact_4812_cSup__inter__less__eq,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( condit941137186595557371_above(A,A4)
=> ( condit941137186595557371_above(A,B3)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ) ) ) ).
% cSup_inter_less_eq
tff(fact_4813_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),A3: B] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))) ) ) ) ) ).
% cSUP_insert
tff(fact_4814_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),B3: set(B)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),B3))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),B3))) ) ) ) ) ) ) ).
% cSUP_union
tff(fact_4815_equiv__class__subset,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ).
% equiv_class_subset
tff(fact_4816_subset__equiv__class,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),B2: A,A3: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))))
=> ( pp(member2(A,B2,A4))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3)) ) ) ) ).
% subset_equiv_class
tff(fact_4817_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X: A,A3: A,B2: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3)) ) ) ).
% equiv_class_nondisjoint
tff(fact_4818_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(member2(set(A),X5,equiv_quotient(A,A4,R3)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),X5))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R3)) ) ) ) ).
% in_quotient_imp_in_rel
tff(fact_4819_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),collect(A,aTP_Lamp_pv(set(A),fun(A,bool),S))) ) ) ) ) ).
% cSup_cInf
tff(fact_4820_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A4: fun(C,A),I5: set(C)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A4),I5))
=> ( ( I5 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pw(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I5))))) ) ) ) ) ).
% mono_cSUP
tff(fact_4821_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A24: set(B),R2: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A1: A,A22: B] :
( equiv_equiv(A,A14,R1)
=> ( equiv_equiv(B,A24,R2)
=> ( equiv_congruent2(A,B,set(C),R1,R2,F)
=> ( pp(member2(A,A1,A14))
=> ( pp(member2(B,A22,A24))
=> ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_px(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R2),F),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)),F,A1),A22) ) ) ) ) ) ) ).
% UN_equiv_class2
tff(fact_4822_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A4: set(A),R3: set(product_prod(A,A)),F: fun(A,set(B)),A3: A] :
( equiv_equiv(A,A4,R3)
=> ( equiv_congruent(A,set(B),R3,F)
=> ( pp(member2(A,A3,A4))
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F,A3) ) ) ) ) ).
% UN_equiv_class
tff(fact_4823_disjnt__equiv__class,axiom,
! [A: $tType,A4: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),image(A,A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
<=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3)) ) ) ).
% disjnt_equiv_class
tff(fact_4824_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),S),S))
<=> ( S = bot_bot(set(A)) ) ) ).
% disjnt_self_iff_empty
tff(fact_4825_disjnt__insert2,axiom,
! [A: $tType,Y5: set(A),A3: A,X5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)))
<=> ( ~ pp(member2(A,A3,Y5))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y5),X5)) ) ) ).
% disjnt_insert2
tff(fact_4826_disjnt__insert1,axiom,
! [A: $tType,A3: A,X5: set(A),Y5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)),Y5))
<=> ( ~ pp(member2(A,A3,Y5))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X5),Y5)) ) ) ).
% disjnt_insert1
tff(fact_4827_disjnt__Un2,axiom,
! [A: $tType,C4: set(A),A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),A4))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),B3)) ) ) ).
% disjnt_Un2
tff(fact_4828_disjnt__Un1,axiom,
! [A: $tType,A4: set(A),B3: set(A),C4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)),C4))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),C4))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B3),C4)) ) ) ).
% disjnt_Un1
tff(fact_4829_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C4: set(A),A4: set(B),B3: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,C4,aTP_Lamp_me(set(B),fun(A,set(B)),A4))),product_Sigma(A,B,C4,aTP_Lamp_me(set(B),fun(A,set(B)),B3))))
<=> ( ( C4 = bot_bot(set(A)) )
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),A4),B3)) ) ) ).
% disjnt_Times1_iff
tff(fact_4830_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),C4: set(B),B3: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B3,aTP_Lamp_me(set(B),fun(A,set(B)),C4))))
<=> ( ( C4 = bot_bot(set(B)) )
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),B3)) ) ) ).
% disjnt_Times2_iff
tff(fact_4831_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),C4: fun(A,set(B)),B3: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,C4)),product_Sigma(A,B,B3,C4)))
<=> ( ! [X4: A] :
( pp(member2(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> ( aa(A,set(B),C4,X4) = bot_bot(set(B)) ) )
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),B3)) ) ) ).
% disjnt_Sigma_iff
tff(fact_4832_disjnt__insert,axiom,
! [A: $tType,X: A,N4: set(A),M: set(A)] :
( ~ pp(member2(A,X,N4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),M),N4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),M)),N4)) ) ) ).
% disjnt_insert
tff(fact_4833_disjnt__iff,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),B3))
<=> ! [X4: A] :
~ ( pp(member2(A,X4,A4))
& pp(member2(A,X4,B3)) ) ) ).
% disjnt_iff
tff(fact_4834_disjnt__sym,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B3),A4)) ) ).
% disjnt_sym
tff(fact_4835_disjnt__empty2,axiom,
! [A: $tType,A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),bot_bot(set(A)))) ).
% disjnt_empty2
tff(fact_4836_disjnt__empty1,axiom,
! [A: $tType,A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),bot_bot(set(A))),A4)) ).
% disjnt_empty1
tff(fact_4837_disjnt__subset1,axiom,
! [A: $tType,X5: set(A),Y5: set(A),Z6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X5),Y5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z6),X5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Z6),Y5)) ) ) ).
% disjnt_subset1
tff(fact_4838_disjnt__subset2,axiom,
! [A: $tType,X5: set(A),Y5: set(A),Z6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X5),Y5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z6),Y5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X5),Z6)) ) ) ).
% disjnt_subset2
tff(fact_4839_disjnt__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A4),B3))
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) ) ) ).
% disjnt_def
tff(fact_4840_congruent2__implies__congruent__UN,axiom,
! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A24: set(B),R2: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A3: B] :
( equiv_equiv(A,A14,R1)
=> ( equiv_equiv(B,A24,R2)
=> ( equiv_congruent2(A,B,set(C),R1,R2,F)
=> ( pp(member2(B,A3,A24))
=> 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_px(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R2),F),A3)) ) ) ) ) ).
% congruent2_implies_congruent_UN
tff(fact_4841_cINF__UNION,axiom,
! [D: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [A4: set(C),B3: fun(C,set(D)),F: fun(D,B)] :
( ( A4 != bot_bot(set(C)) )
=> ( ! [X2: C] :
( pp(member2(C,X2,A4))
=> ( aa(C,set(D),B3,X2) != bot_bot(set(D)) ) )
=> ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ps(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B3),F)),A4)))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B3),A4)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_py(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B3),F)),A4)) ) ) ) ) ) ).
% cINF_UNION
tff(fact_4842_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,A4)
=> ( ( A4 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),A4)))) ) ) ) ) ).
% mono_cInf
tff(fact_4843_mono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( condit1219197933456340205attice(B)
& condit1219197933456340205attice(A) )
=> ! [F: fun(A,B),A4: fun(C,A),I5: set(C)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A4),I5))
=> ( ( I5 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pw(fun(A,B),fun(fun(C,A),fun(C,B)),F),A4)),I5)))) ) ) ) ) ).
% mono_cINF
tff(fact_4844_bdd__below__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit1013018076250108175_below(A,bot_bot(set(A))) ) ).
% bdd_below_empty
tff(fact_4845_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [F: fun(B,A),G: fun(B,A),A4: set(B)] :
( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_pz(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4))
<=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
& condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G),A4)) ) ) ) ).
% bdd_below_image_inf
tff(fact_4846_bdd__below__Int2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: set(A),A4: set(A)] :
( condit1013018076250108175_below(A,B3)
=> condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% bdd_below_Int2
tff(fact_4847_bdd__below__Int1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: set(A),B3: set(A)] :
( condit1013018076250108175_below(A,A4)
=> condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% bdd_below_Int1
tff(fact_4848_cInf__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B3: set(A),A4: set(A)] :
( ( B3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,A4)
=> ( ! [B4: A] :
( pp(member2(A,B4,B3))
=> ? [X3: A] :
( pp(member2(A,X3,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B4)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ) ).
% cInf_mono
tff(fact_4849_le__cInf__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,S)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S)))
<=> ! [X4: A] :
( pp(member2(A,X4,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X4)) ) ) ) ) ) ).
% le_cInf_iff
tff(fact_4850_cInf__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X5)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y))
<=> ? [X4: A] :
( pp(member2(A,X4,X5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y)) ) ) ) ) ) ).
% cInf_less_iff
tff(fact_4851_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [B3: set(B),F: fun(C,A),A4: set(C),G: fun(B,A)] :
( ( B3 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,F),A4))
=> ( ! [M4: B] :
( pp(member2(B,M4,B3))
=> ? [X3: C] :
( pp(member2(C,X3,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X3)),aa(B,A,G,M4))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ).
% cINF_mono
tff(fact_4852_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),U: A] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))))
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X4))) ) ) ) ) ) ).
% le_cINF_iff
tff(fact_4853_cInf__superset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B3)),aa(set(A),A,complete_Inf_Inf(A),A4))) ) ) ) ) ).
% cInf_superset_mono
tff(fact_4854_cInf__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: 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),A3),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).
% cInf_insert
tff(fact_4855_cInf__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( condit1013018076250108175_below(A,X5)
=> ( ( ( X5 = bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = A3 ) )
& ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ) ).
% cInf_insert_If
tff(fact_4856_cInf__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,A4)
=> ( ( B3 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,B3)
=> ( 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)),A4),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ) ) ).
% cInf_union_distrib
tff(fact_4857_cINF__less__iff,axiom,
! [B: $tType,A: $tType] :
( condit6923001295902523014norder(A)
=> ! [A4: set(B),F: fun(B,A),A3: A] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),A3))
<=> ? [X4: B] :
( pp(member2(B,X4,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X4)),A3)) ) ) ) ) ) ).
% cINF_less_iff
tff(fact_4858_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),G: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G),A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qa(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A4)) ) ) ) ) ) ).
% cINF_inf_distrib
tff(fact_4859_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),G: fun(B,A),B3: set(B),F: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G),B3))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,B3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G,X2)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4)))) ) ) ) ) ) ).
% cINF_superset_mono
tff(fact_4860_less__eq__cInf__inter,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A),B3: set(A)] :
( condit1013018076250108175_below(A,A4)
=> ( condit1013018076250108175_below(A,B3)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B3))),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)),A4),B3)))) ) ) ) ) ).
% less_eq_cInf_inter
tff(fact_4861_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),A3: B] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))) ) ) ) ) ).
% cINF_insert
tff(fact_4862_cINF__union,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F: fun(B,A),B3: set(B)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F),B3))
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),B3))) ) ) ) ) ) ) ).
% cINF_union
tff(fact_4863_cInf__le__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,A4)
=> ( condit1013018076250108175_below(A,A4)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ) ).
% cInf_le_cSup
tff(fact_4864_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),collect(A,aTP_Lamp_qb(set(A),fun(A,bool),S))) ) ) ) ) ).
% cInf_cSup
tff(fact_4865_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).
% numeral_num_of_nat_unfold
tff(fact_4866_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A4: set(A)] :
( ? [A6: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),bot_bot(set(A)))))
<=> uniq(A,aTP_Lamp_a(set(A),fun(A,bool),A4)) ) ).
% subset_singleton_iff_Uniq
tff(fact_4867_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% uncurry_apply
tff(fact_4868_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : uncurry(A,B,C,F) = product_case_prod(A,B,C,F) ).
% uncurry_def
tff(fact_4869_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : uniq(list(A),aTP_Lamp_qc(set(A),fun(list(A),bool),A4)) ) ).
% strict_sorted_equal_Uniq
tff(fact_4870_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( S != bot_bot(set(A)) )
=> ? [X_1: A] : lattic501386751177426532rg_min(A,B,F,aTP_Lamp_a(set(A),fun(A,bool),S),X_1) ) ) ) ).
% ex_is_arg_min_if_finite
tff(fact_4871_interval__cases,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A)] :
( ! [A5: A,B4: A,X2: A] :
( pp(member2(A,A5,S))
=> ( pp(member2(A,B4,S))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B4))
=> pp(member2(A,X2,S)) ) ) ) )
=> ? [A5: A,B4: A] :
( ( S = bot_bot(set(A)) )
| ( S = top_top(set(A)) )
| ( S = set_ord_lessThan(A,B4) )
| ( S = set_ord_atMost(A,B4) )
| ( S = set_ord_greaterThan(A,A5) )
| ( S = set_ord_atLeast(A,A5) )
| ( S = set_or5935395276787703475ssThan(A,A5,B4) )
| ( S = set_or3652927894154168847AtMost(A,A5,B4) )
| ( S = set_or7035219750837199246ssThan(A,A5,B4) )
| ( S = set_or1337092689740270186AtMost(A,A5,B4) ) ) ) ) ).
% interval_cases
tff(fact_4872_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_qd(A,fun(list(A),fun(nat,list(A))),X),Xs),N) ).
% take_Cons
tff(fact_4873_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_4874_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atLeast(A,A3)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),D3) ) ).
% Int_atLeastAtMostR2
tff(fact_4875_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_ord_atLeast(A,C3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),B2) ) ).
% Int_atLeastAtMostL2
tff(fact_4876_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)))),set_ord_greaterThan(A,L)) = set_ord_atLeast(A,L) ) ).
% ivl_disj_un_singleton(1)
tff(fact_4877_greaterThan__non__empty,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] : set_ord_greaterThan(A,X) != bot_bot(set(A)) ) ).
% greaterThan_non_empty
tff(fact_4878_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_4879_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_4880_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_greaterThan(A,A3)),set_ord_greaterThan(A,B2)) = set_ord_greaterThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).
% lessThan_Int_lessThan
tff(fact_4881_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_4882_atLeastAtMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atLeast(A,L)),set_ord_atMost(A,U)) ) ).
% atLeastAtMost_def
tff(fact_4883_atLeastLessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atLeast(A,L)),set_ord_lessThan(A,U)) ) ).
% atLeastLessThan_def
tff(fact_4884_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_4885_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)),set_ord_greaterThan(A,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(7)
tff(fact_4886_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aTP_Lamp_qe(list(A),fun(nat,list(A)),Xs),N) ).
% drop_Cons
tff(fact_4887_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),I: nat,V: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),I),V) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_qf(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V),I) ).
% list_update.simps(2)
tff(fact_4888_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)),set_ord_greaterThan(A,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(5)
tff(fact_4889_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_greaterThan(A,L)),set_ord_atMost(A,U)) ) ).
% greaterThanAtMost_def
tff(fact_4890_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = case_nat(A,X,nth(A,Xs),N) ).
% nth_Cons
tff(fact_4891_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_greaterThan(A,L)),set_ord_lessThan(A,U)) ) ).
% greaterThanLessThan_def
tff(fact_4892_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] : set_or5935395276787703475ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_greaterThan(A,A3)),set_ord_lessThan(A,B2)) ) ).
% greaterThanLessThan_eq
tff(fact_4893_atMost__Int__atLeast,axiom,
! [A: $tType] :
( order(A)
=> ! [N: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,N)),set_ord_atLeast(A,N)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),N),bot_bot(set(A))) ) ).
% atMost_Int_atLeast
tff(fact_4894_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A4: set(B),G: fun(B,multiset(A)),B3: set(B),F: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),B3))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,B3))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),G,X2)),aa(B,multiset(A),F,X2))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),B3))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)))) ) ) ) ) ).
% subset_mset.cINF_superset_mono
tff(fact_4895_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( transp(A,P)
=> ( sorted_wrt(A,P,Xs)
<=> ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I2)))) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
tff(fact_4896_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list(int)] :
( ( upto(X,Xa) = Y )
=> ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
=> ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
=> ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
=> ( Y = nil(int) ) ) )
=> ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).
% upto.pelims
tff(fact_4897_subset__mset_Obdd__below__Int1,axiom,
! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A4)
=> condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).
% subset_mset.bdd_below_Int1
tff(fact_4898_subset__mset_Obdd__below__Int2,axiom,
! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B3)
=> condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).
% subset_mset.bdd_below_Int2
tff(fact_4899_transp__inf,axiom,
! [A: $tType,R3: fun(A,fun(A,bool)),S4: fun(A,fun(A,bool))] :
( transp(A,R3)
=> ( transp(A,S4)
=> transp(A,aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),inf_inf(fun(A,fun(A,bool))),R3),S4)) ) ) ).
% transp_inf
tff(fact_4900_list_Orel__transp,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( transp(A,R)
=> transp(list(A),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),R)) ) ).
% list.rel_transp
tff(fact_4901_sorted__wrt2,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A,Zs: list(A)] :
( transp(A,P)
=> ( sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
& sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).
% sorted_wrt2
tff(fact_4902_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
=> ( ! [I3: int,J3: int] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J3))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J3))
=> pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J3)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J3)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% upto.pinduct
tff(fact_4903_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B3: set(B),F: fun(C,multiset(A)),A4: set(C),G: fun(B,multiset(A))] :
( ( B3 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),F),A4))
=> ( ! [M4: B] :
( pp(member2(B,M4,B3))
=> ? [X3: C] :
( pp(member2(C,X3,A4))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(C,multiset(A),F,X3)),aa(B,multiset(A),G,M4))) ) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),F),A4))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),B3)))) ) ) ) ).
% subset_mset.cINF_mono
tff(fact_4904_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),U: multiset(A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),U),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))))
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),U),aa(B,multiset(A),F,X4))) ) ) ) ) ).
% subset_mset.le_cINF_iff
tff(fact_4905_upto_Opsimps,axiom,
! [I: int,J: int] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = nil(int) ) ) ) ) ).
% upto.psimps
tff(fact_4906_subset__mset_Omono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),A4: fun(C,multiset(A)),I5: set(C)] :
( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F))
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I5))
=> ( ( I5 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_qg(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),A4)),I5)))) ) ) ) ) ).
% subset_mset.mono_cINF
tff(fact_4907_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),B3: set(B)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),B3))
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = inter_mset(A,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),B3))) ) ) ) ) ) ).
% subset_mset.cINF_union
tff(fact_4908_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),A3: B] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = inter_mset(A,aa(B,multiset(A),F,A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))) ) ) ) ).
% subset_mset.cINF_insert
tff(fact_4909_set__mset__inter,axiom,
! [A: $tType,A4: multiset(A),B3: multiset(A)] : set_mset(A,inter_mset(A,A4,B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_mset(A,A4)),set_mset(A,B3)) ).
% set_mset_inter
tff(fact_4910_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [Less_eq: fun(A,fun(A,bool))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).
% order.mono.cong
tff(fact_4911_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(multiset(A),B),A4: multiset(A),B3: multiset(A)] :
( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F,inter_mset(A,A4,B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(multiset(A),B,F,A4)),aa(multiset(A),B,F,B3)))) ) ) ).
% subset_mset.mono_inf
tff(fact_4912_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),A4)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),B3)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3) != bot_bot(set(multiset(A))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),inter_mset(A,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A4),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B3))),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))),A4),B3)))) ) ) ) ).
% subset_mset.less_eq_cInf_inter
tff(fact_4913_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),G: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4))
=> ( inter_mset(A,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4))) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_qh(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F),G)),A4)) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
tff(fact_4914_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),A4: fun(C,multiset(A)),I5: set(C)] :
( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F))
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I5))
=> ( ( I5 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_qg(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),A4)),I5))),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A4),I5))))) ) ) ) ) ).
% subset_mset.mono_cSUP
tff(fact_4915_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A4: set(B),G: fun(B,multiset(A)),B3: set(B),F: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),B3))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,X2)),aa(B,multiset(A),G,X2))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),B3)))) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
tff(fact_4916_Eps__Opt__def,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
=> ( eps_Opt(A,P) = aa(A,option(A),some(A),fChoice(A,P)) ) )
& ( ~ ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( eps_Opt(A,P) = none(A) ) ) ) ).
% Eps_Opt_def
tff(fact_4917_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,aTP_Lamp_a(set(A),fun(A,bool),S))),S) = S ) ) ).
% some_insert_self
tff(fact_4918_some__elem,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> pp(member2(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),S)),S)) ) ).
% some_elem
tff(fact_4919_some__in__eq,axiom,
! [A: $tType,A4: set(A)] :
( pp(member2(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A4)),A4))
<=> ( A4 != bot_bot(set(A)) ) ) ).
% some_in_eq
tff(fact_4920_inv__on__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),X: B] : aa(B,A,inv_on(A,B,F,A4),X) = fChoice(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nk(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F),A4),X)) ).
% inv_on_def
tff(fact_4921_subset__mset_Obdd__above__Int1,axiom,
! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
=> condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).
% subset_mset.bdd_above_Int1
tff(fact_4922_subset__mset_Obdd__above__Int2,axiom,
! [A: $tType,B3: set(multiset(A)),A4: set(multiset(A))] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
=> condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3)) ) ).
% subset_mset.bdd_above_Int2
tff(fact_4923_some__theI,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ? [A11: A,X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A11),X_13))
=> ( ! [B13: B,B23: B] :
( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B13))
=> ( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B23))
=> ( B13 = B23 ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,fChoice(A,aTP_Lamp_qi(fun(A,fun(B,bool)),fun(A,bool),P))),the(B,aTP_Lamp_qj(fun(A,fun(B,bool)),fun(B,bool),P)))) ) ) ).
% some_theI
tff(fact_4924_subset__mset_OcSUP__le__iff,axiom,
! [B: $tType,A: $tType,A4: set(B),F: fun(B,multiset(A)),U: multiset(A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))),U))
<=> ! [X4: B] :
( pp(member2(B,X4,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,X4)),U)) ) ) ) ) ).
% subset_mset.cSUP_le_iff
tff(fact_4925_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set(B),G: fun(C,multiset(A)),B3: set(C),F: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),G),B3))
=> ( ! [N7: B] :
( pp(member2(B,N7,A4))
=> ? [X3: C] :
( pp(member2(C,X3,B3))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,N7)),aa(C,multiset(A),G,X3))) ) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),G),B3)))) ) ) ) ).
% subset_mset.cSUP_mono
tff(fact_4926_fun__of__rel__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R,X) = fChoice(A,aa(B,fun(A,bool),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,bool)),R),X)) ).
% fun_of_rel_def
tff(fact_4927_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),B3: set(B)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),B3))
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))) = union_mset(A,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),B3))) ) ) ) ) ) ).
% subset_mset.cSUP_union
tff(fact_4928_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),A3: B] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = union_mset(A,aa(B,multiset(A),F,A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))) ) ) ) ).
% subset_mset.cSUP_insert
tff(fact_4929_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A4: set(multiset(A)),B3: set(multiset(A))] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),A4)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),B3)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3) != bot_bot(set(multiset(A))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),A4),B3))),union_mset(A,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A4),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B3)))) ) ) ) ).
% subset_mset.cSup_inter_less_eq
tff(fact_4930_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),G: fun(B,multiset(A))] :
( ( A4 != bot_bot(set(B)) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4))
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4))
=> ( union_mset(A,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F),A4)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G),A4))) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_qk(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F),G)),A4)) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
tff(fact_4931_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% sgn_neg
tff(fact_4932_irrefl__tranclI,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)) = bot_bot(set(product_prod(A,A))) )
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R3))) ) ).
% irrefl_tranclI
tff(fact_4933_su__rel__fun__def,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B)] :
( su_rel_fun(A,B,F2,F)
<=> ( ! [A9: A,B9: B,B14: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B9),F2))
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B14),F2))
=> ( B9 = B14 ) ) )
& ! [A9: A,P3: bool] :
( ! [B9: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B9),F2))
=> pp(P3) )
=> pp(P3) )
& ! [A9: A] : aa(A,B,F,A9) = the(B,aa(A,fun(B,bool),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),F2),A9)) ) ) ).
% su_rel_fun_def
tff(fact_4934_sgn__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).
% sgn_1
tff(fact_4935_pair__set__inverse,axiom,
! [A: $tType,B: $tType,P: fun(B,fun(A,bool))] : converse(B,A,collect(product_prod(B,A),product_case_prod(B,A,bool,P))) = collect(product_prod(A,B),product_case_prod(A,B,bool,aTP_Lamp_ql(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P))) ).
% pair_set_inverse
tff(fact_4936_divide__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] : divide_divide(A,A3,sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),sgn_sgn(A,B2)) ) ).
% divide_sgn
tff(fact_4937_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( sgn_sgn(A,A3) = one_one(A) ) ) ) ).
% sgn_pos
tff(fact_4938_abs__sgn__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = one_one(A) ) ) ) ).
% abs_sgn_eq_1
tff(fact_4939_sgn__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),sgn_sgn(A,A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).
% sgn_mult_self_eq
tff(fact_4940_converse__Int,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S4: 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))),R3),S4)) = 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,R3)),converse(B,A,S4)) ).
% converse_Int
tff(fact_4941_listrel1__converse,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : listrel1(A,converse(A,A,R3)) = converse(list(A),list(A),listrel1(A,R3)) ).
% listrel1_converse
tff(fact_4942_sgn__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),sgn_sgn(A,B2)) ) ).
% sgn_mult
tff(fact_4943_in__listrel1__converse,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
( pp(member2(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),listrel1(A,converse(A,A,R3))))
<=> pp(member2(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),converse(list(A),list(A),listrel1(A,R3)))) ) ).
% in_listrel1_converse
tff(fact_4944_sgn__minus__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( sgn_sgn(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% sgn_minus_1
tff(fact_4945_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B] :
( su_rel_fun(A,B,F2,F)
=> ( ( aa(A,B,F,A4) = B3 )
<=> pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),F2)) ) ) ).
% su_rel_fun.repr
tff(fact_4946_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F2,F)
=> pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),aa(A,B,F,A4)),F2)) ) ).
% su_rel_fun.repr1
tff(fact_4947_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B] :
( su_rel_fun(A,B,F2,F)
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),F2))
=> ( B3 = aa(A,B,F,A4) ) ) ) ).
% su_rel_fun.repr2
tff(fact_4948_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A,B3: B,B15: B] :
( su_rel_fun(A,B,F2,F)
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),F2))
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B15),F2))
=> ( B3 = B15 ) ) ) ) ).
% su_rel_fun.unique
tff(fact_4949_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F2,F)
=> ~ ! [B6: B] : ~ pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B6),F2)) ) ).
% su_rel_fun.surjective
tff(fact_4950_mult__sgn__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,X)),aa(A,A,abs_abs(A),X)) = X ) ).
% mult_sgn_abs
tff(fact_4951_sgn__mult__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).
% sgn_mult_abs
tff(fact_4952_abs__mult__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),sgn_sgn(A,A3)) = A3 ) ).
% abs_mult_sgn
tff(fact_4953_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).
% linordered_idom_class.abs_sgn
tff(fact_4954_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( sgn_sgn(A,A3) = one_one(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% sgn_1_pos
tff(fact_4955_abs__sgn__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( ( A3 = zero_zero(A) )
=> ( aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = zero_zero(A) ) )
& ( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = one_one(A) ) ) ) ) ).
% abs_sgn_eq
tff(fact_4956_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( ( X = zero_zero(A) )
=> ( sgn_sgn(A,X) = zero_zero(A) ) )
& ( ( X != zero_zero(A) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( sgn_sgn(A,X) = one_one(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( sgn_sgn(A,X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).
% sgn_if
tff(fact_4957_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% sgn_1_neg
tff(fact_4958_su__rel__fun_Of__def,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F2,F)
=> ( aa(A,B,F,A4) = the(B,aa(A,fun(B,bool),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),F2),A4)) ) ) ).
% su_rel_fun.f_def
tff(fact_4959_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F2: set(product_prod(A,B)),F: fun(A,B)] :
( ! [A8: A,B6: B,B7: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B6),F2))
=> ( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B7),F2))
=> ( B6 = B7 ) ) )
=> ( ! [A8: A,P2: bool] :
( ! [B16: B] :
( pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B16),F2))
=> pp(P2) )
=> pp(P2) )
=> ( ! [A8: A] : aa(A,B,F,A8) = the(B,aa(A,fun(B,bool),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),F2),A8))
=> su_rel_fun(A,B,F2,F) ) ) ) ).
% su_rel_fun.intro
tff(fact_4960_Image__INT__eq,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),A4: set(C),B3: fun(C,set(B))] :
( single_valued(A,B,converse(B,A,R3))
=> ( ( A4 != bot_bot(set(C)) )
=> ( image(B,A,R3,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A4))) = 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_qm(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B3)),A4)) ) ) ) ).
% Image_INT_eq
tff(fact_4961_length__upto,axiom,
! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),minus_minus(int,J,I)),one_one(int))) ).
% length_upto
tff(fact_4962_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_4963_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( single_valued(A,B,R)
=> 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))),R),S)) ) ).
% single_valued_inter1
tff(fact_4964_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( single_valued(A,B,R)
=> 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),R)) ) ).
% single_valued_inter2
tff(fact_4965_bijective__alt,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( bijective(A,B,R)
<=> ( single_valued(A,B,R)
& single_valued(B,A,converse(A,B,R)) ) ) ).
% bijective_alt
tff(fact_4966_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B3: set(B)] :
( single_valued(A,B,converse(B,A,R))
=> ( image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,A4)),image(B,A,R,B3)) ) ) ).
% Image_Int_eq
tff(fact_4967_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4) ) )
& ( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ) ).
% Gcd_fin.eq_fold
tff(fact_4968_lexn_Osimps_I2_J,axiom,
! [A: $tType,R3: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),product_case_prod(A,list(A),list(A),cons(A)),product_case_prod(A,list(A),list(A),cons(A)))),lex_prod(A,list(A),R3,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)))),collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_qn(nat,fun(list(A),fun(list(A),bool)),N)))) ).
% lexn.simps(2)
tff(fact_4969_Linear__order__Well__order__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R3),R3)
=> ( order_well_order_on(A,field2(A,R3),R3)
<=> ! [A9: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R3)))
=> ( ( A9 != bot_bot(set(A)) )
=> ? [X4: A] :
( pp(member2(A,X4,A9))
& ! [Xa2: A] :
( pp(member2(A,Xa2,A9))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R3)) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
tff(fact_4970_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),one_one(A)) = one_one(A) ) ).
% gcd.bottom_right_bottom
tff(fact_4971_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A3) = one_one(A) ) ).
% gcd.bottom_left_bottom
tff(fact_4972_is__unit__gcd__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A)))
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% is_unit_gcd_iff
tff(fact_4973_Gcd__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).
% Gcd_2
tff(fact_4974_gcd__add__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [M2: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M2)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ).
% gcd_add_mult
tff(fact_4975_gcd__dvd__prod,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,K: A] : pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))) ) ).
% gcd_dvd_prod
tff(fact_4976_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_4977_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,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),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).
% gcd_mult_unit1
tff(fact_4978_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).
% gcd_mult_unit2
tff(fact_4979_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),divide_divide(A,B2,A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).
% gcd_div_unit1
tff(fact_4980_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),divide_divide(A,C3,A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).
% gcd_div_unit2
tff(fact_4981_Well__order__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> order_well_order_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% Well_order_Restr
tff(fact_4982_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,A4: set(A)] :
( pp(member2(A,A3,A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).
% Gcd_fin.remove
tff(fact_4983_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,A4: 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),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).
% Gcd_fin.insert_remove
tff(fact_4984_well__order__on__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R3)))
=> order_well_order_on(A,A4,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,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ) ).
% well_order_on_Restr
tff(fact_4985_ofilter__Restr__Int,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ) ).
% ofilter_Restr_Int
tff(fact_4986_ofilter__Restr__subset,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> order_ofilter(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),A4) ) ) ) ).
% ofilter_Restr_subset
tff(fact_4987_Field__Restr__ofilter,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).
% Field_Restr_ofilter
tff(fact_4988_ofilter__subset__embedS,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( order_ofilter(A,R3,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ).
% ofilter_subset_embedS
tff(fact_4989_ofilter__subset__embed,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( order_ofilter(A,R3,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ).
% ofilter_subset_embed
tff(fact_4990_ofilter__embed,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R3)))
& bNF_Wellorder_embed(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),R3,id(A)) ) ) ) ).
% ofilter_embed
tff(fact_4991_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( order_ofilter(A,R3,B3)
=> ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),id(A)) )
& ( ( A4 = B3 )
<=> bNF_Wellorder_iso(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
tff(fact_4992_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F: fun(B,A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_well_order_on(B,field2(B,R6),R6)
=> ( bNF_Wellorder_embed(B,A,R6,R3,F)
=> bNF_Wellorder_iso(B,A,R6,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),field2(B,R6)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_qo(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R6),F))),F) ) ) ) ).
% embed_implies_iso_Restr
tff(fact_4993_ofilter__subset__ordLess,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( order_ofilter(A,R3,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B3))
<=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).
% ofilter_subset_ordLess
tff(fact_4994_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(bool,product_prod(A,set(B)))] :
~ ! [B4: bool,H3: A,As: set(B)] : X != aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),B4),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As)) ).
% pure_assn_raw.cases
tff(fact_4995_ofilter__ordLess,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),field2(A,R3)))
<=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),R3),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% ofilter_ordLess
tff(fact_4996_underS__Restr__ordLess,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( ( field2(A,R3) != bot_bot(set(A)) )
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,A3),aa(A,fun(A,set(A)),aTP_Lamp_qp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A3)))),R3),bNF_We4044943003108391690rdLess(A,A))) ) ) ).
% underS_Restr_ordLess
tff(fact_4997_ofilter__subset__ordLeq,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( order_ofilter(A,R3,B3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
<=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))),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,B3,aTP_Lamp_mj(set(A),fun(A,set(A)),B3)))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).
% ofilter_subset_ordLeq
tff(fact_4998_ofilter__Restr__under,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),A3: A] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_ofilter(A,R3,A4)
=> ( pp(member2(A,A3,A4))
=> ( order_under(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4))),A3) = order_under(A,R3,A3) ) ) ) ) ).
% ofilter_Restr_under
tff(fact_4999_underS__empty,axiom,
! [A: $tType,A3: A,R3: set(product_prod(A,A))] :
( ~ pp(member2(A,A3,field2(A,R3)))
=> ( order_underS(A,R3,A3) = bot_bot(set(A)) ) ) ).
% underS_empty
tff(fact_5000_Refl__under__underS,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
( refl_on(A,field2(A,R3),R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( order_under(A,R3,A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R3,A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ).
% Refl_under_underS
tff(fact_5001_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_well_order_on(B,field2(B,R6),R6)
=> ( pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6),bNF_Wellorder_ordLeq(A,B)))
<=> ! [X4: A] :
( pp(member2(A,X4,field2(A,R3)))
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_qp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),R6),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
tff(fact_5002_underS__Field3,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
( ( field2(A,R3) != bot_bot(set(A)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R3,A3)),field2(A,R3))) ) ).
% underS_Field3
tff(fact_5003_ordLess__iff__ordIso__Restr,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( order_well_order_on(B,field2(B,R6),R6)
=> ( pp(member2(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3),bNF_We4044943003108391690rdLess(B,A)))
<=> ? [X4: A] :
( pp(member2(A,X4,field2(A,R3)))
& pp(member2(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_qp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
tff(fact_5004_min__ext__compat,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),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,R))) ) ).
% min_ext_compat
tff(fact_5005_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_5006_max__ext__compat,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),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,R))) ) ).
% max_ext_compat
tff(fact_5007_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( pp(member2(A,A3,A4))
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A4) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).
% bounded_quasi_semilattice_set.remove
tff(fact_5008_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
tff(fact_5009_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),bot_bot(set(A))) = Top ) ) ).
% bounded_quasi_semilattice_set.empty
tff(fact_5010_max__ext_Ocases,axiom,
! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
( pp(member2(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,R)))
=> ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
=> ( pp(aa(set(A),bool,finite_finite2(A),A22))
=> ( ( A22 != bot_bot(set(A)) )
=> ~ ! [X3: A] :
( pp(member2(A,X3,A1))
=> ? [Xa4: A] :
( pp(member2(A,Xa4,A22))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa4),R)) ) ) ) ) ) ) ).
% max_ext.cases
tff(fact_5011_max__ext_Osimps,axiom,
! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
( pp(member2(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,R)))
<=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
& pp(aa(set(A),bool,finite_finite2(A),A22))
& ( A22 != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member2(A,X4,A1))
=> ? [Xa2: A] :
( pp(member2(A,Xa2,A22))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R)) ) ) ) ) ).
% max_ext.simps
tff(fact_5012_max__ext_Omax__extI,axiom,
! [A: $tType,X5: set(A),Y5: set(A),R: set(product_prod(A,A))] :
( pp(aa(set(A),bool,finite_finite2(A),X5))
=> ( pp(aa(set(A),bool,finite_finite2(A),Y5))
=> ( ( Y5 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,X5))
=> ? [Xa3: A] :
( pp(member2(A,Xa3,Y5))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),R)) ) )
=> pp(member2(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),Y5),max_ext(A,R))) ) ) ) ) ).
% max_ext.max_extI
tff(fact_5013_min__ext__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : min_ext(A,R3) = collect(product_prod(set(A),set(A)),aTP_Lamp_qq(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R3)) ).
% min_ext_def
tff(fact_5014_refl__on__reflcl__Image,axiom,
! [A: $tType,B3: set(A),A4: set(product_prod(A,A)),C4: set(A)] :
( refl_on(A,B3,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B3))
=> ( 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))),A4),id2(A)),C4) = image(A,A,A4,C4) ) ) ) ).
% refl_on_reflcl_Image
tff(fact_5015_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,A)] : bind2(B,A,A4,aTP_Lamp_ds(fun(B,A),fun(B,set(A)),F)) = aa(set(B),set(A),image2(B,A,F),A4) ).
% bind_singleton_conv_image
tff(fact_5016_bex__empty,axiom,
! [A: $tType,P: fun(A,bool)] :
~ ? [X3: A] :
( pp(member2(A,X3,bot_bot(set(A))))
& pp(aa(A,bool,P,X3)) ) ).
% bex_empty
tff(fact_5017_empty__bind,axiom,
! [B: $tType,A: $tType,F: fun(B,set(A))] : bind2(B,A,bot_bot(set(B)),F) = bot_bot(set(A)) ).
% empty_bind
tff(fact_5018_bijective__Id,axiom,
! [A: $tType] : bijective(A,A,id2(A)) ).
% bijective_Id
tff(fact_5019_bex__UNIV,axiom,
! [A: $tType,P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,top_top(set(A))))
& pp(aa(A,bool,P,X4)) )
<=> ? [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).
% bex_UNIV
tff(fact_5020_below__Id__inv,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),converse(A,A,R)),id2(A)))
<=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A))) ) ).
% below_Id_inv
tff(fact_5021_Set_Obind__bind,axiom,
! [A: $tType,B: $tType,C: $tType,A4: set(A),B3: fun(A,set(C)),C4: fun(C,set(B))] : bind2(C,B,bind2(A,C,A4,B3),C4) = bind2(A,B,A4,aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_qr(fun(A,set(C)),fun(fun(C,set(B)),fun(A,set(B))),B3),C4)) ).
% Set.bind_bind
tff(fact_5022_image__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] : aa(set(A),set(B),image2(A,B,F),A4) = collect(B,aa(set(A),fun(B,bool),aTP_Lamp_qs(fun(A,B),fun(set(A),fun(B,bool)),F),A4)) ).
% image_def
tff(fact_5023_Set_Obind__def,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,set(B))] : bind2(A,B,A4,F) = collect(B,aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_qt(set(A),fun(fun(A,set(B)),fun(B,bool)),A4),F)) ).
% Set.bind_def
tff(fact_5024_bind__const,axiom,
! [B: $tType,A: $tType,A4: set(B),B3: set(A)] :
( ( ( A4 = bot_bot(set(B)) )
=> ( bind2(B,A,A4,aTP_Lamp_an(set(A),fun(B,set(A)),B3)) = bot_bot(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( bind2(B,A,A4,aTP_Lamp_an(set(A),fun(B,set(A)),B3)) = B3 ) ) ) ).
% bind_const
tff(fact_5025_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> ( bind2(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)) = B3 ) ) ).
% nonempty_bind_const
tff(fact_5026_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(A),set(B),image2(A,B,F),A4)) = collect(A,aa(set(A),fun(A,bool),aTP_Lamp_qu(fun(A,B),fun(set(A),fun(A,bool)),F),A4)) ).
% vimage_image_eq
tff(fact_5027_single__valued__below__Id,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
=> single_valued(A,A,R) ) ).
% single_valued_below_Id
tff(fact_5028_nths__nths,axiom,
! [A: $tType,Xs: list(A),A4: set(nat),B3: set(nat)] : aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),A4)),B3) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_qw(set(nat),fun(set(nat),fun(nat,bool)),A4),B3))) ).
% nths_nths
tff(fact_5029_rtrancl__Int__subset,axiom,
! [A: $tType,S4: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id2(A)),S4))
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)),S4),R3)),S4))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R3)),S4)) ) ) ).
% rtrancl_Int_subset
tff(fact_5030_map__project__def,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),A4: set(A)] : map_project(A,B,F,A4) = collect(B,aa(set(A),fun(B,bool),aTP_Lamp_qx(fun(A,option(B)),fun(set(A),fun(B,bool)),F),A4)) ).
% map_project_def
tff(fact_5031_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R3),R3)
=> ( wf(A,minus_minus(set(product_prod(A,A)),R3,id2(A)))
<=> ! [A9: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R3)))
=> ( ( A9 != bot_bot(set(A)) )
=> ? [X4: A] :
( pp(member2(A,X4,A9))
& ! [Xa2: A] :
( pp(member2(A,Xa2,A9))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R3)) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
tff(fact_5032_bsqr__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R3) = collect(product_prod(product_prod(A,A),product_prod(A,A)),product_case_prod(product_prod(A,A),product_prod(A,A),bool,product_case_prod(A,A,fun(product_prod(A,A),bool),aTP_Lamp_qz(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R3)))) ).
% bsqr_def
tff(fact_5033_wf__listrel1__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(list(A),listrel1(A,R3))
<=> wf(A,R3) ) ).
% wf_listrel1_iff
tff(fact_5034_wf__lex,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,R3)
=> wf(list(A),lex(A,R3)) ) ).
% wf_lex
tff(fact_5035_wf__lenlex,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,R3)
=> wf(list(A),lenlex(A,R3)) ) ).
% wf_lenlex
tff(fact_5036_brk__rel__wf,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
=> wf(product_prod(bool,A),brk_rel(A,A,R)) ) ).
% brk_rel_wf
tff(fact_5037_wf__lexn,axiom,
! [A: $tType,R3: set(product_prod(A,A)),N: nat] :
( wf(A,R3)
=> wf(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)) ) ).
% wf_lexn
tff(fact_5038_wf__measures,axiom,
! [A: $tType,Fs: list(fun(A,nat))] : wf(A,measures(A,Fs)) ).
% wf_measures
tff(fact_5039_wf__Int1,axiom,
! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
( wf(A,R3)
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),R6)) ) ).
% wf_Int1
tff(fact_5040_wf__Int2,axiom,
! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
( wf(A,R3)
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R6),R3)) ) ).
% wf_Int2
tff(fact_5041_wfE__min_H,axiom,
! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
( wf(A,R)
=> ( ( Q != bot_bot(set(A)) )
=> ~ ! [Z4: A] :
( pp(member2(A,Z4,Q))
=> ~ ! [Y4: A] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4),R))
=> ~ pp(member2(A,Y4,Q)) ) ) ) ) ).
% wfE_min'
tff(fact_5042_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
=> ( wf(A,converse(A,A,R))
<=> wf(A,R) ) ) ).
% finite_wf_eq_wf_converse
tff(fact_5043_wfI__pf,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [A8: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),image(A,A,R,A8)))
=> ( A8 = bot_bot(set(A)) ) )
=> wf(A,R) ) ).
% wfI_pf
tff(fact_5044_wfE__pf,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
( wf(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),image(A,A,R,A4)))
=> ( A4 = bot_bot(set(A)) ) ) ) ).
% wfE_pf
tff(fact_5045_wf__bounded__supset,axiom,
! [A: $tType,S: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> wf(set(A),collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_ra(set(A),fun(set(A),fun(set(A),bool)),S)))) ) ).
% wf_bounded_supset
tff(fact_5046_wf__eq__minimal2,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( wf(A,R3)
<=> ! [A9: set(A)] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R3)))
& ( A9 != bot_bot(set(A)) ) )
=> ? [X4: A] :
( pp(member2(A,X4,A9))
& ! [Xa2: A] :
( pp(member2(A,Xa2,A9))
=> ~ pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R3)) ) ) ) ) ).
% wf_eq_minimal2
tff(fact_5047_trans__wf__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( trans(A,R3)
=> ( wf(A,R3)
<=> ! [A6: A] : wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,image(A,A,converse(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_rb(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A6)))) ) ) ).
% trans_wf_iff
tff(fact_5048_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R3)))
=> ( ( ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),minus_minus(set(product_prod(A,A)),R3,id2(A))))
| pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),minus_minus(set(product_prod(A,A)),R3,id2(A)))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( ( A3 = B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total3
tff(fact_5049_max__ext__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_rc(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R))) ).
% max_ext_eq
tff(fact_5050_lenlex__transI,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( trans(A,R3)
=> trans(list(A),lenlex(A,R3)) ) ).
% lenlex_transI
tff(fact_5051_ball__UNIV,axiom,
! [A: $tType,P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,top_top(set(A))))
=> pp(aa(A,bool,P,X4)) )
<=> ! [X_12: A] : pp(aa(A,bool,P,X_12)) ) ).
% ball_UNIV
tff(fact_5052_lexn__transI,axiom,
! [A: $tType,R3: set(product_prod(A,A)),N: nat] :
( trans(A,R3)
=> trans(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)) ) ).
% lexn_transI
tff(fact_5053_lexord__transI,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( trans(A,R3)
=> trans(list(A),lexord(A,R3)) ) ).
% lexord_transI
tff(fact_5054_lex__transI,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( trans(A,R3)
=> trans(list(A),lex(A,R3)) ) ).
% lex_transI
tff(fact_5055_listrel__trans,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( trans(A,R3)
=> trans(list(A),listrel(A,A,R3)) ) ).
% listrel_trans
tff(fact_5056_trans__Int,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( trans(A,R3)
=> ( trans(A,S4)
=> trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),S4)) ) ) ).
% trans_Int
tff(fact_5057_lexord__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A)),Z2: list(A)] :
( pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R3)))
=> ( pp(member2(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),Z2),lexord(A,R3)))
=> ( trans(A,R3)
=> pp(member2(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),Z2),lexord(A,R3))) ) ) ) ).
% lexord_trans
tff(fact_5058_lenlex__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A)),Z2: list(A)] :
( pp(member2(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),lenlex(A,R3)))
=> ( pp(member2(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),Z2),lenlex(A,R3)))
=> ( trans(A,R3)
=> pp(member2(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),Z2),lenlex(A,R3))) ) ) ) ).
% lenlex_trans
tff(fact_5059_trans__Restr,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A)] :
( trans(A,R3)
=> trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A4,aTP_Lamp_mj(set(A),fun(A,set(A)),A4)))) ) ).
% trans_Restr
tff(fact_5060_wo__rel_Omax2__among,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( pp(member2(A,B2,field2(A,R3)))
=> pp(member2(A,bNF_We1388413361240627857o_max2(A,R3,A3,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ) ).
% wo_rel.max2_among
tff(fact_5061_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A4: set(product_prod(A,A))] :
( trans(A,A4)
=> ( aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),A4) = 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))),A4),id2(A)) ) ) ).
% trans_rtrancl_eq_reflcl
tff(fact_5062_Least__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_rd(fun(A,bool),fun(A,bool),P)) ) ).
% Least_def
tff(fact_5063_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R3)))
=> ( ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total
tff(fact_5064_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(member2(A,A3,field2(A,R3)))
=> ( pp(member2(A,B2,field2(A,R3)))
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R3,A3,B2)),R3))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R3,A3,B2)),R3))
& pp(member2(A,bNF_We1388413361240627857o_max2(A,R3,A3,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ) ) ).
% wo_rel.max2_greater_among
tff(fact_5065_Image__absorb__rtrancl,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B3: set(A),C4: set(A)] :
( trans(A,A4)
=> ( refl_on(A,B3,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B3))
=> ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),A4),C4) = image(A,A,A4,C4) ) ) ) ) ).
% Image_absorb_rtrancl
tff(fact_5066_Greatest__def,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_re(fun(A,bool),fun(A,bool),P)) ) ).
% Greatest_def
tff(fact_5067_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( B3 != bot_bot(set(A)) )
=> ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R3,B3,X_1) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
tff(fact_5068_ord_OLeast__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_rf(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).
% ord.Least_def
tff(fact_5069_ord_OLeast_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).
% ord.Least.cong
tff(fact_5070_Greatest__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A] :
( pp(aa(A,bool,P,X))
=> ( ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
=> ( order_Greatest(A,P) = X ) ) ) ) ).
% Greatest_equality
tff(fact_5071_GreatestI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y2: A] :
( pp(aa(A,bool,P,Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X2)) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).
% GreatestI2_order
tff(fact_5072_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( B3 != bot_bot(set(A)) )
=> bNF_We4791949203932849705sMinim(A,R3,B3,bNF_We6954850376910717587_minim(A,R3,B3)) ) ) ) ).
% wo_rel.minim_isMinim
tff(fact_5073_wo__rel_Ominim__inField,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( B3 != bot_bot(set(A)) )
=> pp(member2(A,bNF_We6954850376910717587_minim(A,R3,B3),field2(A,R3))) ) ) ) ).
% wo_rel.minim_inField
tff(fact_5074_wo__rel_Ominim__in,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( B3 != bot_bot(set(A)) )
=> pp(member2(A,bNF_We6954850376910717587_minim(A,R3,B3),B3)) ) ) ) ).
% wo_rel.minim_in
tff(fact_5075_list__ex__length,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Xs))
<=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N3))) ) ) ).
% list_ex_length
tff(fact_5076_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_5077_successively__append__iff,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Ys))
& ( ( Xs = nil(A) )
| ( Ys = nil(A) )
| pp(aa(A,bool,aa(A,fun(A,bool),P,last(A,Xs)),aa(list(A),A,hd(A),Ys))) ) ) ) ).
% successively_append_iff
tff(fact_5078_list__ex__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( pp(aa(A,bool,P,X))
| pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Xs)) ) ) ).
% list_ex_simps(1)
tff(fact_5079_list__ex__simps_I2_J,axiom,
! [A: $tType,P: fun(A,bool)] : ~ pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),nil(A))) ).
% list_ex_simps(2)
tff(fact_5080_list__ex__append,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Xs))
| pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Ys)) ) ) ).
% list_ex_append
tff(fact_5081_list__ex__rev,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),aa(list(A),list(A),rev(A),Xs)))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Xs)) ) ).
% list_ex_rev
tff(fact_5082_successively__rev,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),rev(A),Xs)))
<=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P)),Xs)) ) ).
% successively_rev
tff(fact_5083_successively__map,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(A,bool)),F: fun(B,A),Xs: list(B)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)))
<=> pp(aa(list(B),bool,aa(fun(B,fun(B,bool)),fun(list(B),bool),successively(B),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_bb(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),P),F)),Xs)) ) ).
% successively_map
tff(fact_5084_successively__remdups__adjI,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),remdups_adj(A),Xs))) ) ).
% successively_remdups_adjI
tff(fact_5085_successively__cong,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool)),Ys: list(A)] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),Q,X2),Y2)) ) ) )
=> ( ( Xs = Ys )
=> ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
<=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),Q),Ys)) ) ) ) ).
% successively_cong
tff(fact_5086_successively__mono,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),Q: fun(A,fun(A,bool))] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
=> ( ! [X2: A,Y2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Q,X2),Y2)) ) ) )
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),Q),Xs)) ) ) ).
% successively_mono
tff(fact_5087_successively__if__sorted__wrt,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( sorted_wrt(A,P,Xs)
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs)) ) ).
% successively_if_sorted_wrt
tff(fact_5088_successively_Oelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ).
% successively.elims(3)
tff(fact_5089_successively_Osimps_I3_J,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) ) ) ).
% successively.simps(3)
tff(fact_5090_successively_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),nil(A))) ).
% successively.simps(1)
tff(fact_5091_list__ex__cong,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),F: fun(A,bool),G: fun(A,bool)] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Ys)))
=> ( pp(aa(A,bool,F,X2))
<=> pp(aa(A,bool,G,X2)) ) )
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),F),Xs))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),G),Ys)) ) ) ) ).
% list_ex_cong
tff(fact_5092_distinct__adj__def,axiom,
! [A: $tType] : distinct_adj(A) = aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),aTP_Lamp_fh(A,fun(A,bool))) ).
% distinct_adj_def
tff(fact_5093_successively_Oelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
=> ( ( Xa != nil(A) )
=> ( ! [X2: A] : Xa != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ).
% successively.elims(2)
tff(fact_5094_successively_Oelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
<=> pp(Y) )
=> ( ( ( Xa = nil(A) )
=> ~ pp(Y) )
=> ( ( ? [X2: A] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))
=> ~ pp(Y) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( pp(Y)
<=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ).
% successively.elims(1)
tff(fact_5095_successively_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A] : pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ).
% successively.simps(2)
tff(fact_5096_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_order.axioms(1)
tff(fact_5097_successively__iff__sorted__wrt__strong,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
( ! [X2: A,Y2: A,Z4: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member2(A,Z4,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y2),Z4))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Z4)) ) ) ) ) )
=> ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
<=> sorted_wrt(A,P,Xs) ) ) ).
% successively_iff_sorted_wrt_strong
tff(fact_5098_successively__remdups__adj__iff,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X2),X2)) )
=> ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),remdups_adj(A),Xs)))
<=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs)) ) ) ).
% successively_remdups_adj_iff
tff(fact_5099_successively__conv__sorted__wrt,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( transp(A,P)
=> ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs))
<=> sorted_wrt(A,P,Xs) ) ) ).
% successively_conv_sorted_wrt
tff(fact_5100_successively__altdef,axiom,
! [A: $tType,X3: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X3) = aa(fun(A,fun(list(A),fun(bool,bool))),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),fun(bool,bool))),fun(list(A),bool)),rec_list(bool,A),fTrue),aTP_Lamp_rh(fun(A,fun(A,bool)),fun(A,fun(list(A),fun(bool,bool))),X3)) ).
% successively_altdef
tff(fact_5101_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_5102_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).
% inf_top.semilattice_neutr_axioms
tff(fact_5103_Bex__set__list__ex,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ? [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) )
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),P),Xs)) ) ).
% Bex_set_list_ex
tff(fact_5104_successively__Cons,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( ( Xs = nil(A) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),aa(list(A),A,hd(A),Xs)))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),P),Xs)) ) ) ) ).
% successively_Cons
tff(fact_5105_successively_Opelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
<=> pp(Y) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ( pp(Y)
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A))) ) )
=> ( ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ( pp(Y)
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( ( pp(Y)
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)))) ) ) ) ) ) ) ).
% successively.pelims(1)
tff(fact_5106_successively_Opelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),nil(A))) )
=> ( ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
=> ~ accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) )
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ) ).
% successively.pelims(2)
tff(fact_5107_successively_Opelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),Xa))
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa))
=> ~ ! [X2: A,Y2: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
=> ( accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Y2))
& pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ).
% successively.pelims(3)
tff(fact_5108_bij__betw__nth,axiom,
! [A: $tType,Xs: list(A),A4: set(nat),B3: set(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ( A4 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
=> ( ( B3 = aa(list(A),set(A),set2(A),Xs) )
=> bij_betw(nat,A,nth(A,Xs),A4,B3) ) ) ) ).
% bij_betw_nth
tff(fact_5109_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(B),B3: multiset(B),G: fun(B,A)] :
( ( inter_mset(B,A4,B3) = zero_zero(multiset(B)) )
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,union_mset(B,A4,B3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,A4))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,B3))) ) ) ) ).
% prod_mset.union_disjoint
tff(fact_5110_power__int__add__1_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: int] :
( ( ( X != zero_zero(A) )
| ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M2)) ) ) ) ).
% power_int_add_1'
tff(fact_5111_power__int__1__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int] : power_int(A,one_one(A),N) = one_one(A) ) ).
% power_int_1_left
tff(fact_5112_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,W: num,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W)),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,aa(num,A,numeral_numeral(A),W),M2)) ) ).
% power_int_mult_distrib_numeral2
tff(fact_5113_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( field(A)
=> ! [W: num,Y: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),M2)),power_int(A,Y,M2)) ) ).
% power_int_mult_distrib_numeral1
tff(fact_5114_power__int__0__right,axiom,
! [B: $tType] :
( ( inverse(B)
& power(B) )
=> ! [X: B] : power_int(B,X,zero_zero(int)) = one_one(B) ) ).
% power_int_0_right
tff(fact_5115_mset__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : mset(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = image_mset(B,A,F,mset(B,Xs)) ).
% mset_map
tff(fact_5116_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aTP_Lamp_ae(B,A),A4)) = one_one(A) ) ).
% prod_mset.neutral_const
tff(fact_5117_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M2: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)) = one_one(A) ) ).
% power_int_minus_one_mult_self
tff(fact_5118_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M2: int,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),B2)) = B2 ) ).
% power_int_minus_one_mult_self'
tff(fact_5119_power__int__add__numeral2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B2) ) ).
% power_int_add_numeral2
tff(fact_5120_power__int__add__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).
% power_int_add_numeral
tff(fact_5121_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),X: B,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,add_mset(B,X,A4))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,A4))) ) ).
% prod_mset.insert
tff(fact_5122_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F: fun(B,A),P: multiset(B),M12: multiset(A),M23: multiset(A)] :
( ( image_mset(B,A,F,P) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23) )
=> ~ ! [P12: multiset(B),P23: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P23) )
=> ( ( image_mset(B,A,F,P12) = M12 )
=> ( image_mset(B,A,F,P23) != M23 ) ) ) ) ).
% mset_map_split_orig
tff(fact_5123_power__int__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,Y,M2)) ) ).
% power_int_mult_distrib
tff(fact_5124_power__int__commutes,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N)) ) ).
% power_int_commutes
tff(fact_5125_mset__map__id,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B),X5: multiset(A)] :
( ! [X2: A] : aa(B,A,F,aa(A,B,G,X2)) = X2
=> ( image_mset(B,A,F,image_mset(A,B,G,X5)) = X5 ) ) ).
% mset_map_id
tff(fact_5126_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
( bij_betw(A,B,F,bot_bot(set(A)),A4)
=> ( A4 = bot_bot(set(B)) ) ) ).
% bij_betw_empty1
tff(fact_5127_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( bij_betw(A,B,F,A4,bot_bot(set(B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% bij_betw_empty2
tff(fact_5128_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F: fun(B,A),P: multiset(B),M12: multiset(A),M23: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),image_mset(B,A,F,P)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23)))
=> ~ ! [P12: multiset(B),P23: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P23) )
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),image_mset(B,A,F,P12)),M12))
=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),image_mset(B,A,F,P23)),M23)) ) ) ) ).
% mset_map_split_orig_le
tff(fact_5129_power__int__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : power_int(A,divide_divide(A,one_one(A),X),N) = divide_divide(A,one_one(A),power_int(A,X,N)) ) ).
% power_int_one_over
tff(fact_5130_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H: fun(B,A),A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,A4))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,H,A4))) ) ).
% prod_mset.distrib
tff(fact_5131_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A12: set(B)] :
( ~ pp(member2(A,B2,A4))
=> ( ~ pp(member2(B,aa(A,B,F,B2),A12))
=> ( bij_betw(A,B,F,A4,A12)
<=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A12),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw3
tff(fact_5132_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A12: set(B)] :
( ~ pp(member2(A,B2,A4))
=> ( ~ pp(member2(B,aa(A,B,F,B2),A12))
=> ( bij_betw(A,B,F,A4,A12)
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A12),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw
tff(fact_5133_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B3: set(B),C4: set(A),D2: set(B)] :
( bij_betw(A,B,F,A4,B3)
=> ( bij_betw(A,B,F,C4,D2)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D2) = bot_bot(set(B)) )
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D2)) ) ) ) ).
% bij_betw_combine
tff(fact_5134_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C4: set(A),B3: set(B),D2: set(B)] :
( bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D2))
=> ( bij_betw(A,B,F,C4,D2)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C4) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D2) = bot_bot(set(B)) )
=> bij_betw(A,B,F,A4,B3) ) ) ) ) ).
% bij_betw_partition
tff(fact_5135_power__int__0__left__If,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M2: int] :
( ( ( M2 = zero_zero(int) )
=> ( power_int(A,zero_zero(A),M2) = one_one(A) ) )
& ( ( M2 != zero_zero(int) )
=> ( power_int(A,zero_zero(A),M2) = zero_zero(A) ) ) ) ) ).
% power_int_0_left_If
tff(fact_5136_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N4: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N4))) ) ) ) ).
% power_int_increasing
tff(fact_5137_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N4: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N4))) ) ) ) ).
% power_int_strict_increasing
tff(fact_5138_power__int__minus__one__minus,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N) ) ).
% power_int_minus_one_minus
tff(fact_5139_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),minus_minus(int,A3,B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),minus_minus(int,B2,A3)) ) ).
% power_int_minus_one_diff_commute
tff(fact_5140_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C4: set(B),G: fun(A,B),B3: set(A),D2: set(B)] :
( bij_betw(A,B,F,A4,C4)
=> ( bij_betw(A,B,G,B3,D2)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C4),D2) = 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_dt(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C4),D2)) ) ) ) ) ).
% bij_betw_disjoint_Un
tff(fact_5141_infinite__imp__bij__betw2,axiom,
! [A: $tType,A4: set(A),A3: A] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ? [H3: fun(A,A)] : bij_betw(A,A,H3,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw2
tff(fact_5142_infinite__imp__bij__betw,axiom,
! [A: $tType,A4: set(A),A3: A] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ? [H3: fun(A,A)] : bij_betw(A,A,H3,A4,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw
tff(fact_5143_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N4: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N4)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_5144_one__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).
% one_le_power_int
tff(fact_5145_one__less__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N))) ) ) ) ).
% one_less_power_int
tff(fact_5146_power__int__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: int,N: int] :
( ( ( X != zero_zero(A) )
| ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N) != zero_zero(int) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,X,N)) ) ) ) ).
% power_int_add
tff(fact_5147_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),T5: set(C),H: fun(B,C),S: set(B),T: set(C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( pp(aa(set(C),bool,finite_finite2(C),T5))
=> ( bij_betw(B,C,H,minus_minus(set(B),S,S3),minus_minus(set(C),T,T5))
=> ( ! [A5: B] :
( pp(member2(B,A5,S3))
=> ( aa(C,A,G,aa(B,C,H,A5)) = one_one(A) ) )
=> ( ! [B4: C] :
( pp(member2(C,B4,T5))
=> ( aa(C,A,G,B4) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ri(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
tff(fact_5148_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( division_ring(A)
& one(B)
& uminus(B) )
=> ! [X: C,A3: A,N: int] :
( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N)),power_int(A,A3,N)) ) ) ) ).
% power_int_minus_left_distrib
tff(fact_5149_power__int__le__one,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).
% power_int_le_one
tff(fact_5150_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N4: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> ( ( ( A3 != zero_zero(A) )
| ( N4 != zero_zero(int) )
| ( N = zero_zero(int) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N4)),power_int(A,A3,N))) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_5151_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M2: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M2)),power_int(A,X,N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N)) ) ) ) ) ).
% power_int_le_imp_le_exp
tff(fact_5152_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M2: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M2)),power_int(A,X,N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M2),N)) ) ) ) ) ).
% power_int_le_imp_less_exp
tff(fact_5153_power__int__minus__mult,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N: int] :
( ( ( X != zero_zero(A) )
| ( N != zero_zero(int) ) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,minus_minus(int,N,one_one(int)))),X) = power_int(A,X,N) ) ) ) ).
% power_int_minus_mult
tff(fact_5154_power__int__add__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: int] :
( ( ( X != zero_zero(A) )
| ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),X) ) ) ) ).
% power_int_add_1
tff(fact_5155_count__image__mset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: multiset(B),X: A] : aa(A,nat,count(A,image_mset(B,A,F,A4)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),count(B,A4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),set_mset(B,A4))) ).
% count_image_mset
tff(fact_5156_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( semiring_1(B)
=> ! [Y: B,A4: multiset(A)] : comm_m7189776963980413722m_mset(B,image_mset(A,B,aTP_Lamp_rj(B,fun(A,B),Y),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(multiset(A),nat,size_size(multiset(A)),A4))),Y) ) ).
% sum_mset_constant
tff(fact_5157_numeral__BitM,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = minus_minus(A,aa(num,A,numeral_numeral(A),bit0(N)),one_one(A)) ) ).
% numeral_BitM
tff(fact_5158_mset__empty__count,axiom,
! [A: $tType,M: multiset(A)] :
( ! [P5: A] : aa(A,nat,count(A,M),P5) = zero_zero(nat)
<=> ( M = zero_zero(multiset(A)) ) ) ).
% mset_empty_count
tff(fact_5159_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,A3: A] : comm_m7189776963980413722m_mset(A,replicate_mset(A,N,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),A3) ) ).
% sum_mset_replicate_mset
tff(fact_5160_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C3: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_rk(B,fun(A,fun(B,A)),Y),C3),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,count(B,A4),Y))) ) ).
% sum_mset_delta
tff(fact_5161_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C3: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_rl(B,fun(A,fun(B,A)),Y),C3),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,count(B,A4),Y))) ) ).
% sum_mset_delta'
tff(fact_5162_sum__mset__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& times(A)
& semiring_0(B) )
=> ! [F: fun(A,B),A4: multiset(A),G: fun(C,B),B3: multiset(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_m7189776963980413722m_mset(B,image_mset(A,B,F,A4))),comm_m7189776963980413722m_mset(B,image_mset(C,B,G,B3))) = comm_m7189776963980413722m_mset(B,image_mset(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_rn(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),F),G),B3),A4)) ) ).
% sum_mset_product
tff(fact_5163_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),M: multiset(B),C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F,M))),C3) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_gg(fun(B,A),fun(A,fun(B,A)),F),C3),M)) ) ).
% sum_mset_distrib_right
tff(fact_5164_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C3: A,F: fun(B,A),M: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),comm_m7189776963980413722m_mset(A,image_mset(B,A,F,M))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gf(A,fun(fun(B,A),fun(B,A)),C3),F),M)) ) ).
% sum_mset_distrib_left
tff(fact_5165_count__ne__remove,axiom,
! [A: $tType,X: A,T4: A,S: multiset(A)] :
( ( X != T4 )
=> ( aa(A,nat,count(A,S),X) = aa(A,nat,count(A,minus_minus(multiset(A),S,add_mset(A,T4,zero_zero(multiset(A))))),X) ) ) ).
% count_ne_remove
tff(fact_5166_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),S))
=> ( ( pp(member2(A,A3,S))
=> ( aa(A,nat,count(A,mset_set(A,S)),A3) = one_one(nat) ) )
& ( ~ pp(member2(A,A3,S))
=> ( aa(A,nat,count(A,mset_set(A,S)),A3) = zero_zero(nat) ) ) ) ) ).
% count_mset_set_finite_iff
tff(fact_5167_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C3: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ro(B,fun(A,fun(B,A)),Y),C3),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),aa(B,nat,count(B,A4),Y)) ) ).
% prod_mset_delta'
tff(fact_5168_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C3: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_rp(B,fun(A,fun(B,A)),Y),C3),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),aa(B,nat,count(B,A4),Y)) ) ).
% prod_mset_delta
tff(fact_5169_power__diff__conv__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),minus_minus(nat,N,M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),inverse_inverse(A,X)),M2)) ) ) ) ) ).
% power_diff_conv_inverse
tff(fact_5170_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [R3: B,Xs: list(C)] : groups8242544230860333062m_list(B,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),aTP_Lamp_rq(B,fun(C,B),R3)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R3) ) ).
% sum_list_triv
tff(fact_5171_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_5172_inverse__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : inverse_inverse(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A3)),inverse_inverse(A,B2)) ) ).
% inverse_mult_distrib
tff(fact_5173_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( inverse_inverse(A,one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_5174_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_5175_sum__list_ONil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).
% sum_list.Nil
tff(fact_5176_sum__list_OCons,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [X: A,Xs: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),groups8242544230860333062m_list(A,Xs)) ) ).
% sum_list.Cons
tff(fact_5177_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_rr(B,A)),Xs)) = zero_zero(A) ) ).
% sum_list_0
tff(fact_5178_right__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),inverse_inverse(A,A3)) = one_one(A) ) ) ) ).
% right_inverse
tff(fact_5179_left__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A3)),A3) = one_one(A) ) ) ) ).
% left_inverse
tff(fact_5180_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_5181_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_5182_inverse__unique,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
=> ( inverse_inverse(A,A3) = B2 ) ) ) ).
% inverse_unique
tff(fact_5183_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).
% group.inverse_distrib_swap
tff(fact_5184_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% group.group_left_neutral
tff(fact_5185_group_Oinverse__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,Z2) = Z2 ) ) ).
% group.inverse_neutral
tff(fact_5186_group_Oinverse__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).
% group.inverse_inverse
tff(fact_5187_group_Oinverse__unique,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
=> ( aa(A,A,Inverse,A3) = B2 ) ) ) ).
% group.inverse_unique
tff(fact_5188_group_Oright__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).
% group.right_inverse
tff(fact_5189_group_Oright__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C3: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,B2),A3) = aa(A,A,aa(A,fun(A,A),F,C3),A3) )
<=> ( B2 = C3 ) ) ) ).
% group.right_cancel
tff(fact_5190_group_Oleft__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).
% group.left_inverse
tff(fact_5191_group_Oleft__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C3: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,A3),C3) )
<=> ( B2 = C3 ) ) ) ).
% group.left_cancel
tff(fact_5192_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_5193_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( inverse_inverse(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,B2)),inverse_inverse(A,A3)) ) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_5194_divide__inverse__commute,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,B2)),A3) ) ).
% divide_inverse_commute
tff(fact_5195_divide__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),inverse_inverse(A,B2)) ) ).
% divide_inverse
tff(fact_5196_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),inverse_inverse(A,B2)) ) ).
% field_class.field_divide_inverse
tff(fact_5197_inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : inverse_inverse(A,A3) = divide_divide(A,one_one(A),A3) ) ).
% inverse_eq_divide
tff(fact_5198_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),inverse_inverse(A,X)),N)) = 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)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).
% power_mult_power_inverse_commute
tff(fact_5199_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),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),M2)) ) ).
% power_mult_inverse_distrib
tff(fact_5200_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_5201_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_5202_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_rs(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs))) ) ).
% sum_list_addf
tff(fact_5203_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),C3: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(A,fun(B,A),aTP_Lamp_gg(fun(B,A),fun(A,fun(B,A)),F),C3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))),C3) ) ).
% sum_list_mult_const
tff(fact_5204_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C3: A,F: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gf(A,fun(fun(B,A),fun(B,A)),C3),F)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))) ) ).
% sum_list_const_mult
tff(fact_5205_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_rt(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = minus_minus(A,groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs))) ) ).
% sum_list_subtractf
tff(fact_5206_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),inverse_inverse(A,X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).
% inverse_le_1_iff
tff(fact_5207_one__less__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),inverse_inverse(A,A3))) ) ) ) ).
% one_less_inverse
tff(fact_5208_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),inverse_inverse(A,X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).
% one_less_inverse_iff
tff(fact_5209_inverse__add,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),inverse_inverse(A,A3)),inverse_inverse(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),inverse_inverse(A,A3))),inverse_inverse(A,B2)) ) ) ) ) ).
% inverse_add
tff(fact_5210_division__ring__inverse__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),inverse_inverse(A,A3)),inverse_inverse(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),inverse_inverse(A,B2)) ) ) ) ) ).
% division_ring_inverse_add
tff(fact_5211_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A3)),A3) = one_one(A) ) ) ) ).
% field_class.field_inverse
tff(fact_5212_division__ring__inverse__diff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( minus_minus(A,inverse_inverse(A,A3),inverse_inverse(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),inverse_inverse(A,A3)),minus_minus(A,B2,A3))),inverse_inverse(A,B2)) ) ) ) ) ).
% division_ring_inverse_diff
tff(fact_5213_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( inverse_inverse(A,A3) = divide_divide(A,one_one(A),A3) ) ) ) ).
% nonzero_inverse_eq_divide
tff(fact_5214_sum__list__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),abs_abs(A)),Xs)))) ) ).
% sum_list_abs
tff(fact_5215_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),comp(A,A,B,uminus_uminus(A),F)),Xs)) ) ).
% uminus_sum_list_map
tff(fact_5216_sum__list__replicate,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,C3: A] : groups8242544230860333062m_list(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C3) ) ).
% sum_list_replicate
tff(fact_5217_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& ordere6658533253407199908up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs)))) ) ) ).
% sum_list_mono
tff(fact_5218_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( monoid_add(A)
=> ! [F: fun(B,A),P: fun(B,bool),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ru(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P)),Xs)) ) ).
% sum_list_map_filter'
tff(fact_5219_inverse__le__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),inverse_inverse(A,A3)),inverse_inverse(A,B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).
% inverse_le_iff
tff(fact_5220_inverse__less__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),inverse_inverse(A,A3)),inverse_inverse(A,B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).
% inverse_less_iff
tff(fact_5221_one__le__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),inverse_inverse(A,A3))) ) ) ) ).
% one_le_inverse
tff(fact_5222_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),inverse_inverse(A,X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).
% inverse_less_1_iff
tff(fact_5223_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),inverse_inverse(A,X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).
% one_le_inverse_iff
tff(fact_5224_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& strict9044650504122735259up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ( Xs != nil(A) )
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,G,X2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs)))) ) ) ) ).
% sum_list_strict_mono
tff(fact_5225_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( monoid_add(A)
=> ! [Xs: list(B),P: fun(B,bool),F: fun(B,A)] :
( ! [X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),Xs)))
=> ( ~ pp(aa(B,bool,P,X2))
=> ( aa(B,A,F,X2) = zero_zero(A) ) ) )
=> ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ) ) ) ).
% sum_list_map_filter
tff(fact_5226_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( comm_monoid_add(C)
=> ! [Xs: list(B),F: fun(B,C)] :
( pp(aa(list(B),bool,distinct(B),Xs))
=> ( groups8242544230860333062m_list(C,aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F),Xs)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F),aa(list(B),set(B),set2(B),Xs)) ) ) ) ).
% sum_list_distinct_conv_sum_set
tff(fact_5227_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Xs: list(B),G: fun(B,A)] :
( pp(aa(list(B),bool,distinct(B),Xs))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) ) ) ) ).
% sum.distinct_set_conv_list
tff(fact_5228_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [X: B,Xs: list(B),F: fun(B,A)] :
( pp(member2(B,X,aa(list(B),set(B),set2(B),Xs)))
=> ( groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),X),Xs)))) ) ) ) ).
% sum_list_map_remove1
tff(fact_5229_sum__code,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xs))) ) ).
% sum_code
tff(fact_5230_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : uncurry(A,B,C,product_curry(A,B,C,F)) = F ).
% uncurry_curry_id
tff(fact_5231_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,uncurry(A,B,C,F)) = F ).
% curry_uncurry_id
tff(fact_5232_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set(A),I5: set(A),E5: set(product_prod(A,A))] :
( pp(aa(set(A),bool,finite_finite2(A),Q))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I5),Q))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),range(A,A,E5)),Q))
=> pp(aa(set(A),bool,finite_finite2(A),image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E5),I5))) ) ) ) ).
% finite_reachable_restrictedI
tff(fact_5233_Range__empty,axiom,
! [B: $tType,A: $tType] : range(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(A)) ).
% Range_empty
tff(fact_5234_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A))] :
( ( range(B,A,R3) = bot_bot(set(A)) )
<=> ( R3 = bot_bot(set(product_prod(B,A))) ) ) ).
% Range_empty_iff
tff(fact_5235_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B3: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),range(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))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),range(B,A,A4)),range(B,A,B3)))) ).
% Range_Int_subset
tff(fact_5236_trancl__Image__in__Range,axiom,
! [A: $tType,R: set(product_prod(A,A)),V3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,R),V3)),range(A,A,R))) ).
% trancl_Image_in_Range
tff(fact_5237_Range__rel__restrict,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),range(A,A,rel_restrict(A,R,A4))),minus_minus(set(A),range(A,A,R),A4))) ).
% Range_rel_restrict
tff(fact_5238_wf__UN,axiom,
! [B: $tType,A: $tType,I5: set(A),R3: fun(A,set(product_prod(B,B)))] :
( ! [I3: A] :
( pp(member2(A,I3,I5))
=> wf(B,aa(A,set(product_prod(B,B)),R3,I3)) )
=> ( ! [I3: A,J3: A] :
( pp(member2(A,I3,I5))
=> ( pp(member2(A,J3,I5))
=> ( ( aa(A,set(product_prod(B,B)),R3,I3) != aa(A,set(product_prod(B,B)),R3,J3) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),domain(B,B,aa(A,set(product_prod(B,B)),R3,I3))),range(B,B,aa(A,set(product_prod(B,B)),R3,J3))) = 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)),R3),I5))) ) ) ).
% wf_UN
tff(fact_5239_dom__ran__disj__comp,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,R)),range(A,A,R)) = bot_bot(set(A)) )
=> ( relcomp(A,A,A,R,R) = bot_bot(set(product_prod(A,A))) ) ) ).
% dom_ran_disj_comp
tff(fact_5240_wf__max,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,converse(A,A,R))
=> ( ( R != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M4: A] : ~ pp(member2(A,M4,minus_minus(set(A),range(A,A,R),domain(A,A,R)))) ) ) ).
% wf_max
tff(fact_5241_Domain__empty,axiom,
! [B: $tType,A: $tType] : domain(A,B,bot_bot(set(product_prod(A,B)))) = bot_bot(set(A)) ).
% Domain_empty
tff(fact_5242_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
( ( domain(A,B,R3) = bot_bot(set(A)) )
<=> ( R3 = bot_bot(set(product_prod(A,B))) ) ) ).
% Domain_empty_iff
tff(fact_5243_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B3: set(product_prod(A,B))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,B,A4)),domain(A,B,B3)))) ).
% Domain_Int_subset
tff(fact_5244_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R: set(product_prod(A,B))] :
( pp(member2(A,X,domain(A,B,R)))
=> pp(member2(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,R,X)),R)) ) ).
% for_in_RI
tff(fact_5245_Domain__rel__restrict,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),domain(A,A,rel_restrict(A,R,A4))),minus_minus(set(A),domain(A,A,R),A4))) ).
% Domain_rel_restrict
tff(fact_5246_wf__no__path,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,R)),range(A,A,R)) = bot_bot(set(A)) )
=> wf(A,R) ) ).
% wf_no_path
tff(fact_5247_wf__min,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
=> ( ( R != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M4: A] : ~ pp(member2(A,M4,minus_minus(set(A),domain(A,A,R),range(A,A,R)))) ) ) ).
% wf_min
tff(fact_5248_wf__Un,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( wf(A,R3)
=> ( wf(A,S4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,R3)),range(A,A,S4)) = 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))),R3),S4)) ) ) ) ).
% wf_Un
tff(fact_5249_wf__Union,axiom,
! [A: $tType,R: set(set(product_prod(A,A)))] :
( ! [X2: set(product_prod(A,A))] :
( pp(member2(set(product_prod(A,A)),X2,R))
=> wf(A,X2) )
=> ( ! [X2: set(product_prod(A,A))] :
( pp(member2(set(product_prod(A,A)),X2,R))
=> ! [Xa4: set(product_prod(A,A))] :
( pp(member2(set(product_prod(A,A)),Xa4,R))
=> ( ( X2 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,X2)),range(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))),R)) ) ) ).
% wf_Union
tff(fact_5250_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_5251_folding__insort__key_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> foldin3648464208017769352axioms(B,A,S,F) ) ).
% folding_insort_key.axioms(2)
tff(fact_5252_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: bool,Xa: product_prod(A,set(B))] :
( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ( ( As = bot_bot(set(B)) )
& pp(X) ) ) ) ).
% pure_assn_raw.elims(3)
tff(fact_5253_monoid_Oleft__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% monoid.left_neutral
tff(fact_5254_monoid_Oright__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% monoid.right_neutral
tff(fact_5255_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_5256_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_5257_append_Omonoid__axioms,axiom,
! [A: $tType] : monoid(list(A),append(A),nil(A)) ).
% append.monoid_axioms
tff(fact_5258_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.monoid_axioms
tff(fact_5259_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.monoid_axioms
tff(fact_5260_folding__insort__key__axioms_Ointro,axiom,
! [A: $tType,B: $tType,F: fun(B,A),S: set(B)] :
( inj_on(B,A,F,S)
=> foldin3648464208017769352axioms(B,A,S,F) ) ).
% folding_insort_key_axioms.intro
tff(fact_5261_folding__insort__key__axioms__def,axiom,
! [A: $tType,B: $tType,S: set(B),F: fun(B,A)] :
( foldin3648464208017769352axioms(B,A,S,F)
<=> inj_on(B,A,F,S) ) ).
% folding_insort_key_axioms_def
tff(fact_5262_pure__assn__raw_Osimps,axiom,
! [B: $tType,A: $tType,B2: bool,H: A,As2: set(B)] :
( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,B2),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2)))
<=> ( ( As2 = bot_bot(set(B)) )
& pp(B2) ) ) ).
% pure_assn_raw.simps
tff(fact_5263_pure__assn__raw_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: bool,Xa: product_prod(A,set(B)),Y: bool] :
( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
<=> pp(Y) )
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ( ( As = bot_bot(set(B)) )
& pp(X) ) ) ) ) ).
% pure_assn_raw.elims(1)
tff(fact_5264_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: bool,Xa: product_prod(A,set(B))] :
( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ~ ( ( As = bot_bot(set(B)) )
& pp(X) ) ) ) ).
% pure_assn_raw.elims(2)
tff(fact_5265_disjoint__image__subset,axiom,
! [A: $tType,A15: set(set(A)),F: fun(set(A),set(A))] :
( pairwise(set(A),disjnt(A),A15)
=> ( ! [X10: set(A)] :
( pp(member2(set(A),X10,A15))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),F,X10)),X10)) )
=> pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F),A15)) ) ) ).
% disjoint_image_subset
tff(fact_5266_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 )
=> ( 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) )
=> ~ 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) )
=> ! [X2: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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),X2),Xs12)),Xs22) )
=> ~ 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),X2),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) )
=> ! [X1: A,X23: A,Xs2: list(A)] :
( ( Xa = 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),X23),Xs2)) )
=> ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs2) )
=> ~ 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),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
tff(fact_5267_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).
% max_of_antimono
tff(fact_5268_pairwise__trivial,axiom,
! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_fo(A,fun(A,bool)),I5) ).
% pairwise_trivial
tff(fact_5269_pairwise__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),S: set(A)] :
( pairwise(A,R,S)
<=> ! [X4: A] :
( pp(member2(A,X4,S))
=> ! [Xa2: A] :
( pp(member2(A,Xa2,S))
=> ( ( X4 != Xa2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Xa2)) ) ) ) ) ).
% pairwise_def
tff(fact_5270_pairwiseI,axiom,
! [A: $tType,S: set(A),R: fun(A,fun(A,bool))] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,S))
=> ( pp(member2(A,Y2,S))
=> ( ( X2 != Y2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Y2)) ) ) )
=> pairwise(A,R,S) ) ).
% pairwiseI
tff(fact_5271_pairwiseD,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),S: set(A),X: A,Y: A] :
( pairwise(A,R,S)
=> ( pp(member2(A,X,S))
=> ( pp(member2(A,Y,S))
=> ( ( X != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X),Y)) ) ) ) ) ).
% pairwiseD
tff(fact_5272_pairwise__insert,axiom,
! [A: $tType,R3: fun(A,fun(A,bool)),X: A,S4: set(A)] :
( pairwise(A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S4))
<=> ( ! [Y3: A] :
( ( pp(member2(A,Y3,S4))
& ( Y3 != X ) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R3,X),Y3))
& pp(aa(A,bool,aa(A,fun(A,bool),R3,Y3),X)) ) )
& pairwise(A,R3,S4) ) ) ).
% pairwise_insert
tff(fact_5273_pairwise__subset,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),S: set(A),T: set(A)] :
( pairwise(A,P,S)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T),S))
=> pairwise(A,P,T) ) ) ).
% pairwise_subset
tff(fact_5274_pairwise__mono,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),A4: set(A),Q: fun(A,fun(A,bool)),B3: set(A)] :
( pairwise(A,P,A4)
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Q,X2),Y2)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A4))
=> pairwise(A,Q,B3) ) ) ) ).
% pairwise_mono
tff(fact_5275_pairwise__empty,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : pairwise(A,P,bot_bot(set(A))) ).
% pairwise_empty
tff(fact_5276_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),P: fun(B,fun(B,bool))] :
( ! [X2: A,Y2: A] :
( pp(member2(A,X2,A4))
=> ( pp(member2(A,Y2,A4))
=> ( ( X2 != Y2 )
=> ( ( aa(A,B,F,X2) != aa(A,B,F,Y2) )
=> pp(aa(B,bool,aa(B,fun(B,bool),P,aa(A,B,F,X2)),aa(A,B,F,Y2))) ) ) ) )
=> pairwise(B,P,aa(set(A),set(B),image2(A,B,F),A4)) ) ).
% pairwise_imageI
tff(fact_5277_pairwise__image,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(A,bool)),F: fun(B,A),S4: set(B)] :
( pairwise(A,R3,aa(set(B),set(A),image2(B,A,F),S4))
<=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_rv(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R3),F),S4) ) ).
% pairwise_image
tff(fact_5278_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_antimono(A,B,F)
<=> ! [X4: A,Y3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y3))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y3)),aa(A,B,F,X4))) ) ) ) ).
% antimono_def
tff(fact_5279_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y2)),aa(A,B,F,X2))) )
=> order_antimono(A,B,F) ) ) ).
% antimonoI
tff(fact_5280_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X))) ) ) ) ).
% antimonoE
tff(fact_5281_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X))) ) ) ) ).
% antimonoD
tff(fact_5282_pairwise__disjnt__iff,axiom,
! [A: $tType,A15: set(set(A))] :
( pairwise(set(A),disjnt(A),A15)
<=> ! [X4: A] : uniq(set(A),aa(A,fun(set(A),bool),aTP_Lamp_rw(set(set(A)),fun(A,fun(set(A),bool)),A15),X4)) ) ).
% pairwise_disjnt_iff
tff(fact_5283_pairwise__singleton,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),A4: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ).
% pairwise_singleton
tff(fact_5284_pairwise__alt,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),S: set(A)] :
( pairwise(A,R,S)
<=> ! [X4: A] :
( pp(member2(A,X4,S))
=> ! [Xa2: A] :
( pp(member2(A,Xa2,minus_minus(set(A),S,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))))))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Xa2)) ) ) ) ).
% pairwise_alt
tff(fact_5285_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).
% min_of_antimono
tff(fact_5286_in__chain__finite,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A4: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> pp(member2(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4)) ) ) ) ) ).
% in_chain_finite
tff(fact_5287_drop__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se4197421643247451524op_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% drop_bit_of_1
tff(fact_5288_bit__double__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3),N))
<=> ( pp(bit_se5641148757651400278ts_bit(A,A3,minus_minus(nat,N,one_one(nat))))
& ( N != zero_zero(nat) )
& pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).
% bit_double_iff
tff(fact_5289_bit__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A)),N))
<=> pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).
% bit_minus_1_iff
tff(fact_5290_chain__empty,axiom,
! [A: $tType,Ord: fun(A,fun(A,bool))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).
% chain_empty
tff(fact_5291_bit__minus__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& ~ pp(bit_se5641148757651400278ts_bit(A,minus_minus(A,A3,one_one(A)),N)) ) ) ) ).
% bit_minus_iff
tff(fact_5292_ccpo__Sup__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A4: set(A),B3: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
=> ( comple1602240252501008431_chain(A,ord_less_eq(A),B3)
=> ( ! [X2: A] :
( pp(member2(A,X2,A4))
=> ? [Xa3: A] :
( pp(member2(A,Xa3,B3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).
% ccpo_Sup_mono
tff(fact_5293_drop__bit__exp__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M2: nat,N: nat] : bit_se4197421643247451524op_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N,M2))) ) ).
% drop_bit_exp_eq
tff(fact_5294_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_5295_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] : divide_divide(A,A3,bit_se4730199178511100633sh_bit(A,N,one_one(A))) = bit_se4197421643247451524op_bit(A,N,A3) ) ).
% div_push_bit_of_1_eq_drop_bit
tff(fact_5296_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,A3,N))
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,N,A3)),one_one(A)) = one_one(A) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
tff(fact_5297_bit__mask__sub__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] :
( pp(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))),M2),one_one(A)),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ).
% bit_mask_sub_iff
tff(fact_5298_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( list_ex1(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
<=> ( ( pp(aa(A,bool,P,X))
=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),aa(A,fun(A,bool),aTP_Lamp_rx(fun(A,bool),fun(A,fun(A,bool)),P),X)),Xs)) )
& ( ~ pp(aa(A,bool,P,X))
=> list_ex1(A,P,Xs) ) ) ) ).
% list_ex1_simps(2)
tff(fact_5299_list__ex__iff__not__all__inverval__int,axiom,
! [P: fun(int,bool),I: int,J: int] :
( pp(aa(list(int),bool,aa(fun(int,bool),fun(list(int),bool),list_ex(int),P),upto(I,J)))
<=> ~ all_interval_int(comp(bool,bool,int,fNot,P),I,J) ) ).
% list_ex_iff_not_all_inverval_int
tff(fact_5300_all__interval__int__def,axiom,
! [P: fun(int,bool),I: int,J: int] :
( all_interval_int(P,I,J)
<=> ! [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,I,J)))
=> pp(aa(int,bool,P,X4)) ) ) ).
% all_interval_int_def
tff(fact_5301_list_Opred__inject_I2_J,axiom,
! [A: $tType,P: fun(A,bool),A3: A,Aa2: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),Aa2)))
<=> ( pp(aa(A,bool,P,A3))
& pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Aa2)) ) ) ).
% list.pred_inject(2)
tff(fact_5302_list__all__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( pp(aa(A,bool,P,X))
& pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Xs)) ) ) ).
% list_all_simps(1)
tff(fact_5303_list__all__simps_I2_J,axiom,
! [A: $tType,P: fun(A,bool)] : pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),nil(A))) ).
% list_all_simps(2)
tff(fact_5304_list__all__append,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Xs))
& pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Ys)) ) ) ).
% list_all_append
tff(fact_5305_list__all__rev,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),aa(list(A),list(A),rev(A),Xs)))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Xs)) ) ).
% list_all_rev
tff(fact_5306_list_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,bool),X: list(A),Pa: fun(A,bool)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),X))
=> ( ! [Z4: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),X)))
=> ( pp(aa(A,bool,P,Z4))
=> pp(aa(A,bool,Pa,Z4)) ) )
=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),Pa),X)) ) ) ).
% list.pred_mono_strong
tff(fact_5307_list__all__cong,axiom,
! [A: $tType,X: list(A),Ya: list(A),P: fun(A,bool),Pa: fun(A,bool)] :
( ( X = Ya )
=> ( ! [Z4: A] :
( pp(member2(A,Z4,aa(list(A),set(A),set2(A),Ya)))
=> ( pp(aa(A,bool,P,Z4))
<=> pp(aa(A,bool,Pa,Z4)) ) )
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),X))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),Pa),Ya)) ) ) ) ).
% list_all_cong
tff(fact_5308_list_Opred__True,axiom,
! [A: $tType,X3: list(A)] : pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),aTP_Lamp_kv(A,bool)),X3)) ).
% list.pred_True
tff(fact_5309_list__all__iff__all__interval__int,axiom,
! [P: fun(int,bool),I: int,J: int] :
( pp(aa(list(int),bool,aa(fun(int,bool),fun(list(int),bool),list_all(int),P),upto(I,J)))
<=> all_interval_int(P,I,J) ) ).
% list_all_iff_all_interval_int
tff(fact_5310_list_Opred__inject_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),nil(A))) ).
% list.pred_inject(1)
tff(fact_5311_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list(A),Ya: list(A),F: fun(A,B),G: fun(A,B)] :
( ( X = Ya )
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ry(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ya) ) ) ) ).
% list.map_cong_pred
tff(fact_5312_list_Opred__mono,axiom,
! [A: $tType,P: fun(A,bool),Pa: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Pa))
=> pp(aa(fun(list(A),bool),bool,aa(fun(list(A),bool),fun(fun(list(A),bool),bool),ord_less_eq(fun(list(A),bool)),aa(fun(A,bool),fun(list(A),bool),list_all(A),P)),aa(fun(A,bool),fun(list(A),bool),list_all(A),Pa))) ) ).
% list.pred_mono
tff(fact_5313_prop__matchD,axiom,
! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),E2: A,Bl2: list(A),P: fun(A,bool)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl2)) )
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Al))
=> ( ~ pp(aa(A,bool,P,E3))
=> ( ~ pp(aa(A,bool,P,E2))
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Bl))
=> ( ( Al = Al2 )
& ( E3 = E2 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
tff(fact_5314_prop__match,axiom,
! [A: $tType,P: fun(A,bool),Al: list(A),E3: A,E2: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Al))
=> ( ~ pp(aa(A,bool,P,E3))
=> ( ~ pp(aa(A,bool,P,E2))
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),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),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl2)) )
<=> ( ( Al = Al2 )
& ( E3 = E2 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
tff(fact_5315_Ball__set__list__all,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X4)) )
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Xs)) ) ).
% Ball_set_list_all
tff(fact_5316_list__all__iff,axiom,
! [A: $tType,P: fun(A,bool),X: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),X))
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),X)))
=> pp(aa(A,bool,P,X4)) ) ) ).
% list_all_iff
tff(fact_5317_list_Opred__set,axiom,
! [A: $tType,P: fun(A,bool),X3: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),X3))
<=> ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),X3)))
=> pp(aa(A,bool,P,Xa2)) ) ) ).
% list.pred_set
tff(fact_5318_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: list(A)] :
( pp(aa(list(B),bool,aa(fun(B,bool),fun(list(B),bool),list_all(B),Q),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),X)))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),comp(B,bool,A,Q,F)),X)) ) ).
% list.pred_map
tff(fact_5319_list__all__length,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),P),Xs))
<=> ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N3))) ) ) ).
% list_all_length
tff(fact_5320_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: fun(A,B),C4: set(A),B3: set(A),X: A] :
( inj_on(A,B,G,C4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))
=> pp(member2(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_rz(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C4),X),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),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_5321_is__num_Osimps,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
<=> ( ( A3 = one_one(A) )
| ? [X4: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X4) )
& neg_numeral_is_num(A,X4) )
| ? [X4: A,Y3: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y3) )
& neg_numeral_is_num(A,X4)
& neg_numeral_is_num(A,Y3) ) ) ) ) ).
% is_num.simps
tff(fact_5322_Func__is__emp,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( bNF_Wellorder_Func(A,B,A4,B3) = bot_bot(set(fun(A,B))) )
<=> ( ( A4 != bot_bot(set(A)) )
& ( B3 = bot_bot(set(B)) ) ) ) ).
% Func_is_emp
tff(fact_5323_Func__non__emp,axiom,
! [A: $tType,B: $tType,B3: set(A),A4: set(B)] :
( ( B3 != bot_bot(set(A)) )
=> ( bNF_Wellorder_Func(B,A,A4,B3) != bot_bot(set(fun(B,A))) ) ) ).
% Func_non_emp
tff(fact_5324_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_5325_Func__empty,axiom,
! [B: $tType,A: $tType,B3: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B3) = 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_sa(A,B)),bot_bot(set(fun(A,B)))) ).
% Func_empty
tff(fact_5326_is__num_Ocases,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
=> ( ( A3 != one_one(A) )
=> ( ! [X2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X2) )
=> ~ neg_numeral_is_num(A,X2) )
=> ~ ! [X2: A,Y2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y2) )
=> ( neg_numeral_is_num(A,X2)
=> ~ neg_numeral_is_num(A,Y2) ) ) ) ) ) ) ).
% is_num.cases
tff(fact_5327_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A14: set(B),B17: set(A),F22: fun(C,D),B24: set(C),A24: set(D)] :
( ( aa(set(B),set(A),image2(B,A,F1),A14) = B17 )
=> ( inj_on(C,D,F22,B24)
=> ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B24)),A24))
=> ( ( ( B24 = bot_bot(set(C)) )
=> ( A24 = bot_bot(set(D)) ) )
=> ( bNF_Wellorder_Func(C,A,B24,B17) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B24,F1,F22)),bNF_Wellorder_Func(D,B,A24,A14)) ) ) ) ) ) ).
% Func_map_surj
tff(fact_5328_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,B2))) = divide_divide(A,A3,unit_f5069060285200089521factor(A,B2)) ) ).
% mult_one_div_unit_factor
tff(fact_5329_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),unit_f5069060285200089521factor(A,B2)) ) ) ) ).
% unit_factor_mult_unit_left
tff(fact_5330_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_5331_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% inv_unit_factor_eq_0_iff
tff(fact_5332_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B2)),A3) ) ) ) ).
% unit_factor_mult_unit_right
tff(fact_5333_unit__factor__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A3: A,B2: A] : unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ).
% unit_factor_mult
tff(fact_5334_is__unit__unit__factor,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).
% is_unit_unit_factor
tff(fact_5335_mult__gcd__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ).
% mult_gcd_left
tff(fact_5336_mult__gcd__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C3) = 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),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),unit_f5069060285200089521factor(A,C3)) ) ).
% mult_gcd_right
tff(fact_5337_gcd__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = 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),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).
% gcd_mult_distrib
tff(fact_5338_unit__factor__is__unit,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> pp(dvd_dvd(A,unit_f5069060285200089521factor(A,A3),one_one(A))) ) ) ).
% unit_factor_is_unit
tff(fact_5339_unit__factor__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = zero_zero(A) ) )
& ( ~ ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = one_one(A) ) ) ) ) ).
% unit_factor_gcd
tff(fact_5340_unit__factor__Gcd,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( ( gcd_Gcd(A,A4) = zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = zero_zero(A) ) )
& ( ( gcd_Gcd(A,A4) != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = one_one(A) ) ) ) ) ).
% unit_factor_Gcd
tff(fact_5341_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_5342_Lcm__no__units,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] : gcd_Lcm(A,A4) = gcd_Lcm(A,minus_minus(set(A),A4,collect(A,aTP_Lamp_sb(A,bool)))) ) ).
% Lcm_no_units
tff(fact_5343_list__all__iff__all__interval__nat,axiom,
! [P: fun(nat,bool),I: nat,J: nat] :
( pp(aa(list(nat),bool,aa(fun(nat,bool),fun(list(nat),bool),list_all(nat),P),upt(I,J)))
<=> all_interval_nat(P,I,J) ) ).
% list_all_iff_all_interval_nat
tff(fact_5344_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% normalize_mult_normalize_left
tff(fact_5345_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% normalize_mult_normalize_right
tff(fact_5346_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_5347_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_5348_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_5349_Lcm__1__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( gcd_Lcm(A,A4) = one_one(A) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(dvd_dvd(A,X4,one_one(A))) ) ) ) ).
% Lcm_1_iff
tff(fact_5350_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),aa(A,A,normal6383669964737779283malize(A),A3)) = A3 ) ).
% unit_factor_mult_normalize
tff(fact_5351_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),unit_f5069060285200089521factor(A,A3)) = A3 ) ).
% normalize_mult_unit_factor
tff(fact_5352_normalize__mult__unit__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).
% normalize_mult_unit_left
tff(fact_5353_normalize__mult__unit__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).
% normalize_mult_unit_right
tff(fact_5354_unit__factor__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).
% unit_factor_normalize
tff(fact_5355_normalize__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).
% normalize_unit_factor
tff(fact_5356_Lcm__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% Lcm_singleton
tff(fact_5357_normalize__div,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A3),A3) = divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) ) ).
% normalize_div
tff(fact_5358_Gcd__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% Gcd_singleton
tff(fact_5359_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A),C3: A] :
( ( A4 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Lcm(A,A4))) ) ) ) ).
% Lcm_mult
tff(fact_5360_gcd__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).
% gcd_mult_left
tff(fact_5361_gcd__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = 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),B2),A3)),C3)) ) ).
% gcd_mult_right
tff(fact_5362_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).
% gcd_mult_distrib'
tff(fact_5363_normalize__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% normalize_mult
tff(fact_5364_associated__unit,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> ( pp(dvd_dvd(A,A3,one_one(A)))
=> pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% associated_unit
tff(fact_5365_normalize__1__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% normalize_1_iff
tff(fact_5366_is__unit__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).
% is_unit_normalize
tff(fact_5367_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( pp(dvd_dvd(A,A3,one_one(A)))
<=> ( A3 = one_one(A) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
tff(fact_5368_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C3: A,A4: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Gcd(A,A4))) ) ).
% Gcd_mult
tff(fact_5369_unit__factor__Lcm,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( ( gcd_Lcm(A,A4) = zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = zero_zero(A) ) )
& ( ( gcd_Lcm(A,A4) != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = one_one(A) ) ) ) ) ).
% unit_factor_Lcm
tff(fact_5370_all__interval__nat__def,axiom,
! [P: fun(nat,bool),I: nat,J: nat] :
( all_interval_nat(P,I,J)
<=> ! [X4: nat] :
( pp(member2(nat,X4,set_or7035219750837199246ssThan(nat,I,J)))
=> pp(aa(nat,bool,P,X4)) ) ) ).
% all_interval_nat_def
tff(fact_5371_Gcd__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A),B2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( 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),B2)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4))) ) ) ) ).
% Gcd_fin_mult
tff(fact_5372_list__ex__iff__not__all__inverval__nat,axiom,
! [P: fun(nat,bool),I: nat,J: nat] :
( pp(aa(list(nat),bool,aa(fun(nat,bool),fun(list(nat),bool),list_ex(nat),P),upt(I,J)))
<=> ~ all_interval_nat(comp(bool,bool,nat,fNot,P),I,J) ) ).
% list_ex_iff_not_all_inverval_nat
tff(fact_5373_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_5374_Lcm__coprime,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [A5: A,B4: A] :
( pp(member2(A,A5,A4))
=> ( pp(member2(A,B4,A4))
=> ( ( A5 != B4 )
=> algebr8660921524188924756oprime(A,A5,B4) ) ) )
=> ( gcd_Lcm(A,A4) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_sc(A,A)),A4)) ) ) ) ) ) ).
% Lcm_coprime
tff(fact_5375_Lcm__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A),B2: A] :
( ( A4 != 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),B2)),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A4))) ) ) ) ).
% Lcm_fin_mult
tff(fact_5376_coprime__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( algebr8660921524188924756oprime(A,C3,A3)
& algebr8660921524188924756oprime(A,C3,B2) ) ) ) ).
% coprime_mult_right_iff
tff(fact_5377_coprime__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3)
<=> ( algebr8660921524188924756oprime(A,A3,C3)
& algebr8660921524188924756oprime(A,B2,C3) ) ) ) ).
% coprime_mult_left_iff
tff(fact_5378_coprime__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,A3)
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_self
tff(fact_5379_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% coprime_imp_gcd_eq_1
tff(fact_5380_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_5381_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( pp(dvd_dvd(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A4),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) ) ) ) ).
% is_unit_Lcm_fin_iff
tff(fact_5382_coprime__0__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,zero_zero(A))
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_0_right_iff
tff(fact_5383_coprime__0__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,zero_zero(A),A3)
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_0_left_iff
tff(fact_5384_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
<=> ( pp(dvd_dvd(A,C3,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_5385_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
<=> ( pp(dvd_dvd(A,C3,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_5386_is__unit__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A)))
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_gcd
tff(fact_5387_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [A3: A,N: A,M2: A,B2: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N),M2) )
=> ( algebr8660921524188924756oprime(A,M2,N)
=> ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B2,M2) ) ) ) ) ).
% mult_mod_cancel_right
tff(fact_5388_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [N: A,A3: A,M2: A,B2: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A3),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B2),M2) )
=> ( algebr8660921524188924756oprime(A,M2,N)
=> ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B2,M2) ) ) ) ) ).
% mult_mod_cancel_left
tff(fact_5389_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_right_imp_coprime
tff(fact_5390_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_left_imp_coprime
tff(fact_5391_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C3: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( pp(dvd_dvd(A,C3,A3))
=> ( pp(dvd_dvd(A,C3,B2))
=> pp(dvd_dvd(A,C3,one_one(A))) ) ) ) ) ).
% coprime_common_divisor
tff(fact_5392_coprime__absorb__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Y: A,X: A] :
( pp(dvd_dvd(A,Y,X))
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> pp(dvd_dvd(A,Y,one_one(A))) ) ) ) ).
% coprime_absorb_right
tff(fact_5393_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,D3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C3,D3)
=> ( ! [E: A] :
( ~ pp(dvd_dvd(A,E,one_one(A)))
=> ( pp(dvd_dvd(A,E,A3))
=> ( pp(dvd_dvd(A,E,B2))
=> pp(dvd_dvd(A,E,C3)) ) ) )
=> ( ! [E: A] :
( ~ pp(dvd_dvd(A,E,one_one(A)))
=> ( pp(dvd_dvd(A,E,A3))
=> ( pp(dvd_dvd(A,E,B2))
=> pp(dvd_dvd(A,E,D3)) ) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% coprime_imp_coprime
tff(fact_5394_coprime__absorb__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,Y: A] :
( pp(dvd_dvd(A,X,Y))
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> pp(dvd_dvd(A,X,one_one(A))) ) ) ) ).
% coprime_absorb_left
tff(fact_5395_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C3: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C3,A3))
=> ( pp(dvd_dvd(A,C3,B2))
=> ( ~ pp(dvd_dvd(A,C3,one_one(A)))
=> ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% not_coprimeI
tff(fact_5396_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ~ algebr8660921524188924756oprime(A,A3,B2)
=> ~ ! [C2: A] :
( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> pp(dvd_dvd(A,C2,one_one(A))) ) ) ) ) ).
% not_coprimeE
tff(fact_5397_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ! [C6: A] :
( pp(dvd_dvd(A,C6,A3))
=> ( pp(dvd_dvd(A,C6,B2))
=> pp(dvd_dvd(A,C6,one_one(A))) ) ) ) ) ).
% coprime_def
tff(fact_5398_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ! [C2: A] :
( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> pp(dvd_dvd(A,C2,one_one(A))) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprimeI
tff(fact_5399_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).
% coprime_1_left
tff(fact_5400_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).
% coprime_1_right
tff(fact_5401_coprime__doff__one__right,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,minus_minus(A,A3,one_one(A))) ) ).
% coprime_doff_one_right
tff(fact_5402_coprime__diff__one__left,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,minus_minus(A,A3,one_one(A)),A3) ) ).
% coprime_diff_one_left
tff(fact_5403_coprime__add__one__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).
% coprime_add_one_right
tff(fact_5404_coprime__add__one__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).
% coprime_add_one_left
tff(fact_5405_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C3)
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% coprime_dvd_mult_right_iff
tff(fact_5406_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C3)
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% coprime_dvd_mult_left_iff
tff(fact_5407_divides__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] :
( pp(dvd_dvd(A,A3,C3))
=> ( pp(dvd_dvd(A,B2,C3))
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3)) ) ) ) ) ).
% divides_mult
tff(fact_5408_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C3: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_left_cancel
tff(fact_5409_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C3: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_right_cancel
tff(fact_5410_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_left_cancel
tff(fact_5411_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C3)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_right_cancel
tff(fact_5412_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% gcd_eq_1_imp_coprime
tff(fact_5413_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% coprime_iff_gcd_eq_1
tff(fact_5414_coprime__crossproduct,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,D3: A,B2: A,C3: A] :
( algebr8660921524188924756oprime(A,A3,D3)
=> ( algebr8660921524188924756oprime(A,B2,C3)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B2)),aa(A,A,normal6383669964737779283malize(A),D3)) )
<=> ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
& ( aa(A,A,normal6383669964737779283malize(A),C3) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).
% coprime_crossproduct
tff(fact_5415_invertible__coprime,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = one_one(A) )
=> algebr8660921524188924756oprime(A,A3,C3) ) ) ).
% invertible_coprime
tff(fact_5416_gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,A10: A,B8: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> algebr8660921524188924756oprime(A,A10,B8) ) ) ) ) ).
% gcd_coprime
tff(fact_5417_gcd__coprime__exists,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
=> ? [A16: A,B10: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A16),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
& ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
& algebr8660921524188924756oprime(A,A16,B10) ) ) ) ).
% gcd_coprime_exists
tff(fact_5418_coprime__crossproduct_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [B2: A,D3: A,A3: A,C3: A] :
( ( B2 != zero_zero(A) )
=> ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> ( algebr8660921524188924756oprime(A,C3,D3)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
<=> ( ( A3 = C3 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
tff(fact_5419_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) )
<=> ( ! [X4: A] :
( pp(member2(A,X4,A4))
=> pp(dvd_dvd(A,X4,one_one(A))) )
& pp(aa(set(A),bool,finite_finite2(A),A4)) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_5420_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = finite_fold(A,A,gcd_lcm(A),one_one(A),A4) ) )
& ( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) ) ) ) ) ).
% Lcm_fin.eq_fold
tff(fact_5421_gcd__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).
% gcd_lcm
tff(fact_5422_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_5423_lcm__eq__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = one_one(A) )
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
& pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% lcm_eq_1_iff
tff(fact_5424_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% lcm.top_left_normalize
tff(fact_5425_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% lcm.top_right_normalize
tff(fact_5426_unit__factor__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = zero_zero(A) ) )
& ( ~ ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = one_one(A) ) ) ) ) ).
% unit_factor_lcm
tff(fact_5427_gcd__mult__lcm,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% gcd_mult_lcm
tff(fact_5428_lcm__mult__gcd,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% lcm_mult_gcd
tff(fact_5429_Lcm__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,B2: A] : gcd_Lcm(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).
% Lcm_2
tff(fact_5430_lcm__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).
% lcm_mult_left
tff(fact_5431_lcm__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = 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),B2),A3)),C3)) ) ).
% lcm_mult_right
tff(fact_5432_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).
% lcm_mult_distrib'
tff(fact_5433_lcm__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = 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),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).
% lcm_mult_distrib
tff(fact_5434_mult__lcm__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C3) = 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),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),unit_f5069060285200089521factor(A,C3)) ) ).
% mult_lcm_right
tff(fact_5435_mult__lcm__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ).
% mult_lcm_left
tff(fact_5436_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,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),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).
% lcm_mult_unit1
tff(fact_5437_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).
% lcm_mult_unit2
tff(fact_5438_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),divide_divide(A,B2,A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).
% lcm_div_unit1
tff(fact_5439_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),divide_divide(A,C3,A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).
% lcm_div_unit2
tff(fact_5440_lcm__gcd__prod,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% lcm_gcd_prod
tff(fact_5441_lcm__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% lcm_coprime
tff(fact_5442_lcm__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).
% lcm_gcd
tff(fact_5443_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,A4: set(A)] :
( pp(member2(A,A3,A4))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).
% Lcm_fin.remove
tff(fact_5444_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,A4: 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),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).
% Lcm_fin.insert_remove
tff(fact_5445_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_lcm(A)),Xs),one_one(A)) ) ).
% Lcm_set_eq_fold
tff(fact_5446_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_5447_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)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_lcm(A)),Xs),one_one(A)) ) ).
% Lcm_fin.set_eq_fold
tff(fact_5448_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_5449_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))
=> pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),G))) ) ) ).
% acyclic_insert_cyclic
tff(fact_5450_FI__RESULT__def,axiom,
! [M: list(product_prod(assn,assn)),UP: assn,UQ: assn,F2: assn] :
( fI_RESULT(M,UP,UQ,F2)
<=> ( ! [X4: product_prod(assn,assn)] :
( pp(member2(product_prod(assn,assn),X4,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M)))
=> pp(aa(product_prod(assn,assn),bool,product_case_prod(assn,assn,bool,entails),X4)) )
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M)),one_one(assn))),UP)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M)),one_one(assn))),UQ)),F2))) ) ) ).
% FI_RESULT_def
tff(fact_5451_listsp__mono,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),A4),B3))
=> pp(aa(fun(list(A),bool),bool,aa(fun(list(A),bool),fun(fun(list(A),bool),bool),ord_less_eq(fun(list(A),bool)),listsp(A,A4)),listsp(A,B3))) ) ).
% listsp_mono
tff(fact_5452_listsp__conj__eq,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),X3: list(A)] :
( pp(aa(list(A),bool,listsp(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bu(fun(A,bool),fun(fun(A,bool),fun(A,bool)),A4),B3)),X3))
<=> ( pp(aa(list(A),bool,listsp(A,A4),X3))
& pp(aa(list(A),bool,listsp(A,B3),X3)) ) ) ).
% listsp_conj_eq
tff(fact_5453_listsp__simps_I1_J,axiom,
! [A: $tType,A4: fun(A,bool)] : pp(aa(list(A),bool,listsp(A,A4),nil(A))) ).
% listsp_simps(1)
tff(fact_5454_in__listspI,axiom,
! [A: $tType,Xs: list(A),A4: fun(A,bool)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,A4,X2)) )
=> pp(aa(list(A),bool,listsp(A,A4),Xs)) ) ).
% in_listspI
tff(fact_5455_append__in__listsp__conv,axiom,
! [A: $tType,A4: fun(A,bool),Xs: list(A),Ys: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)))
<=> ( pp(aa(list(A),bool,listsp(A,A4),Xs))
& pp(aa(list(A),bool,listsp(A,A4),Ys)) ) ) ).
% append_in_listsp_conv
tff(fact_5456_acyclic__empty,axiom,
! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).
% acyclic_empty
tff(fact_5457_listsp__inf__eq,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool)] : listsp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3)) = aa(fun(list(A),bool),fun(list(A),bool),aa(fun(list(A),bool),fun(fun(list(A),bool),fun(list(A),bool)),inf_inf(fun(list(A),bool)),listsp(A,A4)),listsp(A,B3)) ).
% listsp_inf_eq
tff(fact_5458_ent__pure__pre__iff,axiom,
! [P: assn,B2: bool,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2))),Q))
<=> ( pp(B2)
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ) ).
% ent_pure_pre_iff
tff(fact_5459_ent__pure__pre__iff__sng,axiom,
! [B2: bool,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,pure_assn(B2)),Q))
<=> ( pp(B2)
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,one_one(assn)),Q)) ) ) ).
% ent_pure_pre_iff_sng
tff(fact_5460_cyclic__subset,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S))
=> ~ transitive_acyclic(A,S) ) ) ).
% cyclic_subset
tff(fact_5461_lists__def,axiom,
! [A: $tType,X3: set(A)] : aa(set(A),set(list(A)),lists(A),X3) = collect(list(A),listsp(A,aTP_Lamp_a(set(A),fun(A,bool),X3))) ).
% lists_def
tff(fact_5462_listsp__lists__eq,axiom,
! [A: $tType,A4: set(A),X3: list(A)] :
( pp(aa(list(A),bool,listsp(A,aTP_Lamp_a(set(A),fun(A,bool),A4)),X3))
<=> pp(member2(list(A),X3,aa(set(A),set(list(A)),lists(A),A4))) ) ).
% listsp_lists_eq
tff(fact_5463_listsp__infI,axiom,
! [A: $tType,A4: fun(A,bool),L: list(A),B3: fun(A,bool)] :
( pp(aa(list(A),bool,listsp(A,A4),L))
=> ( pp(aa(list(A),bool,listsp(A,B3),L))
=> pp(aa(list(A),bool,listsp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),A4),B3)),L)) ) ) ).
% listsp_infI
tff(fact_5464_ent__conjI,axiom,
! [A4: assn,B3: assn,C4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),B3))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B3),C4))) ) ) ).
% ent_conjI
tff(fact_5465_ent__conjE1,axiom,
! [A4: assn,C4: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),B3)),C4)) ) ).
% ent_conjE1
tff(fact_5466_ent__conjE2,axiom,
! [B3: assn,C4: assn,A4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),B3)),C4)) ) ).
% ent_conjE2
tff(fact_5467_in__listsp__conv__set,axiom,
! [A: $tType,A4: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),Xs))
<=> ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,A4,X4)) ) ) ).
% in_listsp_conv_set
tff(fact_5468_in__listspD,axiom,
! [A: $tType,A4: fun(A,bool),Xs: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),Xs))
=> ! [X3: A] :
( pp(member2(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,A4,X3)) ) ) ).
% in_listspD
tff(fact_5469_listspE,axiom,
! [A: $tType,A4: fun(A,bool),X: A,L: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L)))
=> ~ ( pp(aa(A,bool,A4,X))
=> ~ pp(aa(list(A),bool,listsp(A,A4),L)) ) ) ).
% listspE
tff(fact_5470_listsp_OCons,axiom,
! [A: $tType,A4: fun(A,bool),A3: A,L: list(A)] :
( pp(aa(A,bool,A4,A3))
=> ( pp(aa(list(A),bool,listsp(A,A4),L))
=> pp(aa(list(A),bool,listsp(A,A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),L))) ) ) ).
% listsp.Cons
tff(fact_5471_listsp__simps_I2_J,axiom,
! [A: $tType,A4: fun(A,bool),X: A,Xs: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))
<=> ( pp(aa(A,bool,A4,X))
& pp(aa(list(A),bool,listsp(A,A4),Xs)) ) ) ).
% listsp_simps(2)
tff(fact_5472_listsp_ONil,axiom,
! [A: $tType,A4: fun(A,bool)] : pp(aa(list(A),bool,listsp(A,A4),nil(A))) ).
% listsp.Nil
tff(fact_5473_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Q),R))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),R)) ) ) ).
% ent_trans
tff(fact_5474_ent__refl,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P)) ).
% ent_refl
tff(fact_5475_ent__iffI,axiom,
! [A4: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),B3))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),A4))
=> ( A4 = B3 ) ) ) ).
% ent_iffI
tff(fact_5476_is__entails,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ).
% is_entails
tff(fact_5477_ent__star__mono,axiom,
! [P: assn,P6: assn,Q: assn,Q5: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P6))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Q),Q5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P6),Q5))) ) ) ).
% ent_star_mono
tff(fact_5478_ent__frame__fwd,axiom,
! [P: assn,R: assn,Ps3: assn,F2: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),R))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Ps3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F2)))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F2)),Q))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Ps3),Q)) ) ) ) ).
% ent_frame_fwd
tff(fact_5479_fr__rot__rhs,axiom,
! [A4: assn,B3: assn,C4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C4)))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),B3))) ) ).
% fr_rot_rhs
tff(fact_5480_fr__refl,axiom,
! [A4: assn,B3: assn,C4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),B3))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),C4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C4))) ) ).
% fr_refl
tff(fact_5481_fr__rot,axiom,
! [A4: assn,B3: assn,C4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),A4)),C4)) ) ).
% fr_rot
tff(fact_5482_ent__true,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),top_top(assn))) ).
% ent_true
tff(fact_5483_acyclic__union_I1_J,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B3: 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))),A4),B3))
=> transitive_acyclic(A,A4) ) ).
% acyclic_union(1)
tff(fact_5484_acyclic__union_I2_J,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B3: 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))),A4),B3))
=> transitive_acyclic(A,B3) ) ).
% acyclic_union(2)
tff(fact_5485_ent__disjI2__direct,axiom,
! [B3: assn,A4: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3))) ).
% ent_disjI2_direct
tff(fact_5486_ent__disjI1__direct,axiom,
! [A4: assn,B3: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3))) ).
% ent_disjI1_direct
tff(fact_5487_ent__disjI2_H,axiom,
! [A4: assn,C4: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C4))) ) ).
% ent_disjI2'
tff(fact_5488_ent__disjI1_H,axiom,
! [A4: assn,B3: assn,C4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),B3))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C4))) ) ).
% ent_disjI1'
tff(fact_5489_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),R))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Q),R)) ) ).
% ent_disjI2
tff(fact_5490_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),R))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),R)) ) ).
% ent_disjI1
tff(fact_5491_ent__disjE,axiom,
! [A4: assn,C4: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C4))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),C4))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)),C4)) ) ) ).
% ent_disjE
tff(fact_5492_ent__false,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,bot_bot(assn)),P)) ).
% ent_false
tff(fact_5493_ent__star__mono__true,axiom,
! [A4: assn,A12: assn,B3: assn,B15: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A12),top_top(assn))))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B15),top_top(assn))))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A12),B15)),top_top(assn)))) ) ) ).
% ent_star_mono_true
tff(fact_5494_ent__refl__true,axiom,
! [A4: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),top_top(assn)))) ).
% ent_refl_true
tff(fact_5495_ent__true__drop_I1_J,axiom,
! [P: assn,Q: assn,R: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))) ) ).
% ent_true_drop(1)
tff(fact_5496_ent__true__drop_I2_J,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))) ) ).
% ent_true_drop(2)
tff(fact_5497_listsp_Ocases,axiom,
! [A: $tType,A4: fun(A,bool),A3: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),A3))
=> ( ( A3 != nil(A) )
=> ~ ! [A5: A,L2: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),L2) )
=> ( pp(aa(A,bool,A4,A5))
=> ~ pp(aa(list(A),bool,listsp(A,A4),L2)) ) ) ) ) ).
% listsp.cases
tff(fact_5498_listsp_Osimps,axiom,
! [A: $tType,A4: fun(A,bool),A3: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),A3))
<=> ( ( A3 = nil(A) )
| ? [A6: A,L4: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),L4) )
& pp(aa(A,bool,A4,A6))
& pp(aa(list(A),bool,listsp(A,A4),L4)) ) ) ) ).
% listsp.simps
tff(fact_5499_fi__match__entails,axiom,
! [M2: list(product_prod(assn,assn))] :
( ! [X2: product_prod(assn,assn)] :
( pp(member2(product_prod(assn,assn),X2,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M2)))
=> pp(aa(product_prod(assn,assn),bool,product_case_prod(assn,assn,bool,entails),X2)) )
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M2)),one_one(assn))),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M2)),one_one(assn)))) ) ).
% fi_match_entails
tff(fact_5500_FI__QUERY__def,axiom,
! [P: assn,Q: assn,F2: assn] :
( fI_QUERY(P,Q,F2)
<=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F2))) ) ).
% FI_QUERY_def
tff(fact_5501_frame__inference__init,axiom,
! [P: assn,Q: assn,F2: assn] :
( fI_QUERY(P,Q,F2)
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F2))) ) ).
% frame_inference_init
tff(fact_5502_cyclicE,axiom,
! [A: $tType,G: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,G)
=> ~ ! [X2: A] : ~ pp(member2(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,G))) ) ).
% cyclicE
tff(fact_5503_entails__solve__init_I2_J,axiom,
! [P: assn,Q: assn] :
( fI_QUERY(P,Q,one_one(assn))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ).
% entails_solve_init(2)
tff(fact_5504_wf__set,axiom,
! [A: $tType,Xs: list(product_prod(A,A))] :
( wf(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))
<=> transitive_acyclic(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ) ).
% wf_set
tff(fact_5505_entails__solve__init_I1_J,axiom,
! [P: assn,Q: assn] :
( fI_QUERY(P,Q,top_top(assn))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))) ) ).
% entails_solve_init(1)
tff(fact_5506_FI__match,axiom,
! [P4: assn,Q2: assn,M2: list(product_prod(assn,assn)),Ps: assn,Up: assn,Qs: assn,Uq: assn,F: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P4),Q2))
=> ( fi(aa(list(product_prod(assn,assn)),list(product_prod(assn,assn)),aa(product_prod(assn,assn),fun(list(product_prod(assn,assn)),list(product_prod(assn,assn))),cons(product_prod(assn,assn)),aa(assn,product_prod(assn,assn),aa(assn,fun(assn,product_prod(assn,assn)),product_Pair(assn,assn),P4),Q2)),M2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps),Up),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Uq),sln,sln,F)
=> fi(M2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps),P4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F) ) ) ).
% FI_match
tff(fact_5507_FI__def,axiom,
! [M2: list(product_prod(assn,assn)),P4: assn,Q2: assn,Up: assn,Uq: assn,F: assn] :
( fi(M2,P4,Q2,Up,Uq,F)
<=> ( ! [X4: product_prod(assn,assn)] :
( pp(member2(product_prod(assn,assn),X4,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),set2(product_prod(assn,assn)),M2)))
=> pp(aa(product_prod(assn,assn),bool,product_case_prod(assn,assn,bool,entails),X4)) )
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_fst(assn,assn)),M2)),one_one(assn))),P4)),Up)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(list(assn),fun(assn,assn),aa(fun(assn,fun(assn,assn)),fun(list(assn),fun(assn,assn)),foldr(assn,assn),times_times(assn)),aa(list(product_prod(assn,assn)),list(assn),aa(fun(product_prod(assn,assn),assn),fun(list(product_prod(assn,assn)),list(assn)),map(product_prod(assn,assn),assn),product_snd(assn,assn)),M2)),one_one(assn))),Q2)),Uq)),F))) ) ) ).
% FI_def
tff(fact_5508_ent__wand__frameI,axiom,
! [Q: assn,R: assn,F2: assn,S: assn,P: assn,X5: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),wand_assn(Q,R)),F2)),S))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),F2),X5)))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),X5)),R))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),S)) ) ) ) ).
% ent_wand_frameI
tff(fact_5509_ent__wandI,axiom,
! [Q: assn,P: assn,R: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P)),R))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),wand_assn(Q,R))) ) ).
% ent_wandI
tff(fact_5510_ent__mp,axiom,
! [P: assn,Q: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),wand_assn(P,Q))),Q)) ).
% ent_mp
tff(fact_5511_prec__frame,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),R12: assn,R22: assn,H: product_prod(heap_ext(product_unit),set(nat)),X: A,P4: B,F12: assn,Y: A,F23: assn] :
( precise(A,B,P)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R22)),H))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R12),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,X),P4)),F12)))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,Y),P4)),F23)))
=> ( X = Y ) ) ) ) ) ).
% prec_frame
tff(fact_5512_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ( groups1962203154675924110t_prod(C,A) = groups_comm_monoid_G(A,C,times_times(A),one_one(A)) ) ) ).
% prod'_def
tff(fact_5513_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_5514_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,X) = aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Y) )
<=> ( X = Y ) ) ).
% Rep_assn_inject
tff(fact_5515_mod__or__dist,axiom,
! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),H))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
| pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H)) ) ) ).
% mod_or_dist
tff(fact_5516_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat)))))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat)))))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(5)
tff(fact_5517_mod__pure__star__dist,axiom,
! [P: assn,B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2))),H))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
& pp(B2) ) ) ).
% mod_pure_star_dist
tff(fact_5518_mod__h__bot__normalize,axiom,
! [A: $tType,H: heap_ext(product_unit),P: assn] :
( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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_5519_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(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)),H),bot_bot(set(nat)))))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat)))))
| pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(7)
tff(fact_5520_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(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)),H),bot_bot(set(nat)))))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat)))))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(6)
tff(fact_5521_mod__pure,axiom,
! [B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,pure_assn(B2)),H))
<=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) )
& pp(B2) ) ) ).
% mod_pure
tff(fact_5522_mod__h__bot__iff_I1_J,axiom,
! [B2: bool,H: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,pure_assn(B2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
<=> pp(B2) ) ).
% mod_h_bot_iff(1)
tff(fact_5523_ent__false__iff,axiom,
! [P: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),bot_bot(assn)))
<=> ! [H4: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H4)) ) ).
% ent_false_iff
tff(fact_5524_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( heap(B)
=> ! [Q2: array(B),Y: list(B),H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,snga_assn(B,Q2,Y)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(4)
tff(fact_5525_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [P4: ref(A),X: A,H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,sngr_assn(A,P4,X)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(3)
tff(fact_5526_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B2: bool] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn(B2))))
<=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H4))
=> pp(B2) )
& pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ) ).
% ent_pure_post_iff
tff(fact_5527_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: bool] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),pure_assn(B2)))
<=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H4))
=> pp(B2) )
& pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),one_one(assn))) ) ) ).
% ent_pure_post_iff_sng
tff(fact_5528_mod__h__bot__indep,axiom,
! [P: assn,H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H5),bot_bot(set(nat))))) ) ).
% mod_h_bot_indep
tff(fact_5529_mod__and__dist,axiom,
! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),H))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H)) ) ) ).
% mod_and_dist
tff(fact_5530_mod__starD,axiom,
! [A4: assn,B3: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),H))
=> ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,A4),H1))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,B3),H22)) ) ) ).
% mod_starD
tff(fact_5531_mod__starE,axiom,
! [A3: assn,B2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),H))
=> ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,A3),X_1))
=> ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,B2),H_2)) ) ) ).
% mod_starE
tff(fact_5532_mod__false,axiom,
! [H: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,bot_bot(assn)),H)) ).
% mod_false
tff(fact_5533_ent__fwd,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat)),Q: assn] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H)) ) ) ).
% ent_fwd
tff(fact_5534_entailsD,axiom,
! [P: assn,Q: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H)) ) ) ).
% entailsD
tff(fact_5535_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H3))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H3)) )
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ).
% entailsI
tff(fact_5536_entails__def,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
<=> ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H4))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q),H4)) ) ) ).
% entails_def
tff(fact_5537_mod__frame__fwd,axiom,
! [Ps3: assn,H: product_prod(heap_ext(product_unit),set(nat)),P: assn,R: assn,F2: assn] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Ps3),H))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),R))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Ps3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F2)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F2)),H)) ) ) ) ).
% mod_frame_fwd
tff(fact_5538_star__assnI,axiom,
! [P: assn,H: heap_ext(product_unit),As2: set(nat),Q: assn,As4: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),As2)))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),As4)))
=> ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As4) = bot_bot(set(nat)) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4)))) ) ) ) ).
% star_assnI
tff(fact_5539_mod__star__conv,axiom,
! [A4: assn,B3: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B3)),H))
<=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
( ( H = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,A4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,B3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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_5540_mod__star__trueE_H,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H))
=> ~ ! [H6: product_prod(heap_ext(product_unit),set(nat))] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H6) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H) )
=> ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H6)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H)))
=> ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H6)) ) ) ) ).
% mod_star_trueE'
tff(fact_5541_mod__star__trueI,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H)) ) ).
% mod_star_trueI
tff(fact_5542_mod__star__trueE,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H))
=> ~ ! [H6: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H6)) ) ).
% mod_star_trueE
tff(fact_5543_mod__h__bot__iff_I2_J,axiom,
! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,top_top(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).
% mod_h_bot_iff(2)
tff(fact_5544_mod__emp,axiom,
! [H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,one_one(assn)),H))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) ) ) ).
% mod_emp
tff(fact_5545_mod__emp__simp,axiom,
! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,one_one(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).
% mod_emp_simp
tff(fact_5546_preciseD_H,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P4: B,F2: assn,H: product_prod(heap_ext(product_unit),set(nat)),A10: A,F7: assn] :
( precise(A,B,R)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A3),P4)),F2)),H))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A10),P4)),F7)),H))
=> ( A3 = A10 ) ) ) ) ).
% preciseD'
tff(fact_5547_prec__frame_H,axiom,
! [A: $tType,P: fun(A,assn),X: A,F12: assn,Y: A,F23: assn,H: product_prod(heap_ext(product_unit),set(nat)),R12: assn,R22: assn] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,X)),F12)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,Y)),F23))),H))
=> ( X = Y ) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R22)),H))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R12),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,X)),F12)))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,Y)),F23)))
=> ( X = Y ) ) ) ) ) ).
% prec_frame'
tff(fact_5548_prec__frame__expl,axiom,
! [A: $tType,P: fun(A,assn),F12: assn,F23: assn,H: product_prod(heap_ext(product_unit),set(nat)),R12: assn,R22: assn,X: A,Y: A] :
( ! [X2: A,Y2: A] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,X2)),F12)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,Y2)),F23))),H))
=> ( X2 = Y2 ) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R22)),H))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R12),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,X)),F12)))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,R22),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P,Y)),F23)))
=> ( X = Y ) ) ) ) ) ).
% prec_frame_expl
tff(fact_5549_preciseD,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P4: B,F2: assn,A10: A,F7: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( precise(A,B,R)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A3),P4)),F2)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A10),P4)),F7))),H))
=> ( A3 = A10 ) ) ) ).
% preciseD
tff(fact_5550_preciseI,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
( ! [A5: A,A16: A,H3: product_prod(heap_ext(product_unit),set(nat)),P7: B,F3: assn,F8: assn] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A5),P7)),F3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A16),P7)),F8))),H3))
=> ( A5 = A16 ) )
=> precise(A,B,R) ) ).
% preciseI
tff(fact_5551_precise__def,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
( precise(A,B,R)
<=> ! [A6: A,A17: A,H4: product_prod(heap_ext(product_unit),set(nat)),P5: B,F9: assn,F10: assn] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A6),P5)),F9)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A17),P5)),F10))),H4))
=> ( A6 = A17 ) ) ) ).
% precise_def
tff(fact_5552_comm__monoid__list__set_Oset__conv__list,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A),Xs: list(B)] :
( groups4802862169904069756st_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups_monoid_F(A,F,Z2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),aa(list(B),list(B),remdups(B),Xs))) ) ) ).
% comm_monoid_list_set.set_conv_list
tff(fact_5553_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Xs: list(B),G: fun(B,A)] :
( groups4802862169904069756st_set(A,F,Z2)
=> ( pp(aa(list(B),bool,distinct(B),Xs))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups_monoid_F(A,F,Z2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) ) ) ) ).
% comm_monoid_list_set.distinct_set_conv_list
tff(fact_5554_mod__not__dist,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(assn,assn,uminus_uminus(assn),P)),H))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H))
& ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H)) ) ) ).
% mod_not_dist
tff(fact_5555_in__range__empty,axiom,
! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).
% in_range_empty
tff(fact_5556_mod__true,axiom,
! [H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,top_top(assn)),H))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).
% mod_true
tff(fact_5557_in__range__dist__union,axiom,
! [H: heap_ext(product_unit),As2: set(nat),As4: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4))))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))) ) ) ).
% in_range_dist_union
tff(fact_5558_models__in__range,axiom,
! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),H))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).
% models_in_range
tff(fact_5559_in__range__subset,axiom,
! [As2: set(nat),As4: set(nat),H: heap_ext(product_unit)] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As2),As4))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ) ).
% in_range_subset
tff(fact_5560_one__assn__raw_Ocases,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) ).
% one_assn_raw.cases
tff(fact_5561_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))))] :
~ ! [R5: array(A),X2: list(A),H3: heap_ext(product_unit),As: 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)))),R5),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))),X2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) ) ).
% snga_assn_raw.cases
tff(fact_5562_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))))] :
~ ! [R5: ref(A),X2: A,H3: heap_ext(product_unit),As: 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)))),R5),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))),X2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) ) ).
% sngr_assn_raw.cases
tff(fact_5563_times__assn__raw_Ocases,axiom,
! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q6: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H3: heap_ext(product_unit),As: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),P2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Q6),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) ).
% times_assn_raw.cases
tff(fact_5564_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ( groups7121269368397514597t_prod(B,A) = groups_comm_monoid_F(A,B,times_times(A),one_one(A)) ) ) ).
% comm_monoid_mult_class.prod_def
tff(fact_5565_wand__assnI,axiom,
! [H: heap_ext(product_unit),As2: set(nat),Q: assn,R: assn] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> ( ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As5) = bot_bot(set(nat)) )
=> ( relH(As2,H,H6)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As2)))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),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)),H6),As5)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,R),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As5)))) ) ) ) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,wand_assn(Q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ) ).
% wand_assnI
tff(fact_5566_times__assn__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),As2)))
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),As22))) ) ) ).
% times_assn_raw.simps
tff(fact_5567_times__assn__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As22))) ) ) ) ) ).
% times_assn_raw.elims(1)
tff(fact_5568_relH__dist__union,axiom,
! [As2: set(nat),As4: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4),H,H5)
<=> ( relH(As2,H,H5)
& relH(As4,H,H5) ) ) ).
% relH_dist_union
tff(fact_5569_relH__subset,axiom,
! [Bs: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),As2: set(nat)] :
( relH(Bs,H,H5)
=> ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As2),Bs))
=> relH(As2,H,H5) ) ) ).
% relH_subset
tff(fact_5570_relH__trans,axiom,
! [As2: set(nat),H12: heap_ext(product_unit),H23: heap_ext(product_unit),H32: heap_ext(product_unit)] :
( relH(As2,H12,H23)
=> ( relH(As2,H23,H32)
=> relH(As2,H12,H32) ) ) ).
% relH_trans
tff(fact_5571_relH__sym,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H,H5)
=> relH(As2,H5,H) ) ).
% relH_sym
tff(fact_5572_mod__relH,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),P: assn] :
( relH(As2,H,H5)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))) ) ) ).
% mod_relH
tff(fact_5573_relH__refl,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> relH(As2,H,H) ) ).
% relH_refl
tff(fact_5574_relH__in__rangeI_I1_J,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H,H5)
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ).
% relH_in_rangeI(1)
tff(fact_5575_relH__in__rangeI_I2_J,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H,H5)
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))) ) ).
% relH_in_rangeI(2)
tff(fact_5576_times__assn__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ? [As12: set(nat),As23: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As12)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As23))) ) ) ) ).
% times_assn_raw.elims(3)
tff(fact_5577_times__assn__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ~ ? [As13: set(nat),As24: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As13)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As24))) ) ) ) ).
% times_assn_raw.elims(2)
tff(fact_5578_times__assn__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As))))
=> ? [As12: set(nat),As23: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As12)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As23))) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
tff(fact_5579_times__assn__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As))))
=> ~ ? [As13: set(nat),As24: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As13)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As24))) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
tff(fact_5580_times__assn__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa),Xb))
<=> pp(Y) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H3),As22))) ) )
=> ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As)))) ) ) ) ) ).
% times_assn_raw.pelims(1)
tff(fact_5581_wand__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
& relH(As,H3,H6)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),As),As5)))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_5582_wand__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H7: 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,H3,H7)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As6))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) ) ).
% wand_raw.elims(2)
tff(fact_5583_wand__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
& relH(As,H3,H8)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7)))) ) ) ) ) ) ).
% wand_raw.elims(1)
tff(fact_5584_wand__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H),As2)))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As7) = bot_bot(set(nat)) )
& relH(As2,H,H8)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As2)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7)))) ) ) ) ).
% wand_raw.simps
tff(fact_5585_wand__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As))))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
& relH(As,H3,H6)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),As),As5)))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_5586_wand__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As))))
=> ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H7: 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,H3,H7)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As6))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
tff(fact_5587_wand__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa),Xb))
<=> pp(Y) )
=> ( accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
& relH(As,H3,H8)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7)))) ) ) )
=> ~ accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel,aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),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)),H3),As)))) ) ) ) ) ).
% wand_raw.pelims(1)
tff(fact_5588_uminus__assn__def,axiom,
! [P: assn] : aa(assn,assn,uminus_uminus(assn),P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,aTP_Lamp_sd(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).
% uminus_assn_def
tff(fact_5589_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,C4: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C4))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C4))
=> ( ( X2 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X2),Xa4) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C4)) = aa(set(set(B)),A,aa(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),F,Z2),groups_comm_monoid_F(A,B,F,Z2)),G),C4) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
tff(fact_5590_comm__monoid__set_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,I5: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),I5))
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I5))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I5))
=> ( ( X2 != Xa4 )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A4,X2)),aa(B,set(C),A4,Xa4)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,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_se(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F),Z2),A4),G)),I5) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
tff(fact_5591_Rep__assn__inverse,axiom,
! [X: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,X)) = X ).
% Rep_assn_inverse
tff(fact_5592_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty
tff(fact_5593_pure__assn__def,axiom,
! [B2: bool] : pure_assn(B2) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,pure_assn_raw(heap_ext(product_unit),nat,B2)) ).
% pure_assn_def
tff(fact_5594_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_5595_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z2: A,P4: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z2),P4),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty'
tff(fact_5596_Abs__assn__eqI_I2_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,H3))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Pr),H3)) )
=> ( Pr = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,P) ) ) ).
% Abs_assn_eqI(2)
tff(fact_5597_Abs__assn__eqI_I1_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,H3))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Pr),H3)) )
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,P) = Pr ) ) ).
% Abs_assn_eqI(1)
tff(fact_5598_bot__assn__def,axiom,
bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,aTP_Lamp_sf(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
% bot_assn_def
tff(fact_5599_comm__monoid__set_Ointer__restrict,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),G: fun(B,A),B3: set(B)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(set(B),fun(B,A),aa(fun(B,A),fun(set(B),fun(B,A)),aTP_Lamp_sg(A,fun(fun(B,A),fun(set(B),fun(B,A))),Z2),G),B3)),A4) ) ) ) ).
% comm_monoid_set.inter_restrict
tff(fact_5600_top__assn__def,axiom,
top_top(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,in_range) ).
% top_assn_def
tff(fact_5601_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)),bool),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sh(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).
% sup_assn_def
tff(fact_5602_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)),bool),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_si(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).
% inf_assn_def
tff(fact_5603_comm__monoid__set_Ounion__inter,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),B3)) ) ) ) ) ).
% comm_monoid_set.union_inter
tff(fact_5604_comm__monoid__set_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)))
=> ( aa(B,A,G,X2) = Z2 ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),B3)) ) ) ) ) ) ).
% comm_monoid_set.union_inter_neutral
tff(fact_5605_comm__monoid__set_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,T: set(B),S: set(B),H: fun(B,A),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),T))
=> ( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H,I3) = Z2 ) )
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),S,T)))
=> ( aa(B,A,G,I3) = Z2 ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),T)))
=> ( aa(B,A,G,X2) = aa(B,A,H,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),T) ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong
tff(fact_5606_comm__monoid__set_OInt__Diff,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),G: fun(B,A),B3: set(B)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A4) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),minus_minus(set(B),A4,B3))) ) ) ) ).
% comm_monoid_set.Int_Diff
tff(fact_5607_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_al(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A4) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% comm_monoid_set.If_cases
tff(fact_5608_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),G: fun(B,A),X: B] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A4)) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),minus_minus(set(B),A4,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_5609_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),X: B,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(member2(B,X,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A4) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),minus_minus(set(B),A4,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_5610_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),B3)) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
tff(fact_5611_comm__monoid__set_Ounion__diff2,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A4))
=> ( pp(aa(set(B),bool,finite_finite2(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),minus_minus(set(B),A4,B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),minus_minus(set(B),B3,A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% comm_monoid_set.union_diff2
tff(fact_5612_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,S: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ( pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_sj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))) ) )
& ( ~ pp(member2(B,A3,S))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_sj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),C3),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
tff(fact_5613_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)),bool),assn,abs_assn,times_assn_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q))) ).
% times_assn_def
tff(fact_5614_wand__assn__def,axiom,
! [P: assn,Q: assn] : wand_assn(P,Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,wand_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Q))) ).
% wand_assn_def
tff(fact_5615_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: fun(C,assn),H: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,ex_assn(C,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
<=> ? [X4: C] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(C,assn,R,X4)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(8)
tff(fact_5616_sngr__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),X: A] : sngr_assn(A,R3,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,sngr_assn_raw(A,R3,X)) ) ).
% sngr_assn_def
tff(fact_5617_snga__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: array(A),A3: list(A)] : snga_assn(A,R3,A3) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,snga_assn_raw(A,R3,A3)) ) ).
% snga_assn_def
tff(fact_5618_ex__assn__const,axiom,
! [A: $tType,C3: assn] : ex_assn(A,aTP_Lamp_sk(assn,fun(A,assn),C3)) = C3 ).
% ex_assn_const
tff(fact_5619_norm__assertion__simps_I17_J,axiom,
! [B: $tType,R: assn,Q: fun(B,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),ex_assn(B,Q)) = ex_assn(B,aa(fun(B,assn),fun(B,assn),aTP_Lamp_sl(assn,fun(fun(B,assn),fun(B,assn)),R),Q)) ).
% norm_assertion_simps(17)
tff(fact_5620_norm__assertion__simps_I16_J,axiom,
! [A: $tType,Q: fun(A,assn),R: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,Q)),R) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sm(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).
% norm_assertion_simps(16)
tff(fact_5621_mod__ex__dist,axiom,
! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,ex_assn(A,P)),H))
<=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(A,assn,P,X4)),H)) ) ).
% mod_ex_dist
tff(fact_5622_triv__exI,axiom,
! [A: $tType,Q: fun(A,assn),X: A] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(A,assn,Q,X)),ex_assn(A,Q))) ).
% triv_exI
tff(fact_5623_norm__assertion__simps_I21_J,axiom,
! [F6: $tType,Q: assn,P: fun(F6,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Q),ex_assn(F6,P)) = ex_assn(F6,aa(fun(F6,assn),fun(F6,assn),aTP_Lamp_sn(assn,fun(fun(F6,assn),fun(F6,assn)),Q),P)) ).
% norm_assertion_simps(21)
tff(fact_5624_norm__assertion__simps_I20_J,axiom,
! [E4: $tType,Q: fun(E4,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(E4,Q)),P) = ex_assn(E4,aa(assn,fun(E4,assn),aTP_Lamp_so(fun(E4,assn),fun(assn,fun(E4,assn)),Q),P)) ).
% norm_assertion_simps(20)
tff(fact_5625_norm__assertion__simps_I18_J,axiom,
! [C: $tType,Q: fun(C,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(C,Q)),P) = ex_assn(C,aa(assn,fun(C,assn),aTP_Lamp_sp(fun(C,assn),fun(assn,fun(C,assn)),Q),P)) ).
% norm_assertion_simps(18)
tff(fact_5626_norm__assertion__simps_I19_J,axiom,
! [D: $tType,Q: assn,P: fun(D,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Q),ex_assn(D,P)) = ex_assn(D,aa(fun(D,assn),fun(D,assn),aTP_Lamp_sq(assn,fun(fun(D,assn),fun(D,assn)),Q),P)) ).
% norm_assertion_simps(19)
tff(fact_5627_ex__one__point__gen,axiom,
! [A: $tType,P: fun(A,assn),V: A] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat)),X2: A] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(A,assn,P,X2)),H3))
=> ( X2 = V ) )
=> ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).
% ex_one_point_gen
tff(fact_5628_mod__exI,axiom,
! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
( ? [X3: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(A,assn,P,X3)),H))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,ex_assn(A,P)),H)) ) ).
% mod_exI
tff(fact_5629_mod__exE,axiom,
! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,ex_assn(A,P)),H))
=> ~ ! [X2: A] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(A,assn,P,X2)),H)) ) ).
% mod_exE
tff(fact_5630_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: fun(A,assn),X: A] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(A,assn,Q,X)))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),ex_assn(A,Q))) ) ).
% ent_ex_postI
tff(fact_5631_ent__ex__preI,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] :
( ! [X2: A] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(A,assn,P,X2)),Q))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,ex_assn(A,P)),Q)) ) ).
% ent_ex_preI
tff(fact_5632_enorm__exI_H,axiom,
! [A: $tType,Z6: fun(A,bool),P: assn,Q: fun(A,assn)] :
( ! [X2: A] :
( pp(aa(A,bool,Z6,X2))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(A,assn,Q,X2))) )
=> ( ? [X_13: A] : pp(aa(A,bool,Z6,X_13))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),ex_assn(A,Q))) ) ) ).
% enorm_exI'
tff(fact_5633_ex__distrib__star,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sm(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_5634_ex__distrib__and,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_sr(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_5635_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_ss(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_st(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).
% ex_join_or
tff(fact_5636_ex__distrib__or,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_su(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_5637_ex__assn__def,axiom,
! [A: $tType,P: fun(A,assn)] : ex_assn(A,P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,aTP_Lamp_sv(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).
% ex_assn_def
tff(fact_5638_one__assn__def,axiom,
one_one(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,one_assn_raw) ).
% one_assn_def
tff(fact_5639_one__assn__raw_Osimps,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> ( As2 = bot_bot(set(nat)) ) ) ).
% one_assn_raw.simps
tff(fact_5640_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ( As != bot_bot(set(nat)) ) ) ) ) ).
% one_assn_raw.elims(1)
tff(fact_5641_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( As = bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(3)
tff(fact_5642_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( As != bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(2)
tff(fact_5643_properI,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( ! [As: set(nat),H3: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As))) )
=> ( ! [As: set(nat),H3: heap_ext(product_unit),H6: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As)))
=> ( relH(As,H3,H6)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H6),As))) ) ) )
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P)) ) ) ).
% properI
tff(fact_5644_properD2,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat),H5: heap_ext(product_unit)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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))) ) ) ) ) ).
% properD2
tff(fact_5645_proper__def,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
<=> ! [H4: heap_ext(product_unit),H8: heap_ext(product_unit),As8: set(nat)] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As8)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As8))) )
& ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As8)))
& relH(As8,H4,H8)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As8))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H8),As8))) ) ) ) ).
% proper_def
tff(fact_5646_bool__assn__proper_I4_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,Q))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sw(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q))) ) ) ).
% bool_assn_proper(4)
tff(fact_5647_bool__assn__proper_I3_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,Q))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sx(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q))) ) ) ).
% bool_assn_proper(3)
tff(fact_5648_bool__assn__proper_I2_J,axiom,
pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aTP_Lamp_sf(product_prod(heap_ext(product_unit),set(nat)),bool))) ).
% bool_assn_proper(2)
tff(fact_5649_bool__assn__proper_I1_J,axiom,
pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,in_range)) ).
% bool_assn_proper(1)
tff(fact_5650_ex__assn__proper,axiom,
! [A: $tType,P: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool))] :
( ! [X2: A] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool),P,X2)))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aTP_Lamp_sy(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P))) ) ).
% ex_assn_proper
tff(fact_5651_snga__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: array(A),X: list(A)] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,snga_assn_raw(A,R3,X))) ) ).
% snga_assn_proper
tff(fact_5652_sngr__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),X: A] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,sngr_assn_raw(A,R3,X))) ) ).
% sngr_assn_proper
tff(fact_5653_times__assn__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,Q))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,times_assn_raw(P,Q))) ) ) ).
% times_assn_proper
tff(fact_5654_wand__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,wand_raw(P,Q))) ).
% wand_proper
tff(fact_5655_one__assn__proper,axiom,
pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,one_assn_raw)) ).
% one_assn_proper
tff(fact_5656_bool__assn__proper_I5_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aTP_Lamp_sz(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P))) ) ).
% bool_assn_proper(5)
tff(fact_5657_pure__assn__proper,axiom,
! [B2: bool] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,pure_assn_raw(heap_ext(product_unit),nat,B2))) ).
% pure_assn_proper
tff(fact_5658_Abs__assn__inject,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),X,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> ( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> ( ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_assn_inject
tff(fact_5659_Abs__assn__induct,axiom,
! [P: fun(assn,bool),X: assn] :
( ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y2,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> pp(aa(assn,bool,P,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,Y2))) )
=> pp(aa(assn,bool,P,X)) ) ).
% Abs_assn_induct
tff(fact_5660_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( ( X = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,Y2) )
=> ~ pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y2,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper))) ) ).
% Abs_assn_cases
tff(fact_5661_Rep__assn__induct,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),bool),P: fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool)] :
( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> ( ! [X2: assn] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,P,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,X2)))
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,P,Y)) ) ) ).
% Rep_assn_induct
tff(fact_5662_Rep__assn__cases,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> ~ ! [X2: assn] : Y != aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,X2) ) ).
% Rep_assn_cases
tff(fact_5663_Rep__assn,axiom,
! [X: assn] : pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,X),collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper))) ).
% Rep_assn
tff(fact_5664_properD1,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))) ) ) ).
% properD1
tff(fact_5665_Abs__assn__inverse,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( pp(member2(fun(product_prod(heap_ext(product_unit),set(nat)),bool),Y,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)))
=> ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,Y)) = Y ) ) ).
% Abs_assn_inverse
tff(fact_5666_proper__iff,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P))
=> ( relH(As2,H,H5)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2)))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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))) ) ) ) ) ).
% proper_iff
tff(fact_5667_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_5668_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( order_AboveS(A,R3,B3) != bot_bot(set(A)) )
=> ( pp(member2(A,B2,B3))
=> ( ( bNF_Wellorder_wo_suc(A,R3,B3) != B2 )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,B3)),R3)) ) ) ) ) ) ).
% wo_rel.suc_greater
tff(fact_5669_monoid__list_ONil,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),nil(A)) = Z2 ) ) ).
% monoid_list.Nil
tff(fact_5670_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( comm_monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% comm_monoid.comm_neutral
tff(fact_5671_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% semilattice_neutr.axioms(2)
tff(fact_5672_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_5673_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_5674_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_5675_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> comm_monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.comm_monoid_axioms
tff(fact_5676_wo__rel_Osuc__inField,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( order_AboveS(A,R3,B3) != bot_bot(set(A)) )
=> pp(member2(A,bNF_Wellorder_wo_suc(A,R3,B3),field2(A,R3))) ) ) ) ).
% wo_rel.suc_inField
tff(fact_5677_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R3: set(product_prod(A,A)),B3: set(A)] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R3)))
=> ( ( order_AboveS(A,R3,B3) != bot_bot(set(A)) )
=> pp(member2(A,bNF_Wellorder_wo_suc(A,R3,B3),order_AboveS(A,R3,B3))) ) ) ) ).
% wo_rel.suc_AboveS
tff(fact_5678_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set(product_prod(A,A)),A4: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R3)
=> ( order_ofilter(A,R3,A4)
=> ( ( order_AboveS(A,R3,A4) != bot_bot(set(A)) )
=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,A4)),R3))
=> ( ( B2 != bNF_Wellorder_wo_suc(A,R3,A4) )
=> pp(member2(A,B2,A4)) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
tff(fact_5679_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_5680_AboveS__disjoint,axiom,
! [A: $tType,A4: 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)),A4),order_AboveS(A,R3,A4)) = bot_bot(set(A)) ).
% AboveS_disjoint
tff(fact_5681_monoid__list_OCons,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,X: A,Xs: list(A)] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),F,X),aa(list(A),A,groups_monoid_F(A,F,Z2),Xs)) ) ) ).
% monoid_list.Cons
tff(fact_5682_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_max(A),aTP_Lamp_ao(A,fun(A,bool)),aTP_Lamp_ta(A,fun(A,bool))) ) ).
% max.semilattice_order_axioms
tff(fact_5683_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_5684_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice_order(A,sup_sup(A),aTP_Lamp_tb(A,fun(A,bool)),aTP_Lamp_tc(A,fun(A,bool))) ) ).
% sup.semilattice_order_axioms
tff(fact_5685_semilattice__order_Omono,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C3: A,B2: A,D3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),aa(A,A,aa(A,fun(A,A),F,C3),D3))) ) ) ) ).
% semilattice_order.mono
tff(fact_5686_semilattice__order_OorderE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.orderE
tff(fact_5687_semilattice__order_OorderI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).
% semilattice_order.orderI
tff(fact_5688_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb1
tff(fact_5689_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb2
tff(fact_5690_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb3
tff(fact_5691_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb4
tff(fact_5692_semilattice__order_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3)) ) ) ) ).
% semilattice_order.boundedE
tff(fact_5693_semilattice__order_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3))) ) ) ) ).
% semilattice_order.boundedI
tff(fact_5694_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.order_iff
tff(fact_5695_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),A3)) ) ).
% semilattice_order.cobounded1
tff(fact_5696_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2)) ) ).
% semilattice_order.cobounded2
tff(fact_5697_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb_iff1
tff(fact_5698_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb_iff2
tff(fact_5699_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3)) ) ) ) ).
% semilattice_order.bounded_iff
tff(fact_5700_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3)) ) ) ).
% semilattice_order.coboundedI1
tff(fact_5701_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C3: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3)) ) ) ).
% semilattice_order.coboundedI2
tff(fact_5702_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).
% semilattice_order.strict_boundedE
tff(fact_5703_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
tff(fact_5704_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3)) ) ) ).
% semilattice_order.strict_coboundedI1
tff(fact_5705_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C3: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3)) ) ) ).
% semilattice_order.strict_coboundedI2
tff(fact_5706_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ).
% semilattice_neutr_order.axioms(2)
tff(fact_5707_semilattice__neutr__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
<=> ( semilattice_neutr(A,F,Z2)
& semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_neutr_order_def
tff(fact_5708_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_neutr(A,F,Z2)
=> ( semilattice_order(A,F,Less_eq,Less)
=> semila1105856199041335345_order(A,F,Z2,Less_eq,Less) ) ) ).
% semilattice_neutr_order.intro
tff(fact_5709_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice_order(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).
% inf.semilattice_order_axioms
tff(fact_5710_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).
% min.semilattice_order_axioms
tff(fact_5711_entt__def__true,axiom,
! [P: assn,Q: assn] :
( entailst(P,Q)
<=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))) ) ).
% entt_def_true
tff(fact_5712_entailst__def,axiom,
! [A4: assn,B3: assn] :
( entailst(A4,B3)
<=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),top_top(assn)))) ) ).
% entailst_def
tff(fact_5713_enttI__true,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))))
=> entailst(P,Q) ) ).
% enttI_true
tff(fact_5714_entt__refl,axiom,
! [A4: assn] : entailst(A4,A4) ).
% entt_refl
tff(fact_5715_entt__true,axiom,
! [A4: assn] : entailst(A4,top_top(assn)) ).
% entt_true
tff(fact_5716_entt__emp,axiom,
! [A4: assn] : entailst(A4,one_one(assn)) ).
% entt_emp
tff(fact_5717_entt__star__true__simp_I2_J,axiom,
! [A4: assn,B3: assn] :
( entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),top_top(assn)),B3)
<=> entailst(A4,B3) ) ).
% entt_star_true_simp(2)
tff(fact_5718_entt__star__true__simp_I1_J,axiom,
! [A4: assn,B3: assn] :
( entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),top_top(assn)))
<=> entailst(A4,B3) ) ).
% entt_star_true_simp(1)
tff(fact_5719_entt__disjI2__direct,axiom,
! [B3: assn,A4: assn] : entailst(B3,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)) ).
% entt_disjI2_direct
tff(fact_5720_entt__disjI1__direct,axiom,
! [A4: assn,B3: assn] : entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)) ).
% entt_disjI1_direct
tff(fact_5721_entt__disjI2_H,axiom,
! [A4: assn,C4: assn,B3: assn] :
( entailst(A4,C4)
=> entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C4)) ) ).
% entt_disjI2'
tff(fact_5722_entt__disjI1_H,axiom,
! [A4: assn,B3: assn,C4: assn] :
( entailst(A4,B3)
=> entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C4)) ) ).
% entt_disjI1'
tff(fact_5723_entt__disjD2,axiom,
! [A4: assn,B3: assn,C4: assn] :
( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3),C4)
=> entailst(B3,C4) ) ).
% entt_disjD2
tff(fact_5724_entt__disjD1,axiom,
! [A4: assn,B3: assn,C4: assn] :
( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3),C4)
=> entailst(A4,C4) ) ).
% entt_disjD1
tff(fact_5725_entt__disjE,axiom,
! [A4: assn,M: assn,B3: assn] :
( entailst(A4,M)
=> ( entailst(B3,M)
=> entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3),M) ) ) ).
% entt_disjE
tff(fact_5726_entt__frame__fwd,axiom,
! [P: assn,Q: assn,A4: assn,F2: assn,B3: assn] :
( entailst(P,Q)
=> ( entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F2))
=> ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F2),B3)
=> entailst(A4,B3) ) ) ) ).
% entt_frame_fwd
tff(fact_5727_entt__star__mono,axiom,
! [A4: assn,B3: assn,C4: assn,D2: assn] :
( entailst(A4,B3)
=> ( entailst(C4,D2)
=> entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),C4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),D2)) ) ) ).
% entt_star_mono
tff(fact_5728_entt__fr__refl,axiom,
! [F2: assn,F7: assn,A4: assn] :
( entailst(F2,F7)
=> entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),F2),A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),F7),A4)) ) ).
% entt_fr_refl
tff(fact_5729_entt__fr__drop,axiom,
! [F2: assn,F7: assn,A4: assn] :
( entailst(F2,F7)
=> entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),F2),A4),F7) ) ).
% entt_fr_drop
tff(fact_5730_entt__trans,axiom,
! [A4: assn,B3: assn,C4: assn] :
( entailst(A4,B3)
=> ( entailst(B3,C4)
=> entailst(A4,C4) ) ) ).
% entt_trans
tff(fact_5731_ent__imp__entt,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
=> entailst(P,Q) ) ).
% ent_imp_entt
tff(fact_5732_enttD,axiom,
! [A4: assn,B3: assn] :
( entailst(A4,B3)
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),top_top(assn)))) ) ).
% enttD
tff(fact_5733_enttI,axiom,
! [A4: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),top_top(assn))))
=> entailst(A4,B3) ) ).
% enttI
tff(fact_5734_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Xs: list(A),A3: B] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F),Xs),A3) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_td(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),A3),Xs) ) ) ).
% comp_fun_commute.foldr_conv_foldl
tff(fact_5735_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_5736_in__range_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ! [X2: nat] :
( pp(member2(nat,X2,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),lim(product_unit,H3))) ) ) ) ).
% in_range.elims(3)
tff(fact_5737_comp__fun__commute__insort,axiom,
! [A: $tType] :
( linorder(A)
=> finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_az(A,A))) ) ).
% comp_fun_commute_insort
tff(fact_5738_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),A3: A,B2: B,Xs: list(A)] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(B,B,aa(A,fun(B,B),F,A3),aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_td(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),B2),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),aTP_Lamp_td(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),aa(B,B,aa(A,fun(B,B),F,A3),B2)),Xs) ) ) ).
% comp_fun_commute.foldl_f_commute
tff(fact_5739_in__range_Osimps,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> ! [X4: nat] :
( pp(member2(nat,X4,As2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H))) ) ) ).
% in_range.simps
tff(fact_5740_in__range_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ! [X4: nat] :
( pp(member2(nat,X4,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H3))) ) ) ) ) ).
% in_range.elims(1)
tff(fact_5741_in__range_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ~ ! [X3: nat] :
( pp(member2(nat,X3,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H3))) ) ) ) ).
% in_range.elims(2)
tff(fact_5742_czeroE,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A))] :
( pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)))
=> ( field2(A,R3) = bot_bot(set(A)) ) ) ).
% czeroE
tff(fact_5743_type__definition__assn,axiom,
type_definition(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,abs_assn,collect(fun(product_prod(heap_ext(product_unit),set(nat)),bool),proper)) ).
% type_definition_assn
tff(fact_5744_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))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) ) ) ) ).
% sngr_assn_raw.elims(3)
tff(fact_5745_relH__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: ref(A)] :
( relH(As2,H,H5)
=> ( pp(member2(nat,addr_of_ref(A,R3),As2))
=> ( ref_get(A,H,R3) = ref_get(A,H5,R3) ) ) ) ) ).
% relH_ref
tff(fact_5746_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),X: A,H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,R3,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> ( ( ref_get(A,H,R3) = X )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,R3)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R3)),lim(product_unit,H))) ) ) ) ).
% sngr_assn_raw.simps
tff(fact_5747_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: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
tff(fact_5748_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))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) ) ) ) ).
% sngr_assn_raw.elims(2)
tff(fact_5749_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: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
<=> pp(Y) )
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) )
=> ~ 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)),H3),As)))) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
tff(fact_5750_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))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))))
=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
tff(fact_5751_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))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa),Xb))
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))))
=> ( ( ref_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3))) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
tff(fact_5752_relH__set__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),As2: set(nat),H: heap_ext(product_unit),X: A] :
( ~ pp(member2(nat,addr_of_ref(A,R3),As2))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> relH(As2,H,ref_set(A,R3,X,H)) ) ) ) ).
% relH_set_ref
tff(fact_5753_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ( ~ pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3) )
=> ( field2(A,R3) != bot_bot(set(A)) ) ) ).
% Cnotzero_imp_not_empty
tff(fact_5754_czeroI,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> ( ( field2(A,R3) = bot_bot(set(A)) )
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))) ) ) ).
% czeroI
tff(fact_5755_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R3),R3)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
<=> ! [X4: A] :
( pp(member2(A,X4,field2(A,R3)))
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_qp(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% Card_order_iff_Restr_underS
tff(fact_5756_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q2: set(product_prod(C,C))] :
( pp(member2(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))),P24),R2),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q2),Q2)
=> ( ( ( field2(A,P24) = bot_bot(set(A)) )
=> ( field2(B,R2) = bot_bot(set(B)) ) )
=> pp(member2(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,Q2,P24)),bNF_Cardinal_cexp(C,B,Q2,R2)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).
% cexp_mono2'
tff(fact_5757_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),P4: set(product_prod(B,B))] :
( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> ( ( field2(B,P4) != bot_bot(set(B)) )
=> ( pp(member2(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))),P4),R3),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member2(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R3),aTP_Lamp_te(set(product_prod(B,B)),fun(A,set(B)),P4)))),R3),bNF_Wellorder_ordIso(product_prod(A,B),A)))
& pp(member2(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P4),aTP_Lamp_tf(set(product_prod(A,A)),fun(B,set(A)),R3)))),R3),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).
% Card_order_Times_infinite
tff(fact_5758_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> pp(member2(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(bool,top_top(set(bool)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),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(bool,A))) ) ).
% card_of_bool
tff(fact_5759_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : pp(member2(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_5760_card__of__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_empty2
tff(fact_5761_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B3,aTP_Lamp_an(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).
% card_of_Times_infinite_simps(4)
tff(fact_5762_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member2(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,B3,aTP_Lamp_an(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ).
% card_of_Times_infinite_simps(3)
tff(fact_5763_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).
% card_of_Times_infinite_simps(2)
tff(fact_5764_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member2(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,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).
% card_of_Times_infinite_simps(1)
tff(fact_5765_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member2(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,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
& pp(member2(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,B3,aTP_Lamp_an(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ).
% card_of_Times_infinite
tff(fact_5766_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set(B)] : pp(member2(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,A4)),bNF_Wellorder_ordLeq(A,B))) ).
% card_of_empty
tff(fact_5767_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_empty3
tff(fact_5768_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B3,aTP_Lamp_an(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).
% card_of_Times1
tff(fact_5769_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> pp(member2(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,B3)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).
% card_of_Times2
tff(fact_5770_czero__def,axiom,
! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).
% czero_def
tff(fact_5771_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set(A),B3: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> ( ? [G3: fun(B,A)] : aa(set(B),set(A),image2(B,A,G3),B3) = A4
<=> pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Wellorder_ordLeq(A,B))) ) ) ).
% card_of_ordLeq2
tff(fact_5772_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set(A),B2: B] :
( ( A4 != bot_bot(set(A)) )
=> pp(member2(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),B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))) ) ).
% card_of_singl_ordLeq
tff(fact_5773_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B3: set(A),A4: set(B)] :
( ( B3 != bot_bot(set(A)) )
=> ( ~ ? [F5: fun(B,A)] : aa(set(B),set(A),image2(B,A,F5),A4) = B3
<=> pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(A,B3)),bNF_We4044943003108391690rdLess(B,A))) ) ) ).
% card_of_ordLess2
tff(fact_5774_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),B3: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> ( ( B3 != bot_bot(set(B)) )
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R3),aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).
% Card_order_Times1
tff(fact_5775_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A4: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> ( ( A4 != bot_bot(set(B)) )
=> pp(member2(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A4,aTP_Lamp_tf(set(product_prod(A,A)),fun(B,set(A)),R3)))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).
% Card_order_Times2
tff(fact_5776_Card__order__empty,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> pp(member2(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3),bNF_Wellorder_ordLeq(B,A))) ) ).
% Card_order_empty
tff(fact_5777_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member2(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,A4)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)))
<=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_ordIso_czero_iff_empty
tff(fact_5778_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),B2: B] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3)
=> ( ( field2(A,R3) != bot_bot(set(A)) )
=> pp(member2(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),B2),bot_bot(set(B))))),R3),bNF_Wellorder_ordLeq(B,A))) ) ) ).
% Card_order_singl_ordLeq
tff(fact_5779_card__of__empty1,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
( ( order_well_order_on(A,field2(A,R3),R3)
| bNF_Ca8970107618336181345der_on(A,field2(A,R3),R3) )
=> pp(member2(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3),bNF_Wellorder_ordLeq(B,A))) ) ).
% card_of_empty1
tff(fact_5780_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set(product_prod(A,A)),R1: set(product_prod(B,B)),P24: set(product_prod(C,C)),R2: set(product_prod(D,D))] :
( pp(member2(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)))
=> ( pp(member2(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))),P24),R2),bNF_Wellorder_ordLeq(C,D)))
=> ( ( ( field2(C,P24) = bot_bot(set(C)) )
=> ( field2(D,R2) = bot_bot(set(D)) ) )
=> pp(member2(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,P24)),bNF_Cardinal_cexp(B,D,R1,R2)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ).
% cexp_mono'
tff(fact_5781_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B3: set(B)] :
( ( ( A1 != A22 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A4)) )
=> ( pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(B,B3)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member2(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,A4,B3))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).
% card_of_Plus_Times_aux
tff(fact_5782_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B1: B,B22: B,B3: set(B)] :
( ( ( A1 != A22 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A22),bot_bot(set(A))))),A4)) )
=> ( ( ( B1 != B22 )
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B1),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B22),bot_bot(set(B))))),B3)) )
=> pp(member2(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,A4,B3))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_me(set(B),fun(A,set(B)),B3)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).
% card_of_Plus_Times
tff(fact_5783_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))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) ) ) ) ).
% snga_assn_raw.elims(3)
tff(fact_5784_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] : pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A4))),bNF_Wellorder_ordIso(A,sum_sum(B,A)))) ).
% card_of_Plus_empty2
tff(fact_5785_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A4: set(A)] : pp(member2(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,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B)))) ).
% card_of_Plus_empty1
tff(fact_5786_relH__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A)] :
( relH(As2,H,H5)
=> ( pp(member2(nat,addr_of_array(A,R3),As2))
=> ( array_get(A,H,R3) = array_get(A,H5,R3) ) ) ) ) ).
% relH_array
tff(fact_5787_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: array(A),X: list(A),H: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,R3,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> ( ( array_get(A,H,R3) = X )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,R3)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,R3)),lim(product_unit,H))) ) ) ) ).
% snga_assn_raw.simps
tff(fact_5788_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: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
<=> pp(Y) )
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( pp(Y)
<=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
tff(fact_5789_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))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) ) ) ) ).
% snga_assn_raw.elims(2)
tff(fact_5790_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B)] :
( ( sum_Plus(A,B,A4,B3) = bot_bot(set(sum_sum(A,B))) )
<=> ( ( A4 = bot_bot(set(A)) )
& ( B3 = bot_bot(set(B)) ) ) ) ).
% Plus_eq_empty_conv
tff(fact_5791_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: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
<=> pp(Y) )
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) )
=> ~ 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)),H3),As)))) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
tff(fact_5792_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))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))))
=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
tff(fact_5793_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))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa),Xb))
=> ( 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)))
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))))
=> ( ( array_get(A,H3,X) = Xa )
& ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3))) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
tff(fact_5794_relH__set__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: array(A),As2: set(nat),H: heap_ext(product_unit),X: list(A)] :
( ~ pp(member2(nat,addr_of_array(A,R3),As2))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> relH(As2,H,array_set(A,R3,X,H)) ) ) ) ).
% relH_set_array
tff(fact_5795_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: bool,Xa: product_prod(A,set(B)),Y: bool] :
( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
<=> pp(Y) )
=> ( accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa))
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ( ( As = bot_bot(set(B)) )
& pp(X) ) )
=> ~ accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As))) ) ) ) ) ).
% pure_assn_raw.pelims(1)
tff(fact_5796_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: bool,Xa: product_prod(A,set(B))] :
( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
=> ( accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa))
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ( accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As)))
=> ~ ( ( As = bot_bot(set(B)) )
& pp(X) ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
tff(fact_5797_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: bool,Xa: product_prod(A,set(B))] :
( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa))
=> ( accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa))
=> ~ ! [H3: A,As: 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)),H3),As) )
=> ( accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As)))
=> ( ( As = bot_bot(set(B)) )
& pp(X) ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
tff(fact_5798_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_5799_list__all__zip__alt,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),As2: list(A),Bs: list(B)] :
( list_all_zip(A,B,P,As2,Bs)
<=> ( ( aa(list(A),nat,size_size(list(A)),As2) = aa(list(B),nat,size_size(list(B)),Bs) )
& ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),As2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,As2),I2)),aa(nat,B,nth(B,Bs),I2))) ) ) ) ).
% list_all_zip_alt
tff(fact_5800_lazI,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),I3)),aa(nat,B,nth(B,B2),I3))) )
=> list_all_zip(A,B,P,A3,B2) ) ) ).
% lazI
tff(fact_5801_laz__conj,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( list_all_zip(A,B,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_tg(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q),A3,B2)
<=> ( list_all_zip(A,B,P,A3,B2)
& list_all_zip(A,B,Q,A3,B2) ) ) ).
% laz_conj
tff(fact_5802_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: fun(B,bool),A4: list(A),B3: list(B)] :
( list_all_zip(A,B,aTP_Lamp_th(fun(B,bool),fun(A,fun(B,bool)),P),A4,B3)
<=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
& ! [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),B3)))
=> pp(aa(B,bool,P,X4)) ) ) ) ).
% laz_weak_Pb
tff(fact_5803_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),A4: list(A),B3: list(B)] :
( list_all_zip(A,B,aTP_Lamp_ti(fun(A,bool),fun(A,fun(B,bool)),P),A4,B3)
<=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),A4)))
=> pp(aa(A,bool,P,X4)) ) ) ) ).
% laz_weak_Pa
tff(fact_5804_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As2: list(A),F: fun(C,B),Bs: list(C)] :
( list_all_zip(A,B,P,As2,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F),Bs))
<=> list_all_zip(A,C,aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_nw(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F),As2,Bs) ) ).
% list_all_zip_map2
tff(fact_5805_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F: fun(C,A),As2: list(C),Bs: list(B)] :
( list_all_zip(A,B,P,aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F),As2),Bs)
<=> list_all_zip(C,B,aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_nx(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F),As2,Bs) ) ).
% list_all_zip_map1
tff(fact_5806_laz__len,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( list_all_zip(A,B,P,A3,B2)
=> ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) ) ) ).
% laz_len
tff(fact_5807_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: fun(A,fun(B,fun(C,bool))),A4: list(A),B3: list(B)] :
( list_all_zip(A,B,aTP_Lamp_tj(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,bool)),P),A4,B3)
=> ~ ! [C8: list(C)] :
( list_all_zip(A,C,aTP_Lamp_tk(fun(A,fun(B,fun(C,bool))),fun(A,fun(C,bool)),P),A4,C8)
=> ~ list_all_zip(B,C,aTP_Lamp_tl(fun(A,fun(B,fun(C,bool))),fun(B,fun(C,bool)),P),B3,C8) ) ) ).
% laz_swap_ex
tff(fact_5808_laz__eq,axiom,
! [A: $tType,A3: list(A),B2: list(A)] :
( list_all_zip(A,A,fequal(A),A3,B2)
<=> ( A3 = B2 ) ) ).
% laz_eq
tff(fact_5809_list__all__zip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : list_all_zip(A,B,P,nil(A),nil(B)) ).
% list_all_zip.simps(1)
tff(fact_5810_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: A,As2: list(A),B2: B,Bs: list(B)] :
( list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2))
& list_all_zip(A,B,P,As2,Bs) ) ) ).
% list_all_zip.simps(2)
tff(fact_5811_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V: A,Va: list(A)] : ~ list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(B)) ).
% list_all_zip.simps(3)
tff(fact_5812_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V: B,Va: 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),Va)) ).
% list_all_zip.simps(4)
tff(fact_5813_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B),Y: bool] :
( ( list_all_zip(A,B,X,Xa,Xb)
<=> pp(Y) )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ pp(Y) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( pp(Y)
<=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) ) ) )
=> ( ( ? [V2: A,Va2: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)
=> ( ( Xb = nil(B) )
=> pp(Y) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V2: B,Va2: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)
=> pp(Y) ) ) ) ) ) ) ).
% list_all_zip.elims(1)
tff(fact_5814_list__all__zip_Oelims_I2_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X,Xa,Xb)
=> ( ( ( Xa = nil(A) )
=> ( Xb != nil(B) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) ) ) ) ) ).
% list_all_zip.elims(2)
tff(fact_5815_list__all__zip_Oelims_I3_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X,Xa,Xb)
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) ) )
=> ( ( ? [V2: A,Va2: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)
=> ( Xb != nil(B) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va2: list(B)] : Xb != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) ) ) ) ) ).
% list_all_zip.elims(3)
tff(fact_5816_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X,Xa,Xb)
=> ( accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) ) ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xb = nil(B) )
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),V2),Va2)),nil(B)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) )
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),V2),Va2)))) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
tff(fact_5817_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X,Xa,Xb)
=> ( accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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) )
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))))
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
tff(fact_5818_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B),Y: bool] :
( ( list_all_zip(A,B,X,Xa,Xb)
<=> pp(Y) )
=> ( accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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) )
=> ( pp(Y)
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B4: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
=> ( ( pp(Y)
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A5),B4))
& list_all_zip(A,B,X,As,Bs2) ) )
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))) ) ) )
=> ( ! [V2: A,Va2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) )
=> ( ( Xb = nil(B) )
=> ( ~ pp(Y)
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),V2),Va2)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2) )
=> ( ~ pp(Y)
=> ~ accp(product_prod(fun(A,fun(B,bool)),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,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),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),V2),Va2)))) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
tff(fact_5819_lexord__asymmetric,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: list(A),B2: list(A)] :
( asym(A,R)
=> ( pp(member2(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)),A3),B2),lexord(A,R)))
=> ~ pp(member2(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)),B2),A3),lexord(A,R))) ) ) ).
% lexord_asymmetric
tff(fact_5820_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> group_axioms(A,F,Z2,Inverse) ) ).
% group.axioms(2)
tff(fact_5821_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top.axioms(2)
tff(fact_5822_asym__lenlex,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lenlex(A,R)) ) ).
% asym_lenlex
tff(fact_5823_group__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A5)),A5) = Z2
=> group_axioms(A,F,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_5824_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),Top))
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top_axioms.intro
tff(fact_5825_group__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F,Z2,Inverse)
<=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
& ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A6)),A6) = Z2 ) ) ).
% group_axioms_def
tff(fact_5826_ordering__top__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
( ordering_top_axioms(A,Less_eq,Top)
<=> ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),Top)) ) ).
% ordering_top_axioms_def
tff(fact_5827_lexord__asym,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lexord(A,R)) ) ).
% lexord_asym
tff(fact_5828_asym__lex,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lex(A,R)) ) ).
% asym_lex
tff(fact_5829_ordering__top__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),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_5830_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),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_5831_group__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
<=> ( semigroup(A,F)
& group_axioms(A,F,Z2,Inverse) ) ) ).
% group_def
tff(fact_5832_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semigroup(A,sup_sup(A)) ) ).
% sup.semigroup_axioms
tff(fact_5833_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semigroup(A,inf_inf(A)) ) ).
% inf.semigroup_axioms
tff(fact_5834_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering(A,Less_eq,Less) ) ).
% ordering_top.axioms(1)
tff(fact_5835_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> semigroup(A,F) ) ).
% group.axioms(1)
tff(fact_5836_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_5837_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_min(A)) ) ).
% min.semigroup_axioms
tff(fact_5838_ordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
=> ordering(A,Less_eq,Less) ) ).
% ordering_dualI
tff(fact_5839_ordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
| ( A5 = B4 ) ) )
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A5)) )
=> ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),A5))
=> ( ! [A5: A,B4: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),C2)) ) )
=> ordering(A,Less_eq,Less) ) ) ) ) ).
% ordering_strictI
tff(fact_5840_semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
<=> ! [A6: A,B5: A,C6: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A6),B5)),C6) = aa(A,A,aa(A,fun(A,A),F,A6),aa(A,A,aa(A,fun(A,A),F,B5),C6)) ) ).
% semigroup_def
tff(fact_5841_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 != B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2)) ) ) ) ).
% ordering.not_eq_order_implies_strict
tff(fact_5842_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
tff(fact_5843_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
tff(fact_5844_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
tff(fact_5845_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
tff(fact_5846_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 = B2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).
% ordering.eq_iff
tff(fact_5847_semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B4)),C2) = aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B4),C2))
=> semigroup(A,F) ) ).
% semigroup.intro
tff(fact_5848_semigroup_Oassoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
( semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)) ) ) ).
% semigroup.assoc
tff(fact_5849_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_5850_append_Osemigroup__axioms,axiom,
! [A: $tType] : semigroup(list(A),append(A)) ).
% append.semigroup_axioms
tff(fact_5851_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_max(A)) ) ).
% max.semigroup_axioms
tff(fact_5852_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> semigroup(A,F) ) ).
% monoid.axioms(1)
tff(fact_5853_order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.ordering_axioms
tff(fact_5854_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,aTP_Lamp_tm(A,fun(A,bool)),aTP_Lamp_tn(A,fun(A,bool))) ) ).
% dual_order.ordering_axioms
tff(fact_5855_group_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( semigroup(A,F)
=> ( group_axioms(A,F,Z2,Inverse)
=> group(A,F,Z2,Inverse) ) ) ).
% group.intro
tff(fact_5856_monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
<=> ( semigroup(A,F)
& monoid_axioms(A,F,Z2) ) ) ).
% monoid_def
tff(fact_5857_monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semigroup(A,F)
=> ( monoid_axioms(A,F,Z2)
=> monoid(A,F,Z2) ) ) ).
% monoid.intro
tff(fact_5858_ID_Opred__set,axiom,
! [A: $tType,P: fun(A,bool),X3: A] :
( pp(aa(A,bool,bNF_id_bnf(fun(A,bool),P),X3))
<=> ! [Xa2: A] :
( pp(member2(A,Xa2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))))
=> pp(aa(A,bool,P,Xa2)) ) ) ).
% ID.pred_set
tff(fact_5859_monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5
=> monoid_axioms(A,F,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_5860_monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F,Z2)
<=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
& ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6 ) ) ).
% monoid_axioms_def
tff(fact_5861_ID_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,bool),X: A,Pa: fun(A,bool)] :
( pp(aa(A,bool,bNF_id_bnf(fun(A,bool),P),X))
=> ( ! [Z4: A] :
( pp(member2(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
=> ( pp(aa(A,bool,P,Z4))
=> pp(aa(A,bool,Pa,Z4)) ) )
=> pp(aa(A,bool,bNF_id_bnf(fun(A,bool),Pa),X)) ) ) ).
% ID.pred_mono_strong
tff(fact_5862_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya: A,P: fun(A,bool),Pa: fun(A,bool)] :
( ( X = Ya )
=> ( ! [Z4: A] :
( pp(member2(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))))
=> ( pp(aa(A,bool,P,Z4))
<=> pp(aa(A,bool,Pa,Z4)) ) )
=> ( pp(aa(A,bool,bNF_id_bnf(fun(A,bool),P),X))
<=> pp(aa(A,bool,bNF_id_bnf(fun(A,bool),Pa),Ya)) ) ) ) ).
% ID.pred_cong
tff(fact_5863_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> monoid_axioms(A,F,Z2) ) ).
% monoid.axioms(2)
tff(fact_5864_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),L: list(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),hilbert_inv_into(A,B,top_top(set(A)),F)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),L)) = L ) ) ).
% inj_map_inv_f
tff(fact_5865_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
=> ordering_axioms(A,Less_eq,Less) ) ).
% ordering.axioms(2)
tff(fact_5866_nth__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,Xs: list(A),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> hoare_hoare_triple(A,snga_assn(A,A3,Xs),array_nth(A,A3,I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_to(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A3)) ) ) ).
% nth_rule
tff(fact_5867_false__rule,axiom,
! [A: $tType,C3: heap_Time_Heap(A),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C3,Q) ).
% false_rule
tff(fact_5868_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B2: bool,F: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2)),F,Q)
<=> ( pp(B2)
=> hoare_hoare_triple(A,P,F,Q) ) ) ).
% norm_pre_pure_iff
tff(fact_5869_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B2: bool,F: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,pure_assn(B2),F,Q)
<=> ( pp(B2)
=> hoare_hoare_triple(A,one_one(assn),F,Q) ) ) ).
% norm_pre_pure_iff_sng
tff(fact_5870_case__list__rule,axiom,
! [A: $tType,B: $tType,L: list(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fc: fun(A,fun(list(A),heap_Time_Heap(B)))] :
( ( ( L = nil(A) )
=> hoare_hoare_triple(B,P,Fn,Q) )
=> ( ! [X2: A,Xs2: list(A)] :
( ( L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
=> hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),aa(A,fun(list(A),heap_Time_Heap(B)),Fc,X2),Xs2),Q) )
=> hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),aa(fun(A,fun(list(A),heap_Time_Heap(B))),fun(list(A),heap_Time_Heap(B)),aa(heap_Time_Heap(B),fun(fun(A,fun(list(A),heap_Time_Heap(B))),fun(list(A),heap_Time_Heap(B))),case_list(heap_Time_Heap(B),A),Fn),Fc),L),Q) ) ) ).
% case_list_rule
tff(fact_5871_frame__rule__left,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
( hoare_hoare_triple(A,P,C3,Q)
=> hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),P),C3,aa(assn,fun(A,assn),aTP_Lamp_tp(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).
% frame_rule_left
tff(fact_5872_frame__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
( hoare_hoare_triple(A,P,C3,Q)
=> hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R),C3,aa(assn,fun(A,assn),aTP_Lamp_sm(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).
% frame_rule
tff(fact_5873_is__hoare__triple,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,C3,Q)
=> hoare_hoare_triple(A,P,C3,Q) ) ).
% is_hoare_triple
tff(fact_5874_ordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering_axioms(A,Less_eq,Less)
<=> ( ! [A6: A,B5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B5))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B5))
& ( A6 != B5 ) ) )
& ! [A6: A,B5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),A6))
=> ( A6 = B5 ) ) ) ) ) ).
% ordering_axioms_def
tff(fact_5875_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
& ( A5 != B4 ) ) )
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A5))
=> ( A5 = B4 ) ) )
=> ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_axioms.intro
tff(fact_5876_norm__pre__pure__rule1,axiom,
! [A: $tType,B2: bool,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn)] :
( ( pp(B2)
=> hoare_hoare_triple(A,P,F,Q) )
=> hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2)),F,Q) ) ).
% norm_pre_pure_rule1
tff(fact_5877_norm__pre__pure__rule2,axiom,
! [A: $tType,B2: bool,F: heap_Time_Heap(A),Q: fun(A,assn)] :
( ( pp(B2)
=> hoare_hoare_triple(A,one_one(assn),F,Q) )
=> hoare_hoare_triple(A,pure_assn(B2),F,Q) ) ).
% norm_pre_pure_rule2
tff(fact_5878_fi__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap(A),Q: fun(A,assn),Ps3: assn,F2: assn] :
( hoare_hoare_triple(A,P,C3,Q)
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Ps3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F2)))
=> hoare_hoare_triple(A,Ps3,C3,aa(assn,fun(A,assn),aTP_Lamp_sm(fun(A,assn),fun(assn,fun(A,assn)),Q),F2)) ) ) ).
% fi_rule
tff(fact_5879_cons__post__rulet,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap(A),Q: fun(A,assn),Q5: fun(A,assn)] :
( hoare_hoare_triple(A,P,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q))
=> ( ! [X2: A] : entailst(aa(A,assn,Q,X2),aa(A,assn,Q5,X2))
=> hoare_hoare_triple(A,P,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q5)) ) ) ).
% cons_post_rulet
tff(fact_5880_cons__pre__rulet,axiom,
! [A: $tType,P: assn,P6: assn,C3: heap_Time_Heap(A),Q: fun(A,assn)] :
( entailst(P,P6)
=> ( hoare_hoare_triple(A,P6,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q))
=> hoare_hoare_triple(A,P,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q)) ) ) ).
% cons_pre_rulet
tff(fact_5881_cons__rulet,axiom,
! [A: $tType,P: assn,P6: assn,Q: fun(A,assn),Q5: fun(A,assn),C3: heap_Time_Heap(A)] :
( entailst(P,P6)
=> ( ! [X2: A] : entailst(aa(A,assn,Q,X2),aa(A,assn,Q5,X2))
=> ( hoare_hoare_triple(A,P6,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q))
=> hoare_hoare_triple(A,P,C3,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Q5)) ) ) ) ).
% cons_rulet
tff(fact_5882_if__rule__split,axiom,
! [A: $tType,B2: bool,P: assn,F: heap_Time_Heap(A),Q1: fun(A,assn),G: heap_Time_Heap(A),Q22: fun(A,assn),Q: fun(A,assn)] :
( ( pp(B2)
=> hoare_hoare_triple(A,P,F,Q1) )
=> ( ( ~ pp(B2)
=> hoare_hoare_triple(A,P,G,Q22) )
=> ( ! [X2: A] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q1,X2)),pure_assn(B2))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Q22,X2)),pure_assn(aa(bool,bool,fNot,B2))))),aa(A,assn,Q,X2)))
=> hoare_hoare_triple(A,P,if(heap_Time_Heap(A),B2,F,G),Q) ) ) ) ).
% if_rule_split
tff(fact_5883_upd__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,Xs: list(A),A3: array(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> hoare_hoare_triple(array(A),snga_assn(A,A3,Xs),array_upd(A,I,X,A3),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_tr(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A3),X)) ) ) ).
% upd_rule
tff(fact_5884_make__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,F),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_ts(nat,fun(fun(nat,A),fun(array(A),assn)),N),F)) ) ).
% make_rule
tff(fact_5885_length__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(nat,snga_assn(A,A3,Xs),array_len(A,A3),aa(list(A),fun(nat,assn),aTP_Lamp_tt(array(A),fun(list(A),fun(nat,assn)),A3),Xs)) ) ).
% length_rule
tff(fact_5886_new__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,N,X),aa(A,fun(array(A),assn),aTP_Lamp_tu(nat,fun(A,fun(array(A),assn)),N),X)) ) ).
% new_rule
tff(fact_5887_lookup__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [P4: ref(A),X: A] : hoare_hoare_triple(A,sngr_assn(A,P4,X),ref_lookup(A,P4),aa(A,fun(A,assn),aTP_Lamp_tv(ref(A),fun(A,fun(A,assn)),P4),X)) ) ).
% lookup_rule
tff(fact_5888_of__list__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xs),aa(list(A),fun(array(A),assn),aTP_Lamp_km(list(A),fun(array(A),assn)),Xs)) ) ).
% of_list_rule
tff(fact_5889_freeze__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(list(A),snga_assn(A,A3,Xs),array_freeze(A,A3),aa(list(A),fun(list(A),assn),aTP_Lamp_tw(array(A),fun(list(A),fun(list(A),assn)),A3),Xs)) ) ).
% freeze_rule
tff(fact_5890_ref__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: A] : hoare_hoare_triple(ref(A),one_one(assn),ref_ref(A,X),aa(A,fun(ref(A),assn),aTP_Lamp_kn(A,fun(ref(A),assn)),X)) ) ).
% ref_rule
tff(fact_5891_return__sp__rule,axiom,
! [A: $tType,P: assn,X: A] : hoare_hoare_triple(A,P,heap_Time_return(A,X),aa(A,fun(A,assn),aTP_Lamp_tx(assn,fun(A,fun(A,assn)),P),X)) ).
% return_sp_rule
tff(fact_5892_return__cons__rule,axiom,
! [A: $tType,P: assn,Q: fun(A,assn),X: A] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(A,assn,Q,X)))
=> hoare_hoare_triple(A,P,heap_Time_return(A,X),Q) ) ).
% return_cons_rule
tff(fact_5893_raise__iff,axiom,
! [A: $tType,P: assn,S4: list(char),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,heap_Time_raise(A,S4),Q)
<=> ( P = bot_bot(assn) ) ) ).
% raise_iff
tff(fact_5894_raise__rule,axiom,
! [A: $tType,S4: list(char),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),heap_Time_raise(A,S4),Q) ).
% raise_rule
tff(fact_5895_ordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_def
tff(fact_5896_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,ord_less_eq(A)) ) ).
% order.partial_preordering_axioms
tff(fact_5897_partial__preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
<=> ( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),A6))
& ! [A6: A,B5: A,C6: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),C6))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),C6)) ) ) ) ) ).
% partial_preordering_def
tff(fact_5898_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( partial_preordering(A,Less_eq)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3)) ) ) ) ).
% partial_preordering.trans
tff(fact_5899_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),A5))
=> ( ! [A5: A,B4: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),C2)) ) )
=> partial_preordering(A,Less_eq) ) ) ).
% partial_preordering.intro
tff(fact_5900_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A] :
( partial_preordering(A,Less_eq)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),A3)) ) ).
% partial_preordering.refl
tff(fact_5901_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% ordering.axioms(1)
tff(fact_5902_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,aTP_Lamp_ty(A,fun(A,bool))) ) ).
% dual_order.partial_preordering_axioms
tff(fact_5903_ordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
=> ( ordering_axioms(A,Less_eq,Less)
=> ordering(A,Less_eq,Less) ) ) ).
% ordering.intro
tff(fact_5904_wait__rule,axiom,
! [N: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(N),aTP_Lamp_tz(product_unit,assn)) ).
% wait_rule
tff(fact_5905_update__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [P4: ref(A),Y: A,X: A] : hoare_hoare_triple(product_unit,sngr_assn(A,P4,Y),ref_update(A,P4,X),aa(A,fun(product_unit,assn),aTP_Lamp_ua(ref(A),fun(A,fun(product_unit,assn)),P4),X)) ) ).
% update_rule
tff(fact_5906_semilattice__neutr_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice(A,F)
=> ( comm_monoid(A,F,Z2)
=> semilattice_neutr(A,F,Z2) ) ) ).
% semilattice_neutr.intro
tff(fact_5907_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> semilattice(A,F) ) ).
% semilattice_neutr.axioms(1)
tff(fact_5908_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_max(A)) ) ).
% max.semilattice_axioms
tff(fact_5909_semilattice_Oidem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),A3) = A3 ) ) ).
% semilattice.idem
tff(fact_5910_semilattice_Oleft__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.left_idem
tff(fact_5911_semilattice_Oright__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.right_idem
tff(fact_5912_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_min(A)) ) ).
% min.semilattice_axioms
tff(fact_5913_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice(A,inf_inf(A)) ) ).
% inf.semilattice_axioms
tff(fact_5914_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice(A,sup_sup(A)) ) ).
% sup.semilattice_axioms
tff(fact_5915_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
=> semilattice(A,F) ) ).
% semilattice_order.axioms(1)
tff(fact_5916_semilattice__map2,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> semilattice(list(A),aTP_Lamp_ub(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),F)) ) ).
% semilattice_map2
tff(fact_5917_semilattice__neutr__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
<=> ( semilattice(A,F)
& comm_monoid(A,F,Z2) ) ) ).
% semilattice_neutr_def
tff(fact_5918_update__wp__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),Y: A,X: A,Q: fun(product_unit,assn)] : hoare_hoare_triple(product_unit,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,R3,Y)),wand_assn(sngr_assn(A,R3,X),aa(product_unit,assn,Q,product_Unity))),ref_update(A,R3,X),Q) ) ).
% update_wp_rule
tff(fact_5919_semilattice__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
<=> ( semilattice(A,F)
& semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_def
tff(fact_5920_semilattice__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice(A,F)
=> ( semila6385135966242565138axioms(A,F,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_order.intro
tff(fact_5921_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_5922_semilattice__order__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila6385135966242565138axioms(A,F,Less_eq,Less)
<=> ( ! [A6: A,B5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B5))
<=> ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B5) ) )
& ! [A6: A,B5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B5))
<=> ( ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B5) )
& ( A6 != B5 ) ) ) ) ) ).
% semilattice_order_axioms_def
tff(fact_5923_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),F: fun(A,fun(A,A)),Less: fun(A,fun(A,bool))] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
<=> ( A5 = aa(A,A,aa(A,fun(A,A),F,A5),B4) ) )
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
<=> ( ( A5 = aa(A,A,aa(A,fun(A,A),F,A5),B4) )
& ( A5 != B4 ) ) )
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_axioms.intro
tff(fact_5924_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ).
% semilattice_order.axioms(2)
tff(fact_5925_CODE__ABORT__def,axiom,
! [A: $tType,F: fun(product_unit,A)] : cODE_ABORT(A,F) = aa(product_unit,A,F,product_Unity) ).
% CODE_ABORT_def
tff(fact_5926_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid.axioms(2)
tff(fact_5927_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_5928_comm__monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F,Z2)
<=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6 ) ).
% comm_monoid_axioms_def
tff(fact_5929_comm__monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
<=> ( abel_semigroup(A,F)
& comm_monoid_axioms(A,F,Z2) ) ) ).
% comm_monoid_def
tff(fact_5930_comm__monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( abel_semigroup(A,F)
=> ( comm_monoid_axioms(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ) ).
% comm_monoid.intro
tff(fact_5931_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> semilattice_axioms(A,F) ) ).
% semilattice.axioms(2)
tff(fact_5932_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> abel_semigroup(A,inf_inf(A)) ) ).
% inf.abel_semigroup_axioms
tff(fact_5933_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> semigroup(A,F) ) ).
% abel_semigroup.axioms(1)
tff(fact_5934_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> abel_semigroup(A,sup_sup(A)) ) ).
% sup.abel_semigroup_axioms
tff(fact_5935_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_5936_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_min(A)) ) ).
% min.abel_semigroup_axioms
tff(fact_5937_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_max(A)) ) ).
% max.abel_semigroup_axioms
tff(fact_5938_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_5939_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),B2: A,A3: A,C3: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C3)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)) ) ) ).
% abel_semigroup.left_commute
tff(fact_5940_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).
% abel_semigroup.commute
tff(fact_5941_semilattice__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice_axioms(A,F)
<=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),A6) = A6 ) ).
% semilattice_axioms_def
tff(fact_5942_semilattice__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),A5) = A5
=> semilattice_axioms(A,F) ) ).
% semilattice_axioms.intro
tff(fact_5943_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Disj) ) ).
% abstract_boolean_algebra.axioms(2)
tff(fact_5944_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Conj) ) ).
% abstract_boolean_algebra.axioms(1)
tff(fact_5945_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> abel_semigroup(A,F) ) ).
% comm_monoid.axioms(1)
tff(fact_5946_semilattice__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
<=> ( abel_semigroup(A,F)
& semilattice_axioms(A,F) ) ) ).
% semilattice_def
tff(fact_5947_semilattice_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> ( semilattice_axioms(A,F)
=> semilattice(A,F) ) ) ).
% semilattice.intro
tff(fact_5948_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> abel_semigroup(A,F) ) ).
% semilattice.axioms(1)
tff(fact_5949_abel__semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
<=> ( semigroup(A,F)
& abel_s757365448890700780axioms(A,F) ) ) ).
% abel_semigroup_def
tff(fact_5950_abel__semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
=> ( abel_s757365448890700780axioms(A,F)
=> abel_semigroup(A,F) ) ) ).
% abel_semigroup.intro
tff(fact_5951_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( abel_semigroup(A,Conj)
=> ( abel_semigroup(A,Disj)
=> ( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ) ) ).
% abstract_boolean_algebra.intro
tff(fact_5952_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),One: A,Zero: A,Compl: fun(A,A)] :
( ! [X2: A,Y2: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,aa(A,fun(A,A),Disj,Y2),Z4)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),Y2)),aa(A,A,aa(A,fun(A,A),Conj,X2),Z4))
=> ( ! [X2: A,Y2: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,aa(A,fun(A,A),Conj,Y2),Z4)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X2),Y2)),aa(A,A,aa(A,fun(A,A),Disj,X2),Z4))
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),One) = X2
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),Zero) = X2
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,X2)) = Zero
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,Compl,X2)) = One
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
tff(fact_5953_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
<=> ( ! [X4: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,aa(A,fun(A,A),Disj,Y3),Z3)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Conj,X4),Z3))
& ! [X4: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,aa(A,fun(A,A),Conj,Y3),Z3)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Disj,X4),Z3))
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),One) = X4
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),Zero) = X4
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,X4)) = Zero
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,Compl,X4)) = One ) ) ).
% abstract_boolean_algebra_axioms_def
tff(fact_5954_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F,A5),B4) = aa(A,A,aa(A,fun(A,A),F,B4),A5)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup_axioms.intro
tff(fact_5955_abel__semigroup__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F)
<=> ! [A6: A,B5: A] : aa(A,A,aa(A,fun(A,A),F,A6),B5) = aa(A,A,aa(A,fun(A,A),F,B5),A6) ) ).
% abel_semigroup_axioms_def
tff(fact_5956_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup.axioms(2)
tff(fact_5957_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra.axioms(3)
tff(fact_5958_abstract__boolean__algebra__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
<=> ( abel_semigroup(A,Conj)
& abel_semigroup(A,Disj)
& boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ).
% abstract_boolean_algebra_def
tff(fact_5959_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,aTP_Lamp_ty(A,fun(A,bool)),aTP_Lamp_uc(A,fun(A,bool))) ) ).
% dual_order.preordering_axioms
tff(fact_5960_relH__def,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H,H5)
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2)))
& ! [T2: typerep,X4: nat] :
( pp(member2(nat,X4,As2))
=> ( ( refs(product_unit,H,T2,X4) = refs(product_unit,H5,T2,X4) )
& ( arrays(product_unit,H,T2,X4) = arrays(product_unit,H5,T2,X4) ) ) ) ) ) ).
% relH_def
tff(fact_5961_preordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
=> preordering(A,Less_eq,Less) ) ).
% preordering_dualI
tff(fact_5962_preordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
| ( A5 = B4 ) ) )
=> ( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A5)) )
=> ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),A5))
=> ( ! [A5: A,B4: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),C2)) ) )
=> preordering(A,Less_eq,Less) ) ) ) ) ).
% preordering_strictI
tff(fact_5963_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).
% preordering.strict_implies_order
tff(fact_5964_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).
% preordering.strict_iff_not
tff(fact_5965_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).
% preordering.strict_trans2
tff(fact_5966_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).
% preordering.strict_trans1
tff(fact_5967_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).
% preordering.strict_trans
tff(fact_5968_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A] :
( preordering(A,Less_eq,Less)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),A3)) ) ).
% preordering.irrefl
tff(fact_5969_preordering_Oasym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3)) ) ) ).
% preordering.asym
tff(fact_5970_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% preordering.axioms(1)
tff(fact_5971_order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.preordering_axioms
tff(fact_5972_preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
=> ( preordering_axioms(A,Less_eq,Less)
=> preordering(A,Less_eq,Less) ) ) ).
% preordering.intro
tff(fact_5973_preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& preordering_axioms(A,Less_eq,Less) ) ) ).
% preordering_def
tff(fact_5974_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
( ! [A5: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A5)) ) )
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering_axioms.intro
tff(fact_5975_preordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering_axioms(A,Less_eq,Less)
<=> ! [A6: A,B5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B5))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B5))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),A6)) ) ) ) ).
% preordering_axioms_def
tff(fact_5976_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering.axioms(2)
tff(fact_5977_Restr__natLeq,axiom,
! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,collect(nat,aTP_Lamp_bf(nat,fun(nat,bool),N)),aTP_Lamp_ud(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ue(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq
tff(fact_5978_Restr__natLeq2,axiom,
! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_uf(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_ue(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq2
tff(fact_5979_in__range_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
<=> pp(Y) )
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ! [X4: nat] :
( pp(member2(nat,X4,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H3))) ) )
=> ~ 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)),H3),As)) ) ) ) ) ).
% in_range.pelims(1)
tff(fact_5980_in__range_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))
=> ~ ! [X3: nat] :
( pp(member2(nat,X3,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H3))) ) ) ) ) ) ).
% in_range.pelims(2)
tff(fact_5981_in__range_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))
=> ! [X2: nat] :
( pp(member2(nat,X2,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),lim(product_unit,H3))) ) ) ) ) ) ).
% in_range.pelims(3)
tff(fact_5982_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))
=> ( As = bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
tff(fact_5983_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( 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)),H3),As))
=> ( As != bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
tff(fact_5984_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
<=> pp(Y) )
=> ( accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel,X)
=> ~ ! [H3: heap_ext(product_unit),As: 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)),H3),As) )
=> ( ( pp(Y)
<=> ( As = bot_bot(set(nat)) ) )
=> ~ 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)),H3),As)) ) ) ) ) ).
% one_assn_raw.pelims(1)
tff(fact_5985_lenlex__def,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R3)),aTP_Lamp_ug(list(A),product_prod(nat,list(A)))) ).
% lenlex_def
tff(fact_5986_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap(B),R: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
( ! [R5: A] : hoare_hoare_triple(B,P,F,aa(fun(B,assn),fun(B,assn),aTP_Lamp_sl(assn,fun(fun(B,assn),fun(B,assn)),P),R))
=> ( ! [X2: B] : hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,R,X2)),P),aa(B,heap_Time_Heap(C),G,X2),Q)
=> hoare_hoare_triple(C,P,heap_Time_bind(B,C,F,G),Q) ) ) ).
% bind_rule'
tff(fact_5987_measures__def,axiom,
! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_ui(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).
% measures_def
tff(fact_5988_ccpo_Oadmissible__def,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
<=> ! [A9: set(A)] :
( comple1602240252501008431_chain(A,Ord,A9)
=> ( ( A9 != bot_bot(set(A)) )
=> ( ! [X4: A] :
( pp(member2(A,X4,A9))
=> pp(aa(A,bool,P,X4)) )
=> pp(aa(A,bool,P,aa(set(A),A,Lub,A9))) ) ) ) ) ).
% ccpo.admissible_def
tff(fact_5989_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool),A4: set(A)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
=> ( comple1602240252501008431_chain(A,Ord,A4)
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member2(A,X2,A4))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,aa(set(A),A,Lub,A4))) ) ) ) ) ).
% ccpo.admissibleD
tff(fact_5990_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: fun(A,fun(A,bool)),P: fun(A,bool),Lub: fun(set(A),A)] :
( ! [A8: set(A)] :
( comple1602240252501008431_chain(A,Ord,A8)
=> ( ( A8 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( pp(member2(A,X3,A8))
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,aa(set(A),A,Lub,A8))) ) ) )
=> comple1908693960933563346ssible(A,Lub,Ord,P) ) ).
% ccpo.admissibleI
tff(fact_5991_fixp__induct,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,bool),F: fun(A,A)] :
( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
=> ( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F))
=> ( pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A)))))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,P,aa(A,A,F,X2))) )
=> pp(aa(A,bool,P,comple115746919287870866o_fixp(A,F))) ) ) ) ) ) ).
% fixp_induct
tff(fact_5992_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_5993_fixp__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A),G: fun(A,A)] :
( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F))
=> ( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),G))
=> ( ! [Z7: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,Z7)),aa(A,A,G,Z7)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comple115746919287870866o_fixp(A,F)),comple115746919287870866o_fixp(A,G))) ) ) ) ) ).
% fixp_mono
tff(fact_5994_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType,X3: product_prod(A,B)] : basic_fsts(A,B,X3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(product_prod(A,B),A,product_fst(A,B),X3)),bot_bot(set(A))) ).
% prod_set_defs(1)
tff(fact_5995_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Y),bot_bot(set(B))) ).
% prod_set_simps(2)
tff(fact_5996_prod__set__defs_I2_J,axiom,
! [C: $tType,D: $tType,X3: product_prod(C,D)] : basic_snds(C,D,X3) = aa(set(D),set(D),aa(D,fun(set(D),set(D)),insert2(D),aa(product_prod(C,D),D,product_snd(C,D),X3)),bot_bot(set(D))) ).
% prod_set_defs(2)
tff(fact_5997_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,N),Xs) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(fun(product_prod(nat,A),product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),map(product_prod(nat,A),product_prod(nat,A)),product_apfst(nat,nat,A,suc)),enumerate(A,N,Xs)) ).
% enumerate_Suc_eq
tff(fact_5998_select,axiom,
! [A: $tType,Xs: list(A),S4: product_prod(code_natural,code_natural)] :
( ( Xs != nil(A) )
=> pp(member2(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select(A,Xs),S4)),aa(list(A),set(A),set2(A),Xs))) ) ).
% select
tff(fact_5999_select__weight__select,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( select_weight(A,aa(list(A),list(product_prod(code_natural,A)),aa(fun(A,product_prod(code_natural,A)),fun(list(A),list(product_prod(code_natural,A))),map(A,product_prod(code_natural,A)),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs)) = select(A,Xs) ) ) ).
% select_weight_select
tff(fact_6000_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_6001_those_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(option(A)),option(list(A)),those(A),nil(option(A))) = aa(list(A),option(list(A)),some(list(A)),nil(A)) ).
% those.simps(1)
tff(fact_6002_bot__in__iterates,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] : pp(member2(A,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))),comple6359979572994053840erates(A,F))) ) ).
% bot_in_iterates
tff(fact_6003_those_Osimps_I2_J,axiom,
! [A: $tType,X: option(A),Xs: list(option(A))] : aa(list(option(A)),option(list(A)),those(A),aa(list(option(A)),list(option(A)),aa(option(A),fun(list(option(A)),list(option(A))),cons(option(A)),X),Xs)) = case_option(option(list(A)),A,none(list(A)),aTP_Lamp_uj(list(option(A)),fun(A,option(list(A))),Xs),X) ).
% those.simps(2)
tff(fact_6004_sym__Int__converse,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),converse(A,A,R3))) ).
% sym_Int_converse
tff(fact_6005_listrel__sym,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( sym(A,R3)
=> sym(list(A),listrel(A,A,R3)) ) ).
% listrel_sym
tff(fact_6006_sym__Int,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
( sym(A,R3)
=> ( sym(A,S4)
=> 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))),R3),S4)) ) ) ).
% sym_Int
tff(fact_6007_ac__operator_Osafe__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A,Y: A,A3: A,B2: A] :
( syntax_ac_operator(A,F)
=> ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),F,X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ) ).
% ac_operator.safe_commute
tff(fact_6008_Nitpick_OEx1__unfold,axiom,
! [A: $tType,P: fun(A,bool)] :
( ? [X4: A] :
( pp(aa(A,bool,P,X4))
& ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> ( Y3 = X4 ) ) )
<=> ? [X4: A] : collect(A,P) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).
% Nitpick.Ex1_unfold
tff(fact_6009_ac__operator_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B4)),C2) = aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B4),C2))
=> ( ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F,A5),B4) = aa(A,A,aa(A,fun(A,A),F,B4),A5)
=> syntax_ac_operator(A,F) ) ) ).
% ac_operator.intro
tff(fact_6010_ac__operator_Ocommute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).
% ac_operator.commute
tff(fact_6011_ac__operator_Oleft__assoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3) ) ) ).
% ac_operator.left_assoc
tff(fact_6012_ac__operator_Oright__assoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)) ) ) ).
% ac_operator.right_assoc
tff(fact_6013_ac__operator_Oleft__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C3)) = aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C3)) ) ) ).
% ac_operator.left_commute
tff(fact_6014_ac__operator_Oright__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),C3)),B2) ) ) ).
% ac_operator.right_commute
tff(fact_6015_ac__operator__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( syntax_ac_operator(A,F)
<=> ( ! [A6: A,B5: A,C6: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A6),B5)),C6) = aa(A,A,aa(A,fun(A,A),F,A6),aa(A,A,aa(A,fun(A,A),F,B5),C6))
& ! [A6: A,B5: A] : aa(A,A,aa(A,fun(A,A),F,A6),B5) = aa(A,A,aa(A,fun(A,A),F,B5),A6) ) ) ).
% ac_operator_def
tff(fact_6016_mult_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> syntax_ac_operator(A,times_times(A)) ) ).
% mult.ac_operator_axioms
tff(fact_6017_add_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> syntax_ac_operator(A,plus_plus(A)) ) ).
% add.ac_operator_axioms
tff(fact_6018_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A4)) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,minus_minus(set(A),A4,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_6019_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( pp(member2(A,X,A4))
=> ( lattic5214292709420241887eutr_F(A,F,Z2,A4) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,minus_minus(set(A),A4,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_6020_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( ! [X2: A,Y2: A] : pp(member2(A,aa(A,A,aa(A,fun(A,A),F,X2),Y2),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),Y2),bot_bot(set(A))))))
=> pp(member2(A,lattic5214292709420241887eutr_F(A,F,Z2,A4),A4)) ) ) ) ) ).
% semilattice_neutr_set.closed
tff(fact_6021_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( lattic5214292709420241887eutr_F(A,F,Z2,bot_bot(set(A))) = Z2 ) ) ).
% semilattice_neutr_set.empty
tff(fact_6022_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_6023_null__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool)),null(A)),null(B))) ).
% null_transfer
tff(fact_6024_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_6025_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( field_char_0_of_rat(A,A3) = one_one(A) )
<=> ( A3 = one_one(rat) ) ) ) ).
% of_rat_eq_1_iff
tff(fact_6026_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( one_one(A) = field_char_0_of_rat(A,A3) )
<=> ( one_one(rat) = A3 ) ) ) ).
% one_eq_of_rat_iff
tff(fact_6027_one__le__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R3: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R3)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R3)) ) ) ).
% one_le_of_rat_iff
tff(fact_6028_of__rat__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R3: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R3)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),one_one(rat))) ) ) ).
% of_rat_le_1_iff
tff(fact_6029_one__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R3: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),field_char_0_of_rat(A,R3)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R3)) ) ) ).
% one_less_of_rat_iff
tff(fact_6030_of__rat__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R3: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),field_char_0_of_rat(A,R3)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),one_one(rat))) ) ) ).
% of_rat_less_1_iff
tff(fact_6031_list_Omap__transfer,axiom,
! [A: $tType,B: $tType,F6: $tType,E4: $tType,Rb: fun(A,fun(E4,bool)),Sd: fun(B,fun(F6,bool))] : pp(aa(fun(fun(E4,F6),fun(list(E4),list(F6))),bool,aa(fun(fun(A,B),fun(list(A),list(B))),fun(fun(fun(E4,F6),fun(list(E4),list(F6))),bool),bNF_rel_fun(fun(A,B),fun(E4,F6),fun(list(A),list(B)),fun(list(E4),list(F6)),bNF_rel_fun(A,E4,B,F6,Rb,Sd),bNF_rel_fun(list(A),list(E4),list(B),list(F6),aa(fun(A,fun(E4,bool)),fun(list(A),fun(list(E4),bool)),list_all2(A,E4),Rb),aa(fun(B,fun(F6,bool)),fun(list(B),fun(list(F6),bool)),list_all2(B,F6),Sd))),map(A,B)),map(E4,F6))) ).
% list.map_transfer
tff(fact_6032_foldr__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: fun(A,fun(C,bool)),B3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool),bNF_rel_fun(fun(A,fun(B,B)),fun(C,fun(D,D)),fun(list(A),fun(B,B)),fun(list(C),fun(D,D)),bNF_rel_fun(A,C,fun(B,B),fun(D,D),A4,bNF_rel_fun(B,D,B,D,B3,B3)),bNF_rel_fun(list(A),list(C),fun(B,B),fun(D,D),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),A4),bNF_rel_fun(B,D,B,D,B3,B3))),foldr(A,B)),foldr(C,D))) ).
% foldr_transfer
tff(fact_6033_of__rat__mult,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat,B2: rat] : field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B2)) ) ).
% of_rat_mult
tff(fact_6034_foldl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,B3: fun(A,fun(C,bool)),A4: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,C)),fun(C,fun(list(D),C))),bool,aa(fun(fun(A,fun(B,A)),fun(A,fun(list(B),A))),fun(fun(fun(C,fun(D,C)),fun(C,fun(list(D),C))),bool),bNF_rel_fun(fun(A,fun(B,A)),fun(C,fun(D,C)),fun(A,fun(list(B),A)),fun(C,fun(list(D),C)),bNF_rel_fun(A,C,fun(B,A),fun(D,C),B3,bNF_rel_fun(B,D,A,C,A4,B3)),bNF_rel_fun(A,C,fun(list(B),A),fun(list(D),C),B3,bNF_rel_fun(list(B),list(D),A,C,aa(fun(B,fun(D,bool)),fun(list(B),fun(list(D),bool)),list_all2(B,D),A4),B3))),foldl(A,B)),foldl(C,D))) ).
% foldl_transfer
tff(fact_6035_List_Ofold__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A4: fun(A,fun(C,bool)),B3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool),bNF_rel_fun(fun(A,fun(B,B)),fun(C,fun(D,D)),fun(list(A),fun(B,B)),fun(list(C),fun(D,D)),bNF_rel_fun(A,C,fun(B,B),fun(D,D),A4,bNF_rel_fun(B,D,B,D,B3,B3)),bNF_rel_fun(list(A),list(C),fun(B,B),fun(D,D),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),A4),bNF_rel_fun(B,D,B,D,B3,B3))),fold(A,B)),fold(C,D))) ).
% List.fold_transfer
tff(fact_6036_list_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: fun(C,fun(D,bool)),R: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D))),bool,aa(fun(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C))),fun(fun(D,fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D)),S,bNF_rel_fun(fun(A,fun(list(A),fun(C,C))),fun(B,fun(list(B),fun(D,D))),fun(list(A),C),fun(list(B),D),bNF_rel_fun(A,B,fun(list(A),fun(C,C)),fun(list(B),fun(D,D)),R,bNF_rel_fun(list(A),list(B),fun(C,C),fun(D,D),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),bNF_rel_fun(C,D,C,D,S,S))),bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S))),rec_list(C,A)),rec_list(D,B))) ).
% list.rec_transfer
tff(fact_6037_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_mult(B)
& monoid_mult(A) )
=> ! [A4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B)))
=> pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),A4),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).
% prod_list_transfer
tff(fact_6038_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( comm_semiring_0(B)
& comm_semiring_0(A) )
=> ! [A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B)))
=> pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B3,A4),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A4,bNF_rel_fun(list(C),list(D),A,B,aa(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool)),list_all2(C,D),B3),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).
% horner_sum_transfer
tff(fact_6039_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& semiring_numeral(B)
& monoid_add(A)
& semiring_numeral(A) )
=> ! [R: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
=> pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).
% transfer_rule_numeral
tff(fact_6040_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ring_1(B)
& ring_1(A) )
=> ! [R: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
=> ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R,R),uminus_uminus(A)),uminus_uminus(B)))
=> pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).
% transfer_rule_of_int
tff(fact_6041_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( power(B)
& power(A) )
=> ! [R: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),times_times(A)),times_times(B)))
=> pp(aa(fun(B,fun(nat,B)),bool,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),bool),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R,bNF_rel_fun(nat,nat,A,B,fequal(nat),R)),power_power(A)),power_power(B))) ) ) ) ).
% power_transfer
tff(fact_6042_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),fequal(bool)),aTP_Lamp_fj(list(A),bool)),aTP_Lamp_ft(list(B),bool))) ).
% list.disc_transfer(2)
tff(fact_6043_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),fequal(bool)),aTP_Lamp_uk(list(A),bool)),aTP_Lamp_ul(list(B),bool))) ).
% list.disc_transfer(1)
tff(fact_6044_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& semiring_1(A) )
=> ! [R: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
=> pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).
% transfer_rule_of_nat
tff(fact_6045_list__ex__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),bool)),bool,aa(fun(fun(A,bool),fun(list(A),bool)),fun(fun(fun(B,bool),fun(list(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),bool),fun(list(B),bool),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool))),list_ex(A)),list_ex(B))) ).
% list_ex_transfer
tff(fact_6046_list_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(list(B),list(B))),bool,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),bool),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),R,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R))),cons(A)),cons(B))) ).
% list.ctr_transfer(2)
tff(fact_6047_subseqs__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(list(B))),bool,aa(fun(list(A),list(list(A))),fun(fun(list(B),list(list(B))),bool),bNF_rel_fun(list(A),list(B),list(list(A)),list(list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),subseqs(A)),subseqs(B))) ).
% subseqs_transfer
tff(fact_6048_length__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),nat),bool,aa(fun(list(A),nat),fun(fun(list(B),nat),bool),bNF_rel_fun(list(A),list(B),nat,nat,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(nat)),size_size(list(A))),size_size(list(B)))) ).
% length_transfer
tff(fact_6049_successively__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,fun(B,bool)),fun(list(B),bool)),bool,aa(fun(fun(A,fun(A,bool)),fun(list(A),bool)),fun(fun(fun(B,fun(B,bool)),fun(list(B),bool)),bool),bNF_rel_fun(fun(A,fun(A,bool)),fun(B,fun(B,bool)),fun(list(A),bool),fun(list(B),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A4,bNF_rel_fun(A,B,bool,bool,A4,fequal(bool))),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool))),successively(A)),successively(B))) ).
% successively_transfer
tff(fact_6050_splice__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(list(B),list(B))),bool,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),bool),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),splice(A)),splice(B))) ).
% splice_transfer
tff(fact_6051_replicate__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(B,list(B))),bool,aa(fun(nat,fun(A,list(A))),fun(fun(nat,fun(B,list(B))),bool),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),A4,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),replicate(A)),replicate(B))) ).
% replicate_transfer
tff(fact_6052_rotate__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),rotate(A)),rotate(B))) ).
% rotate_transfer
tff(fact_6053_list_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(list(A),fun(list(B),bool)),fun(list(C),fun(list(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(list(A),list(C),fun(list(B),bool),fun(list(D),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),Sa),bNF_rel_fun(list(B),list(D),bool,bool,aa(fun(B,fun(D,bool)),fun(list(B),fun(list(D),bool)),list_all2(B,D),Sc),fequal(bool)))),list_all2(A,B)),list_all2(C,D))) ).
% list.rel_transfer
tff(fact_6054_dropWhile__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),dropWhile(A)),dropWhile(B))) ).
% dropWhile_transfer
tff(fact_6055_list__update__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(nat,fun(B,list(B)))),bool,aa(fun(list(A),fun(nat,fun(A,list(A)))),fun(fun(list(B),fun(nat,fun(B,list(B)))),bool),bNF_rel_fun(list(A),list(B),fun(nat,fun(A,list(A))),fun(nat,fun(B,list(B))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),A4,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)))),list_update(A)),list_update(B))) ).
% list_update_transfer
tff(fact_6056_take__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),take(A)),take(B))) ).
% take_transfer
tff(fact_6057_drop__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),drop(A)),drop(B))) ).
% drop_transfer
tff(fact_6058_append__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(list(B),list(B))),bool,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),bool),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),append(A)),append(B))) ).
% append_transfer
tff(fact_6059_List_Ofilter__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),filter2(A)),filter2(B))) ).
% List.filter_transfer
tff(fact_6060_takeWhile__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),takeWhile(A)),takeWhile(B))) ).
% takeWhile_transfer
tff(fact_6061_tl__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),tl(A)),tl(B))) ).
% tl_transfer
tff(fact_6062_rev__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),rev(A)),rev(B))) ).
% rev_transfer
tff(fact_6063_concat__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(list(B)),list(B)),bool,aa(fun(list(list(A)),list(A)),fun(fun(list(list(B)),list(B)),bool),bNF_rel_fun(list(list(A)),list(list(B)),list(A),list(B),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),concat(A)),concat(B))) ).
% concat_transfer
tff(fact_6064_butlast__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),butlast(A)),butlast(B))) ).
% butlast_transfer
tff(fact_6065_rotate1__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),rotate1(A)),rotate1(B))) ).
% rotate1_transfer
tff(fact_6066_list_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: fun(C,fun(D,bool)),R: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,fun(list(B),D)),fun(list(B),D))),bool,aa(fun(C,fun(fun(A,fun(list(A),C)),fun(list(A),C))),fun(fun(D,fun(fun(B,fun(list(B),D)),fun(list(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,fun(list(A),C)),fun(list(A),C)),fun(fun(B,fun(list(B),D)),fun(list(B),D)),S,bNF_rel_fun(fun(A,fun(list(A),C)),fun(B,fun(list(B),D)),fun(list(A),C),fun(list(B),D),bNF_rel_fun(A,B,fun(list(A),C),fun(list(B),D),R,bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S)),bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S))),case_list(C,A)),case_list(D,B))) ).
% list.case_transfer
tff(fact_6067_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( zero_neq_one(B)
& zero_neq_one(A) )
=> ! [R: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
=> pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).
% transfer_rule_of_bool
tff(fact_6068_product__lists__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(list(B)),list(list(B))),bool,aa(fun(list(list(A)),list(list(A))),fun(fun(list(list(B)),list(list(B))),bool),bNF_rel_fun(list(list(A)),list(list(B)),list(list(A)),list(list(B)),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),product_lists(A)),product_lists(B))) ).
% product_lists_transfer
tff(fact_6069_list_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),bool)),bool,aa(fun(fun(A,bool),fun(list(A),bool)),fun(fun(fun(B,bool),fun(list(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),bool),fun(list(B),bool),bNF_rel_fun(A,B,bool,bool,R,fequal(bool)),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),fequal(bool))),list_all(A)),list_all(B))) ).
% list.pred_transfer
tff(fact_6070_list__all__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),bool)),bool,aa(fun(fun(A,bool),fun(list(A),bool)),fun(fun(fun(B,bool),fun(list(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),bool),fun(list(B),bool),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool))),list_all(A)),list_all(B))) ).
% list_all_transfer
tff(fact_6071_List_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(B,fun(list(B),list(B))),bool,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),bool),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),A4,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),insert(A)),insert(B))) ) ).
% List.insert_transfer
tff(fact_6072_removeAll__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(B,fun(list(B),list(B))),bool,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),bool),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),A4,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),removeAll(A)),removeAll(B))) ) ).
% removeAll_transfer
tff(fact_6073_list_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
( bi_unique(A,B,R)
=> bi_unique(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ) ).
% list.bi_unique_rel
tff(fact_6074_remdups__adj__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),remdups_adj(A)),remdups_adj(B))) ) ).
% remdups_adj_transfer
tff(fact_6075_remdups__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),remdups(A)),remdups(B))) ) ).
% remdups_transfer
tff(fact_6076_distinct__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool)),distinct(A)),distinct(B))) ) ).
% distinct_transfer
tff(fact_6077_distinct__adj__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),fequal(bool)),distinct_adj(A)),distinct_adj(B))) ) ).
% distinct_adj_transfer
tff(fact_6078_remove1__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(B,fun(list(B),list(B))),bool,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),bool),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),A4,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),remove1(A)),remove1(B))) ) ).
% remove1_transfer
tff(fact_6079_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] :
( bi_total(A,B,A4)
=> pp(aa(fun(fun(B,fun(B,bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool))),bool,aa(fun(fun(A,fun(A,bool)),fun(fun(C,fun(C,bool)),fun(fun(A,C),bool))),fun(fun(fun(B,fun(B,bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool))),bool),bNF_rel_fun(fun(A,fun(A,bool)),fun(B,fun(B,bool)),fun(fun(C,fun(C,bool)),fun(fun(A,C),bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool)),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A4,bNF_rel_fun(A,B,bool,bool,A4,fequal(bool))),bNF_rel_fun(fun(C,fun(C,bool)),fun(D,fun(D,bool)),fun(fun(A,C),bool),fun(fun(B,D),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B3,bNF_rel_fun(C,D,bool,bool,B3,fequal(bool))),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A4,B3),fequal(bool)))),comple7038119648293358887notone(A,C)),comple7038119648293358887notone(B,D))) ) ).
% monotone_parametric
tff(fact_6080_partition__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),product_prod(list(B),list(B)))),bool,aa(fun(fun(A,bool),fun(list(A),product_prod(list(A),list(A)))),fun(fun(fun(B,bool),fun(list(B),product_prod(list(B),list(B)))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),product_prod(list(A),list(A))),fun(list(B),product_prod(list(B),list(B))),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),product_prod(list(A),list(A)),product_prod(list(B),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),basic_rel_prod(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)))),partition(A)),partition(B))) ).
% partition_transfer
tff(fact_6081_zip__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] : pp(aa(fun(list(B),fun(list(D),list(product_prod(B,D)))),bool,aa(fun(list(A),fun(list(C),list(product_prod(A,C)))),fun(fun(list(B),fun(list(D),list(product_prod(B,D)))),bool),bNF_rel_fun(list(A),list(B),fun(list(C),list(product_prod(A,C))),fun(list(D),list(product_prod(B,D))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(list(C),list(D),list(product_prod(A,C)),list(product_prod(B,D)),aa(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool)),list_all2(C,D),B3),aa(fun(product_prod(A,C),fun(product_prod(B,D),bool)),fun(list(product_prod(A,C)),fun(list(product_prod(B,D)),bool)),list_all2(product_prod(A,C),product_prod(B,D)),basic_rel_prod(A,B,C,D,A4,B3)))),zip(A,C)),zip(B,D))) ).
% zip_transfer
tff(fact_6082_list_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
( bi_total(A,B,R)
=> bi_total(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ) ).
% list.bi_total_rel
tff(fact_6083_product__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] : pp(aa(fun(list(B),fun(list(D),list(product_prod(B,D)))),bool,aa(fun(list(A),fun(list(C),list(product_prod(A,C)))),fun(fun(list(B),fun(list(D),list(product_prod(B,D)))),bool),bNF_rel_fun(list(A),list(B),fun(list(C),list(product_prod(A,C))),fun(list(D),list(product_prod(B,D))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(list(C),list(D),list(product_prod(A,C)),list(product_prod(B,D)),aa(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool)),list_all2(C,D),B3),aa(fun(product_prod(A,C),fun(product_prod(B,D),bool)),fun(list(product_prod(A,C)),fun(list(product_prod(B,D)),bool)),list_all2(product_prod(A,C),product_prod(B,D)),basic_rel_prod(A,B,C,D,A4,B3)))),product(A,C)),product(B,D))) ).
% product_transfer
tff(fact_6084_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F6: $tType,E4: $tType,C4: fun(A,fun(B,bool)),A4: fun(C,fun(E4,bool)),B3: fun(D,fun(F6,bool))] :
( bi_total(A,B,C4)
=> pp(aa(fun(fun(E4,fun(F6,bool)),fun(fun(B,E4),fun(fun(B,F6),bool))),bool,aa(fun(fun(C,fun(D,bool)),fun(fun(A,C),fun(fun(A,D),bool))),fun(fun(fun(E4,fun(F6,bool)),fun(fun(B,E4),fun(fun(B,F6),bool))),bool),bNF_rel_fun(fun(C,fun(D,bool)),fun(E4,fun(F6,bool)),fun(fun(A,C),fun(fun(A,D),bool)),fun(fun(B,E4),fun(fun(B,F6),bool)),bNF_rel_fun(C,E4,fun(D,bool),fun(F6,bool),A4,bNF_rel_fun(D,F6,bool,bool,B3,fequal(bool))),bNF_rel_fun(fun(A,C),fun(B,E4),fun(fun(A,D),bool),fun(fun(B,F6),bool),bNF_rel_fun(A,B,C,E4,C4,A4),bNF_rel_fun(fun(A,D),fun(B,F6),bool,bool,bNF_rel_fun(A,B,D,F6,C4,B3),fequal(bool)))),partial_fun_ord(C,D,A)),partial_fun_ord(E4,F6,B))) ) ).
% fun_ord_parametric
tff(fact_6085_rtrancl__parametric,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> ( bi_total(A,B,A4)
=> pp(aa(fun(set(product_prod(B,B)),set(product_prod(B,B))),bool,aa(fun(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(set(product_prod(B,B)),set(product_prod(B,B))),bool),bNF_rel_fun(set(product_prod(A,A)),set(product_prod(B,B)),set(product_prod(A,A)),set(product_prod(B,B)),bNF_rel_set(product_prod(A,A),product_prod(B,B),basic_rel_prod(A,B,A,B,A4,A4)),bNF_rel_set(product_prod(A,A),product_prod(B,B),basic_rel_prod(A,B,A,B,A4,A4))),transitive_rtrancl(A)),transitive_rtrancl(B))) ) ) ).
% rtrancl_parametric
tff(fact_6086_list_Oset__transfer,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(list(B),set(B)),bool,aa(fun(list(A),set(A)),fun(fun(list(B),set(B)),bool),bNF_rel_fun(list(A),list(B),set(A),set(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),bNF_rel_set(A,B,R)),set2(A)),set2(B))) ).
% list.set_transfer
tff(fact_6087_listset__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(set(B)),set(list(B))),bool,aa(fun(list(set(A)),set(list(A))),fun(fun(list(set(B)),set(list(B))),bool),bNF_rel_fun(list(set(A)),list(set(B)),set(list(A)),set(list(B)),aa(fun(set(A),fun(set(B),bool)),fun(list(set(A)),fun(list(set(B)),bool)),list_all2(set(A),set(B)),bNF_rel_set(A,B,A4)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),listset(A)),listset(B))) ).
% listset_transfer
tff(fact_6088_set__Cons__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(set(list(B)),set(list(B)))),bool,aa(fun(set(A),fun(set(list(A)),set(list(A)))),fun(fun(set(B),fun(set(list(B)),set(list(B)))),bool),bNF_rel_fun(set(A),set(B),fun(set(list(A)),set(list(A))),fun(set(list(B)),set(list(B))),bNF_rel_set(A,B,A4),bNF_rel_fun(set(list(A)),set(list(B)),set(list(A)),set(list(B)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)))),set_Cons(A)),set_Cons(B))) ).
% set_Cons_transfer
tff(fact_6089_lists__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(set(B),set(list(B))),bool,aa(fun(set(A),set(list(A))),fun(fun(set(B),set(list(B))),bool),bNF_rel_fun(set(A),set(B),set(list(A)),set(list(B)),bNF_rel_set(A,B,A4),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),lists(A)),lists(B))) ).
% lists_transfer
tff(fact_6090_inter__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> pp(aa(fun(set(B),fun(set(B),set(B))),bool,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),bool),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A4),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4))),inf_inf(set(A))),inf_inf(set(B)))) ) ).
% inter_transfer
tff(fact_6091_fun__lub__parametric,axiom,
! [A: $tType,C: $tType,E4: $tType,F6: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool)),C4: fun(E4,fun(F6,bool))] :
( bi_total(A,B,A4)
=> ( bi_unique(A,B,A4)
=> pp(aa(fun(fun(set(B),D),fun(set(fun(F6,B)),fun(F6,D))),bool,aa(fun(fun(set(A),C),fun(set(fun(E4,A)),fun(E4,C))),fun(fun(fun(set(B),D),fun(set(fun(F6,B)),fun(F6,D))),bool),bNF_rel_fun(fun(set(A),C),fun(set(B),D),fun(set(fun(E4,A)),fun(E4,C)),fun(set(fun(F6,B)),fun(F6,D)),bNF_rel_fun(set(A),set(B),C,D,bNF_rel_set(A,B,A4),B3),bNF_rel_fun(set(fun(E4,A)),set(fun(F6,B)),fun(E4,C),fun(F6,D),bNF_rel_set(fun(E4,A),fun(F6,B),bNF_rel_fun(E4,F6,A,B,C4,A4)),bNF_rel_fun(E4,F6,C,D,C4,B3))),partial_fun_lub(A,C,E4)),partial_fun_lub(B,D,F6))) ) ) ).
% fun_lub_parametric
tff(fact_6092_nths__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(set(nat),list(B))),bool,aa(fun(list(A),fun(set(nat),list(A))),fun(fun(list(B),fun(set(nat),list(B))),bool),bNF_rel_fun(list(A),list(B),fun(set(nat),list(A)),fun(set(nat),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(set(nat),set(nat),list(A),list(B),bNF_rel_set(nat,nat,fequal(nat)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4))),nths(A)),nths(B))) ).
% nths_transfer
tff(fact_6093_shuffles__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(list(B),set(list(B)))),bool,aa(fun(list(A),fun(list(A),set(list(A)))),fun(fun(list(B),fun(list(B),set(list(B)))),bool),bNF_rel_fun(list(A),list(B),fun(list(A),set(list(A))),fun(list(B),set(list(B))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_fun(list(A),list(B),set(list(A)),set(list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),bNF_rel_set(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4)))),shuffles(A)),shuffles(B))) ).
% shuffles_transfer
tff(fact_6094_empty__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,A4),bot_bot(set(A))),bot_bot(set(B)))) ).
% empty_transfer
tff(fact_6095_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B3: fun(A,fun(B,bool)),A4: fun(C,fun(D,bool))] :
( bi_unique(A,B,B3)
=> ( right_total(C,D,A4)
=> pp(aa(fun(fun(D,B),fun(set(B),set(D))),bool,aa(fun(fun(C,A),fun(set(A),set(C))),fun(fun(fun(D,B),fun(set(B),set(D))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(set(A),set(C)),fun(set(B),set(D)),bNF_rel_fun(C,D,A,B,A4,B3),bNF_rel_fun(set(A),set(B),set(C),set(D),bNF_rel_set(A,B,B3),bNF_rel_set(C,D,A4))),aTP_Lamp_um(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),A4)),vimage(D,B))) ) ) ).
% vimage_right_total_transfer
tff(fact_6096_find__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),option(B))),bool,aa(fun(fun(A,bool),fun(list(A),option(A))),fun(fun(fun(B,bool),fun(list(B),option(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),option(A)),fun(list(B),option(B)),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(list(A),list(B),option(A),option(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A4),rel_option(A,B,A4))),find(A)),find(B))) ).
% find_transfer
tff(fact_6097_right__total__Inter__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> ( right_total(A,B,A4)
=> pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A4)),bNF_rel_set(A,B,A4)),aTP_Lamp_un(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),A4)),complete_Inf_Inf(set(B)))) ) ) ).
% right_total_Inter_transfer
tff(fact_6098_those__transfer,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : pp(aa(fun(list(option(B)),option(list(B))),bool,aa(fun(list(option(A)),option(list(A))),fun(fun(list(option(B)),option(list(B))),bool),bNF_rel_fun(list(option(A)),list(option(B)),option(list(A)),option(list(B)),aa(fun(option(A),fun(option(B),bool)),fun(list(option(A)),fun(list(option(B)),bool)),list_all2(option(A),option(B)),rel_option(A,B,P)),rel_option(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P))),those(A)),those(B))) ).
% those_transfer
tff(fact_6099_list_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] : domainp(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) = aa(fun(A,bool),fun(list(A),bool),list_all(A),domainp(A,B,R)) ).
% list.Domainp_rel
tff(fact_6100_list_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
( right_total(A,B,R)
=> right_total(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ) ).
% list.right_total_rel
tff(fact_6101_right__total__Compl__transfer,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> ( right_total(A,B,A4)
=> pp(aa(fun(set(B),set(B)),bool,aa(fun(set(A),set(A)),fun(fun(set(B),set(B)),bool),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A4),bNF_rel_set(A,B,A4)),aTP_Lamp_uo(fun(A,fun(B,bool)),fun(set(A),set(A)),A4)),uminus_uminus(set(B)))) ) ) ).
% right_total_Compl_transfer
tff(fact_6102_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),inf_inf(fun(option(A),fun(option(B),bool))),rel_option(A,B,A4)),rel_option(A,B,B3)) = rel_option(A,B,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A4),B3)) ).
% rel_option_inf
tff(fact_6103_wfP__SUP,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,fun(B,bool)))] :
( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,bool)),R3,I3))
=> ( ! [I3: A,J3: A] :
( ( aa(A,fun(B,fun(B,bool)),R3,I3) != aa(A,fun(B,fun(B,bool)),R3,J3) )
=> ( aa(fun(B,bool),fun(B,bool),aa(fun(B,bool),fun(fun(B,bool),fun(B,bool)),inf_inf(fun(B,bool)),domainp(B,B,aa(A,fun(B,fun(B,bool)),R3,I3))),rangep(B,B,aa(A,fun(B,fun(B,bool)),R3,J3))) = bot_bot(fun(B,bool)) ) )
=> wfP(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Sup_Sup(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),image2(A,fun(B,fun(B,bool)),R3),top_top(set(A))))) ) ) ).
% wfP_SUP
tff(fact_6104_option_Osimps_I15_J,axiom,
! [A: $tType,X24: A] : set_option(A,aa(A,option(A),some(A),X24)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X24),bot_bot(set(A))) ).
% option.simps(15)
tff(fact_6105_these__not__empty__eq,axiom,
! [A: $tType,B3: set(option(A))] :
( ( these(A,B3) != bot_bot(set(A)) )
<=> ( ( B3 != bot_bot(set(option(A))) )
& ( B3 != 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_6106_set__empty__eq,axiom,
! [A: $tType,Xo2: option(A)] :
( ( set_option(A,Xo2) = bot_bot(set(A)) )
<=> ( Xo2 = none(A) ) ) ).
% set_empty_eq
tff(fact_6107_these__empty,axiom,
! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).
% these_empty
tff(fact_6108_option_Osimps_I14_J,axiom,
! [A: $tType] : set_option(A,none(A)) = bot_bot(set(A)) ).
% option.simps(14)
tff(fact_6109_these__empty__eq,axiom,
! [A: $tType,B3: set(option(A))] :
( ( these(A,B3) = bot_bot(set(A)) )
<=> ( ( B3 = bot_bot(set(option(A))) )
| ( B3 = 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_6110_partial__function__definitions_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,Leq: fun(A,fun(A,bool)),Lub: fun(set(A),A),U2: fun(C,fun(B,A)),F2: fun(C,C),C4: fun(fun(B,A),C),F: C,P: fun(fun(B,A),bool)] :
( partia7178651479351089652itions(A,Leq,Lub)
=> ( ! [X2: B] : pp(aa(fun(fun(B,A),A),bool,aa(fun(A,fun(A,bool)),fun(fun(fun(B,A),A),bool),aa(fun(fun(B,A),fun(fun(B,A),bool)),fun(fun(A,fun(A,bool)),fun(fun(fun(B,A),A),bool)),comple7038119648293358887notone(fun(B,A),A),aa(fun(A,fun(A,bool)),fun(fun(B,A),fun(fun(B,A),bool)),partial_fun_ord(A,A,B),Leq)),Leq),aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_up(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),U2),F2),C4),X2)))
=> ( ( F = aa(fun(B,A),C,C4,comple187402453842119260l_fixp(fun(B,A),aa(fun(set(A),A),fun(set(fun(B,A)),fun(B,A)),partial_fun_lub(A,A,B),Lub),aa(fun(A,fun(A,bool)),fun(fun(B,A),fun(fun(B,A),bool)),partial_fun_ord(A,A,B),Leq),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_uq(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),U2),F2),C4))) )
=> ( ! [F4: fun(B,A)] : aa(C,fun(B,A),U2,aa(fun(B,A),C,C4,F4)) = F4
=> ( comple1908693960933563346ssible(fun(B,A),aa(fun(set(A),A),fun(set(fun(B,A)),fun(B,A)),partial_fun_lub(A,A,B),Lub),aa(fun(A,fun(A,bool)),fun(fun(B,A),fun(fun(B,A),bool)),partial_fun_ord(A,A,B),Leq),P)
=> ( pp(aa(fun(B,A),bool,P,aTP_Lamp_ur(fun(set(A),A),fun(B,A),Lub)))
=> ( ! [F4: C] :
( pp(aa(fun(B,A),bool,P,aa(C,fun(B,A),U2,F4)))
=> pp(aa(fun(B,A),bool,P,aa(C,fun(B,A),U2,aa(C,C,F2,F4)))) )
=> pp(aa(fun(B,A),bool,P,aa(C,fun(B,A),U2,F))) ) ) ) ) ) ) ) ).
% partial_function_definitions.fixp_induct_uc
tff(fact_6111_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] :
( bi_unique(A,B,A4)
=> ( bi_total(A,B,A4)
=> pp(aa(fun(filter(B),fun(filter(B),filter(B))),bool,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),bool),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A4),rel_filter(A,B,A4))),inf_inf(filter(A))),inf_inf(filter(B)))) ) ) ).
% inf_filter_parametric
tff(fact_6112_INF__filter__bot__base,axiom,
! [A: $tType,B: $tType,I5: set(A),F2: fun(A,filter(B))] :
( ! [I3: A] :
( pp(member2(A,I3,I5))
=> ! [J3: A] :
( pp(member2(A,J3,I5))
=> ? [X3: A] :
( pp(member2(A,X3,I5))
& pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F2,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F2,I3)),aa(A,filter(B),F2,J3)))) ) ) )
=> ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F2),I5)) = bot_bot(filter(B)) )
<=> ? [X4: A] :
( pp(member2(A,X4,I5))
& ( aa(A,filter(B),F2,X4) = bot_bot(filter(B)) ) ) ) ) ).
% INF_filter_bot_base
tff(fact_6113_inf__principal,axiom,
! [A: $tType,A4: set(A),B3: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),principal(A,A4)),principal(A,B3)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ).
% inf_principal
tff(fact_6114_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B3: set(A),F2: fun(A,filter(B)),P: fun(B,bool)] :
( ( B3 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(member2(A,A5,B3))
=> ! [B4: A] :
( pp(member2(A,B4,B3))
=> ? [X3: A] :
( pp(member2(A,X3,B3))
& pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F2,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F2,A5)),aa(A,filter(B),F2,B4)))) ) ) )
=> ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F2),B3)))
<=> ? [X4: A] :
( pp(member2(A,X4,B3))
& eventually(B,P,aa(A,filter(B),F2,X4)) ) ) ) ) ).
% eventually_INF_base
tff(fact_6115_eventually__Inf__base,axiom,
! [A: $tType,B3: set(filter(A)),P: fun(A,bool)] :
( ( B3 != bot_bot(set(filter(A))) )
=> ( ! [F3: filter(A)] :
( pp(member2(filter(A),F3,B3))
=> ! [G5: filter(A)] :
( pp(member2(filter(A),G5,B3))
=> ? [X3: filter(A)] :
( pp(member2(filter(A),X3,B3))
& pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X3),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),G5))) ) ) )
=> ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
<=> ? [X4: filter(A)] :
( pp(member2(filter(A),X4,B3))
& eventually(A,P,X4) ) ) ) ) ).
% eventually_Inf_base
tff(fact_6116_eventually__inf,axiom,
! [A: $tType,P: fun(A,bool),F2: filter(A),F7: filter(A)] :
( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F2),F7))
<=> ? [Q7: fun(A,bool),R8: fun(A,bool)] :
( eventually(A,Q7,F2)
& eventually(A,R8,F7)
& ! [X4: A] :
( ( pp(aa(A,bool,Q7,X4))
& pp(aa(A,bool,R8,X4)) )
=> pp(aa(A,bool,P,X4)) ) ) ) ).
% eventually_inf
tff(fact_6117_eventually__inf__principal,axiom,
! [A: $tType,P: fun(A,bool),F2: filter(A),S4: set(A)] :
( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F2),principal(A,S4)))
<=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_us(fun(A,bool),fun(set(A),fun(A,bool)),P),S4),F2) ) ).
% eventually_inf_principal
tff(fact_6118_principal__eq__bot__iff,axiom,
! [A: $tType,X5: set(A)] :
( ( principal(A,X5) = bot_bot(filter(A)) )
<=> ( X5 = bot_bot(set(A)) ) ) ).
% principal_eq_bot_iff
tff(fact_6119_bot__eq__principal__empty,axiom,
! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).
% bot_eq_principal_empty
tff(fact_6120_filterlim__base__iff,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F2: fun(A,set(B)),F: fun(B,C),G2: fun(D,set(C)),J4: set(D)] :
( ( I5 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I5))
=> ! [J3: A] :
( pp(member2(A,J3,I5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,I3)),aa(A,set(B),F2,J3)))
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,J3)),aa(A,set(B),F2,I3))) ) ) )
=> ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_ut(fun(D,set(C)),fun(D,filter(C)),G2)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_uu(fun(A,set(B)),fun(A,filter(B)),F2)),I5)))
<=> ! [X4: D] :
( pp(member2(D,X4,J4))
=> ? [Xa2: A] :
( pp(member2(A,Xa2,I5))
& ! [Xb4: B] :
( pp(member2(B,Xb4,aa(A,set(B),F2,Xa2)))
=> pp(member2(C,aa(B,C,F,Xb4),aa(D,set(C),G2,X4))) ) ) ) ) ) ) ).
% filterlim_base_iff
tff(fact_6121_inf__filter__def,axiom,
! [A: $tType,F2: filter(A),F7: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F2),F7) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_uv(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F2),F7)) ).
% inf_filter_def
tff(fact_6122_filterlim__inf,axiom,
! [B: $tType,A: $tType,F: fun(A,B),F23: filter(B),F32: filter(B),F12: filter(A)] :
( filterlim(A,B,F,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F32),F12)
<=> ( filterlim(A,B,F,F23,F12)
& filterlim(A,B,F,F32,F12) ) ) ).
% filterlim_inf
tff(fact_6123_filterlim__If,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G2: filter(B),F2: filter(A),P: fun(A,bool),G: fun(A,B)] :
( filterlim(A,B,F,G2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F2),principal(A,collect(A,P))))
=> ( filterlim(A,B,G,G2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F2),principal(A,collect(A,aTP_Lamp_bi(fun(A,bool),fun(A,bool),P)))))
=> filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_uw(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),F),P),G),G2,F2) ) ) ).
% filterlim_If
tff(fact_6124_prod__filter__INF,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J4: set(B),A4: fun(A,filter(C)),B3: fun(B,filter(D))] :
( ( I5 != bot_bot(set(A)) )
=> ( ( J4 != bot_bot(set(B)) )
=> ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A4),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B3),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_uy(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A4),B3)),I5)) ) ) ) ).
% prod_filter_INF
tff(fact_6125_prod__filter__INF1,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set(A),A4: fun(A,filter(B)),B3: filter(C)] :
( ( I5 != bot_bot(set(A)) )
=> ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A4),I5)),B3) = 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_uz(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A4),B3)),I5)) ) ) ).
% prod_filter_INF1
tff(fact_6126_prod__filter__INF2,axiom,
! [C: $tType,B: $tType,A: $tType,J4: set(A),A4: filter(B),B3: fun(A,filter(C))] :
( ( J4 != bot_bot(set(A)) )
=> ( prod_filter(B,C,A4,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B3),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_va(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A4),B3)),J4)) ) ) ).
% prod_filter_INF2
tff(fact_6127_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F2: filter(A),X: B] : prod_filter(A,B,F2,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ht(B,fun(A,product_prod(A,B))),X),F2) ).
% prod_filter_principal_singleton2
tff(fact_6128_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F2: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),F2) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F2) ).
% prod_filter_principal_singleton
tff(fact_6129_filtermap__inf,axiom,
! [A: $tType,B: $tType,F: fun(B,A),F12: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F,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)),filtermap(B,A,F,F12)),filtermap(B,A,F,F23)))) ).
% filtermap_inf
tff(fact_6130_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya: A,Y: B,Xa: B,R: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( pp(member2(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))))
=> ( pp(member2(B,Yb,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B)))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z4),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z4),Yb)) ) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),bNF_id_bnf(fun(A,fun(B,bool)),R),X),Y))
<=> pp(aa(B,bool,aa(A,fun(B,bool),bNF_id_bnf(fun(A,fun(B,bool)),Ra),Ya),Xa)) ) ) ) ) ).
% ID.rel_cong
tff(fact_6131_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: A,Y: B,Ra: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),bNF_id_bnf(fun(A,fun(B,bool)),R),X),Y))
=> ( ! [Z4: A,Yb: B] :
( pp(member2(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
=> ( pp(member2(B,Yb,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Y),bot_bot(set(B)))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z4),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z4),Yb)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),bNF_id_bnf(fun(A,fun(B,bool)),Ra),X),Y)) ) ) ).
% ID.rel_mono_strong
tff(fact_6132_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra: fun(A,fun(A,bool))] :
( ! [Z4: A] :
( pp(member2(A,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z4),Z4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),bNF_id_bnf(fun(A,fun(A,bool)),Ra),X),X)) ) ).
% ID.rel_refl_strong
tff(fact_6133_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set(A),P: fun(A,fun(A,bool)),C4: set(A),Z2: A] :
( pred_chain(A,A4,P,C4)
=> ( pp(member2(A,Z2,A4))
=> ( ! [X2: A] :
( pp(member2(A,X2,C4))
=> pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X2),Z2)) )
=> pred_chain(A,A4,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)))),C4)) ) ) ) ).
% pred_on.chain_extend
tff(fact_6134_sup2CI,axiom,
! [A: $tType,B: $tType,B3: fun(A,fun(B,bool)),X: A,Y: B,A4: fun(A,fun(B,bool))] :
( ( ~ pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y)) )
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A4),B3),X),Y)) ) ).
% sup2CI
tff(fact_6135_sup2I2,axiom,
! [A: $tType,B: $tType,B3: fun(A,fun(B,bool)),X: A,Y: B,A4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A4),B3),X),Y)) ) ).
% sup2I2
tff(fact_6136_sup2I1,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X: A,Y: B,B3: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A4),B3),X),Y)) ) ).
% sup2I1
tff(fact_6137_sup2E,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool)),X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A4),B3),X),Y))
=> ( ~ pp(aa(B,bool,aa(A,fun(B,bool),A4,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),B3,X),Y)) ) ) ).
% sup2E
tff(fact_6138_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set(A),P: fun(A,fun(A,bool))] : pred_chain(A,A4,P,bot_bot(set(A))) ).
% pred_on.chain_empty
tff(fact_6139_ATP_Olambda__1,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ug(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).
% ATP.lambda_1
tff(fact_6140_ATP_Olambda__2,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_ee(nat,nat),Uu) = minus_minus(nat,Uu,aa(nat,nat,suc,zero_zero(nat))) ).
% ATP.lambda_2
tff(fact_6141_ATP_Olambda__3,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_db(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_3
tff(fact_6142_ATP_Olambda__4,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_hn(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_4
tff(fact_6143_ATP_Olambda__5,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_sb(A,bool),Uu))
<=> pp(dvd_dvd(A,Uu,one_one(A))) ) ) ).
% ATP.lambda_5
tff(fact_6144_ATP_Olambda__6,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,list(A),aTP_Lamp_eq(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).
% ATP.lambda_6
tff(fact_6145_ATP_Olambda__7,axiom,
! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_fb(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_7
tff(fact_6146_ATP_Olambda__8,axiom,
! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_ap(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_8
tff(fact_6147_ATP_Olambda__9,axiom,
! [B: $tType,Uu: list(B)] :
( pp(aa(list(B),bool,aTP_Lamp_ul(list(B),bool),Uu))
<=> ( Uu = nil(B) ) ) ).
% ATP.lambda_9
tff(fact_6148_ATP_Olambda__10,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_uk(list(A),bool),Uu))
<=> ( Uu = nil(A) ) ) ).
% ATP.lambda_10
tff(fact_6149_ATP_Olambda__11,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_fs(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),minus_minus(nat,aa(list(B),nat,size_size(list(B)),Uu),aa(nat,nat,suc,zero_zero(nat)))) ).
% ATP.lambda_11
tff(fact_6150_ATP_Olambda__12,axiom,
! [B: $tType,Uu: list(B)] :
( pp(aa(list(B),bool,aTP_Lamp_ft(list(B),bool),Uu))
<=> ( Uu != nil(B) ) ) ).
% ATP.lambda_12
tff(fact_6151_ATP_Olambda__13,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_fj(list(A),bool),Uu))
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_13
tff(fact_6152_ATP_Olambda__14,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_hs(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = product_case_prod(A,C,product_prod(A,product_prod(B,C)),aTP_Lamp_hr(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_14
tff(fact_6153_ATP_Olambda__15,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_fr(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).
% ATP.lambda_15
tff(fact_6154_ATP_Olambda__16,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_dk(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ).
% ATP.lambda_16
tff(fact_6155_ATP_Olambda__17,axiom,
! [A: $tType,B: $tType,Uu: product_prod(product_prod(bool,A),product_prod(bool,B))] :
( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_jy(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu))
<=> ? [X4: A,Y3: B] : Uu = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fTrue),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y3)) ) ).
% ATP.lambda_17
tff(fact_6156_ATP_Olambda__18,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fx(nat,fun(nat,A),Uu),Uua) = if(A,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).
% ATP.lambda_18
tff(fact_6157_ATP_Olambda__19,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_lp(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).
% ATP.lambda_19
tff(fact_6158_ATP_Olambda__20,axiom,
! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,option(A),aTP_Lamp_on(list(A),fun(nat,option(A)),Uu),Uua) = if(option(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).
% ATP.lambda_20
tff(fact_6159_ATP_Olambda__21,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fw(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).
% ATP.lambda_21
tff(fact_6160_ATP_Olambda__22,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ml(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),bot_bot(A)) ) ).
% ATP.lambda_22
tff(fact_6161_ATP_Olambda__23,axiom,
! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_ob(list(list(A)),fun(list(A),bool),Uu),Uua))
<=> pp(aa(list(list(A)),bool,aa(list(A),fun(list(list(A)),bool),aa(fun(A,fun(list(A),bool)),fun(list(A),fun(list(list(A)),bool)),list_all2(A,list(A)),aTP_Lamp_oa(A,fun(list(A),bool))),Uua),Uu)) ) ).
% ATP.lambda_23
tff(fact_6162_ATP_Olambda__24,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_oz(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_oy(A,fun(A,option(A)),Uua),Uu) ) ).
% ATP.lambda_24
tff(fact_6163_ATP_Olambda__25,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_ey(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ex(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).
% ATP.lambda_25
tff(fact_6164_ATP_Olambda__26,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hk(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),binomial(Uu,Uua))) ) ).
% ATP.lambda_26
tff(fact_6165_ATP_Olambda__27,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_ea(set(A),fun(list(A),bool),Uu),Uua))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu))
& pp(aa(list(A),bool,distinct(A),Uua)) ) ) ).
% ATP.lambda_27
tff(fact_6166_ATP_Olambda__28,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_df(set(A),fun(list(A),bool),Uu),Uua))
<=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
& pp(aa(list(A),bool,distinct(A),Uua)) ) ) ).
% ATP.lambda_28
tff(fact_6167_ATP_Olambda__29,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hm(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),binomial(Uu,Uua))) ) ).
% ATP.lambda_29
tff(fact_6168_ATP_Olambda__30,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_po(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_30
tff(fact_6169_ATP_Olambda__31,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gt(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_31
tff(fact_6170_ATP_Olambda__32,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fy(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_32
tff(fact_6171_ATP_Olambda__33,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hb(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_33
tff(fact_6172_ATP_Olambda__34,axiom,
! [A: $tType,Uu: fun(set(A),bool),Uua: set(A)] :
( pp(aa(set(A),bool,aa(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_kt(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
<=> ( ( Uua = bot_bot(set(A)) )
| ? [A9: set(A),A6: A] :
( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A6),A9) )
& pp(aa(set(A),bool,Uu,A9)) ) ) ) ).
% ATP.lambda_34
tff(fact_6173_ATP_Olambda__35,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ls(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_35
tff(fact_6174_ATP_Olambda__36,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_qc(set(A),fun(list(A),bool),Uu),Uua))
<=> ( sorted_wrt(A,ord_less(A),Uua)
& ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).
% ATP.lambda_36
tff(fact_6175_ATP_Olambda__37,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_md(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,comp(product_prod(A,B),B,C,product_snd(A,B),Uu)),top_top(set(C))) ).
% ATP.lambda_37
tff(fact_6176_ATP_Olambda__38,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_et(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_38
tff(fact_6177_ATP_Olambda__39,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ga(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_39
tff(fact_6178_ATP_Olambda__40,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_rd(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ( pp(aa(A,bool,Uu,Uua))
& ! [Y3: A] :
( pp(aa(A,bool,Uu,Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y3)) ) ) ) ) ).
% ATP.lambda_40
tff(fact_6179_ATP_Olambda__41,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_re(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ( pp(aa(A,bool,Uu,Uua))
& ! [Y3: A] :
( pp(aa(A,bool,Uu,Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Uua)) ) ) ) ) ).
% ATP.lambda_41
tff(fact_6180_ATP_Olambda__42,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_bx(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_42
tff(fact_6181_ATP_Olambda__43,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: A] : aa(A,assn,aTP_Lamp_tq(fun(A,assn),fun(A,assn),Uu),Uua) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uua)),top_top(assn)) ).
% ATP.lambda_43
tff(fact_6182_ATP_Olambda__44,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_av(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).
% ATP.lambda_44
tff(fact_6183_ATP_Olambda__45,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_ds(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_45
tff(fact_6184_ATP_Olambda__46,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_dj(fun(B,A),fun(B,bool),Uu),Uua))
<=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).
% ATP.lambda_46
tff(fact_6185_ATP_Olambda__47,axiom,
! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_ui(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),aa(fun(fun(A,nat),nat),fun(list(fun(A,nat)),list(nat)),map(fun(A,nat),nat),aTP_Lamp_uh(A,fun(fun(A,nat),nat),Uua)),Uu) ).
% ATP.lambda_47
tff(fact_6186_ATP_Olambda__48,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_do(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),aa(fun(A,list(A)),fun(list(A),list(list(A))),map(A,list(A)),aa(list(A),fun(A,list(A)),aTP_Lamp_dn(list(A),fun(A,list(A))),Uua)),Uu) ).
% ATP.lambda_48
tff(fact_6187_ATP_Olambda__49,axiom,
! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_sd(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,Uua))
& ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Uu),Uua)) ) ) ).
% ATP.lambda_49
tff(fact_6188_ATP_Olambda__50,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_sz(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,Uua))
& ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Uu,Uua)) ) ) ).
% ATP.lambda_50
tff(fact_6189_ATP_Olambda__51,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_nu(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),collect(product_prod(A,B),product_case_prod(A,B,bool,Uu)))) ) ).
% ATP.lambda_51
tff(fact_6190_ATP_Olambda__52,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_un(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uua)),collect(A,domainp(A,B,Uu))) ).
% ATP.lambda_52
tff(fact_6191_ATP_Olambda__53,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_uo(fun(A,fun(B,bool)),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Uua)),collect(A,domainp(A,B,Uu))) ).
% ATP.lambda_53
tff(fact_6192_ATP_Olambda__54,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_lb(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu)),aa(product_prod(A,B),A,product_fst(A,B),Uua))) ) ) ).
% ATP.lambda_54
tff(fact_6193_ATP_Olambda__55,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_lc(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu))) ) ) ).
% ATP.lambda_55
tff(fact_6194_ATP_Olambda__56,axiom,
! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_uj(list(option(A)),fun(A,option(list(A))),Uu),Uua) = map_option(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),aa(list(option(A)),option(list(A)),those(A),Uu)) ).
% ATP.lambda_56
tff(fact_6195_ATP_Olambda__57,axiom,
! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_dr(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),aa(list(list(A)),list(list(A)),product_lists(A),Uu)) ).
% ATP.lambda_57
tff(fact_6196_ATP_Olambda__58,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_jr(list(A),fun(list(A),bool)),Uu),Uua))
<=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_58
tff(fact_6197_ATP_Olambda__59,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_mn(set(A),fun(list(A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)) ) ).
% ATP.lambda_59
tff(fact_6198_ATP_Olambda__60,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hf(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_60
tff(fact_6199_ATP_Olambda__61,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_dg(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(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_61
tff(fact_6200_ATP_Olambda__62,axiom,
! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_ll(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(member2(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu)) ) ).
% ATP.lambda_62
tff(fact_6201_ATP_Olambda__63,axiom,
! [Uu: set(nat),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_kr(set(nat),fun(nat,bool),Uu),Uua))
<=> pp(member2(nat,aa(nat,nat,suc,Uua),Uu)) ) ).
% ATP.lambda_63
tff(fact_6202_ATP_Olambda__64,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_ms(nat,fun(list(A),bool),Uu),Uua))
<=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).
% ATP.lambda_64
tff(fact_6203_ATP_Olambda__65,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_65
tff(fact_6204_ATP_Olambda__66,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_if(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_66
tff(fact_6205_ATP_Olambda__67,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aa(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_67
tff(fact_6206_ATP_Olambda__68,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_hp(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_68
tff(fact_6207_ATP_Olambda__69,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_pn(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_69
tff(fact_6208_ATP_Olambda__70,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_ng(fun(A,B),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).
% ATP.lambda_70
tff(fact_6209_ATP_Olambda__71,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_nl(set(A),fun(A,bool),Uu),Uua))
<=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).
% ATP.lambda_71
tff(fact_6210_ATP_Olambda__72,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_dp(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).
% ATP.lambda_72
tff(fact_6211_ATP_Olambda__73,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_by(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_73
tff(fact_6212_ATP_Olambda__74,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_fd(nat,fun(list(A),bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).
% ATP.lambda_74
tff(fact_6213_ATP_Olambda__75,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eu(A,fun(nat,A),Uu),Uua) = minus_minus(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_75
tff(fact_6214_ATP_Olambda__76,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_pa(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_76
tff(fact_6215_ATP_Olambda__77,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_oa(A,fun(list(A),bool)),Uu),Uua))
<=> pp(member2(A,Uu,aa(list(A),set(A),set2(A),Uua))) ) ).
% ATP.lambda_77
tff(fact_6216_ATP_Olambda__78,axiom,
! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_ej(list(set(A)),fun(set(A),bool),Uu),Uua))
<=> pp(member2(set(A),Uua,aa(list(set(A)),set(set(A)),set2(set(A)),Uu))) ) ).
% ATP.lambda_78
tff(fact_6217_ATP_Olambda__79,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_fp(list(A),fun(A,bool),Uu),Uua))
<=> pp(member2(A,Uua,aa(list(A),set(A),set2(A),Uu))) ) ).
% ATP.lambda_79
tff(fact_6218_ATP_Olambda__80,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_mr(nat,fun(list(A),bool),Uu),Uua))
<=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_80
tff(fact_6219_ATP_Olambda__81,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_of(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).
% ATP.lambda_81
tff(fact_6220_ATP_Olambda__82,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_oe(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua = last(A,Uu) ) ) ).
% ATP.lambda_82
tff(fact_6221_ATP_Olambda__83,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_bk(set(A),fun(set(A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).
% ATP.lambda_83
tff(fact_6222_ATP_Olambda__84,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ke(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_84
tff(fact_6223_ATP_Olambda__85,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_tb(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_85
tff(fact_6224_ATP_Olambda__86,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pq(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_86
tff(fact_6225_ATP_Olambda__87,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ty(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_87
tff(fact_6226_ATP_Olambda__88,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ao(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_88
tff(fact_6227_ATP_Olambda__89,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_tm(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_89
tff(fact_6228_ATP_Olambda__90,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_bf(nat,fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).
% ATP.lambda_90
tff(fact_6229_ATP_Olambda__91,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_kf(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_91
tff(fact_6230_ATP_Olambda__92,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_tc(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_92
tff(fact_6231_ATP_Olambda__93,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pr(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_93
tff(fact_6232_ATP_Olambda__94,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_uc(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_94
tff(fact_6233_ATP_Olambda__95,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ta(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_95
tff(fact_6234_ATP_Olambda__96,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_tn(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_96
tff(fact_6235_ATP_Olambda__97,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ek(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_97
tff(fact_6236_ATP_Olambda__98,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cl(A,fun(A,A),Uu),Uua) = minus_minus(A,Uua,Uu) ) ).
% ATP.lambda_98
tff(fact_6237_ATP_Olambda__99,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_da(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_99
tff(fact_6238_ATP_Olambda__100,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).
% ATP.lambda_100
tff(fact_6239_ATP_Olambda__101,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).
% ATP.lambda_101
tff(fact_6240_ATP_Olambda__102,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: ref(A)] : aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_kn(A,fun(ref(A),assn)),Uu),Uua) = sngr_assn(A,Uua,Uu) ) ).
% ATP.lambda_102
tff(fact_6241_ATP_Olambda__103,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_km(list(A),fun(array(A),assn)),Uu),Uua) = snga_assn(A,Uua,Uu) ) ).
% ATP.lambda_103
tff(fact_6242_ATP_Olambda__104,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_ht(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_104
tff(fact_6243_ATP_Olambda__105,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ks(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).
% ATP.lambda_105
tff(fact_6244_ATP_Olambda__106,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aTP_Lamp_la(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_106
tff(fact_6245_ATP_Olambda__107,axiom,
! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_dn(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).
% ATP.lambda_107
tff(fact_6246_ATP_Olambda__108,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_kd(set(A),fun(A,bool),Uu),Uua))
<=> pp(member2(A,Uua,Uu)) ) ) ).
% ATP.lambda_108
tff(fact_6247_ATP_Olambda__109,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_mq(set(A),fun(A,bool),Uu),Uua))
<=> pp(member2(A,Uua,Uu)) ) ) ).
% ATP.lambda_109
tff(fact_6248_ATP_Olambda__110,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
<=> pp(member2(A,Uua,Uu)) ) ).
% ATP.lambda_110
tff(fact_6249_ATP_Olambda__111,axiom,
! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_qe(list(A),fun(nat,list(A)),Uu),Uua) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Uua),Uu) ).
% ATP.lambda_111
tff(fact_6250_ATP_Olambda__112,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_fv(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).
% ATP.lambda_112
tff(fact_6251_ATP_Olambda__113,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ad(A,fun(A,bool),Uu),Uua))
<=> ( Uua = Uu ) ) ).
% ATP.lambda_113
tff(fact_6252_ATP_Olambda__114,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_er(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_114
tff(fact_6253_ATP_Olambda__115,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_es(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).
% ATP.lambda_115
tff(fact_6254_ATP_Olambda__116,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_ni(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_116
tff(fact_6255_ATP_Olambda__117,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_uf(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).
% ATP.lambda_117
tff(fact_6256_ATP_Olambda__118,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_lf(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).
% ATP.lambda_118
tff(fact_6257_ATP_Olambda__119,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_lg(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).
% ATP.lambda_119
tff(fact_6258_ATP_Olambda__120,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_lh(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_120
tff(fact_6259_ATP_Olambda__121,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_lj(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_121
tff(fact_6260_ATP_Olambda__122,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_122
tff(fact_6261_ATP_Olambda__123,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).
% ATP.lambda_123
tff(fact_6262_ATP_Olambda__124,axiom,
! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_uh(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).
% ATP.lambda_124
tff(fact_6263_ATP_Olambda__125,axiom,
! [B: $tType,A: $tType,Uu: fun(set(A),A),Uua: B] : aa(B,A,aTP_Lamp_ur(fun(set(A),A),fun(B,A),Uu),Uua) = aa(set(A),A,Uu,bot_bot(set(A))) ).
% ATP.lambda_125
tff(fact_6264_ATP_Olambda__126,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_pd(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).
% ATP.lambda_126
tff(fact_6265_ATP_Olambda__127,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(bool)),aTP_Lamp_pc(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Uu),Uua) = product_case_prod(A,list(A),option(bool),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_pb(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua)) ).
% ATP.lambda_127
tff(fact_6266_ATP_Olambda__128,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_ov(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_ou(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).
% ATP.lambda_128
tff(fact_6267_ATP_Olambda__129,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),list(B)),aTP_Lamp_iq(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu),Uua) = product_case_prod(list(B),list(B),list(B),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_ip(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua)) ) ).
% ATP.lambda_129
tff(fact_6268_ATP_Olambda__130,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_ic(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = product_case_prod(B,A,C,aa(C,fun(B,fun(A,C)),aTP_Lamp_ib(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).
% ATP.lambda_130
tff(fact_6269_ATP_Olambda__131,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_pj(fun(A,option(B)),fun(A,bool),Uu),Uua))
<=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).
% ATP.lambda_131
tff(fact_6270_ATP_Olambda__132,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_lq(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),aa(fun(product_prod(C,B),list(product_prod(A,B))),fun(list(product_prod(C,B)),list(list(product_prod(A,B)))),map(product_prod(C,B),list(product_prod(A,B))),aTP_Lamp_lp(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).
% ATP.lambda_132
tff(fact_6271_ATP_Olambda__133,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fc(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_133
tff(fact_6272_ATP_Olambda__134,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fe(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_134
tff(fact_6273_ATP_Olambda__135,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_fq(list(A),fun(A,bool),Uu),Uua))
<=> ~ pp(member2(A,Uua,aa(list(A),set(A),set2(A),Uu))) ) ).
% ATP.lambda_135
tff(fact_6274_ATP_Olambda__136,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ij(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua != last(A,Uu) ) ) ).
% ATP.lambda_136
tff(fact_6275_ATP_Olambda__137,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_oy(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).
% ATP.lambda_137
tff(fact_6276_ATP_Olambda__138,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_in(A,fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uua)) ) ) ).
% ATP.lambda_138
tff(fact_6277_ATP_Olambda__139,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bh(set(A),fun(A,bool),Uu),Uua))
<=> ~ pp(member2(A,Uua,Uu)) ) ).
% ATP.lambda_139
tff(fact_6278_ATP_Olambda__140,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_fh(A,fun(A,bool)),Uu),Uua))
<=> ( Uu != Uua ) ) ).
% ATP.lambda_140
tff(fact_6279_ATP_Olambda__141,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_fo(A,fun(A,bool)),Uu),Uua))
<=> ( Uua != Uu ) ) ).
% ATP.lambda_141
tff(fact_6280_ATP_Olambda__142,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_mz(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_142
tff(fact_6281_ATP_Olambda__143,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_ge(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_143
tff(fact_6282_ATP_Olambda__144,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_nt(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_144
tff(fact_6283_ATP_Olambda__145,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fu(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_145
tff(fact_6284_ATP_Olambda__146,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_op(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).
% ATP.lambda_146
tff(fact_6285_ATP_Olambda__147,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_pf(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).
% ATP.lambda_147
tff(fact_6286_ATP_Olambda__148,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_hv(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_148
tff(fact_6287_ATP_Olambda__149,axiom,
! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_ut(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).
% ATP.lambda_149
tff(fact_6288_ATP_Olambda__150,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_uu(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_150
tff(fact_6289_ATP_Olambda__151,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_id(fun(A,set(B)),fun(A,nat),Uu),Uua) = finite_card(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_151
tff(fact_6290_ATP_Olambda__152,axiom,
! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_dm(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ).
% ATP.lambda_152
tff(fact_6291_ATP_Olambda__153,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_ia(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_153
tff(fact_6292_ATP_Olambda__154,axiom,
! [B: $tType,Uu: fun(B,bool),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_am(fun(B,bool),fun(B,bool),Uu),Uua))
<=> ~ pp(aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_154
tff(fact_6293_ATP_Olambda__155,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bi(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_155
tff(fact_6294_ATP_Olambda__156,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_qi(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
<=> ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ).
% ATP.lambda_156
tff(fact_6295_ATP_Olambda__157,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ud(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aTP_Lamp_bf(nat,fun(nat,bool),Uu)) ).
% ATP.lambda_157
tff(fact_6296_ATP_Olambda__158,axiom,
! [A: $tType,Uu: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_sy(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
<=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu,X4),Uua)) ) ).
% ATP.lambda_158
tff(fact_6297_ATP_Olambda__159,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_qj(fun(A,fun(B,bool)),fun(B,bool),Uu),Uua))
<=> ? [A6: A] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,A6),Uua)) ) ).
% ATP.lambda_159
tff(fact_6298_ATP_Olambda__160,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_jm(list(A),fun(A,bool),Uu),Uua))
<=> ? [I2: nat] :
( ( Uua = aa(nat,A,nth(A,Uu),I2) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).
% ATP.lambda_160
tff(fact_6299_ATP_Olambda__161,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_jz(set(A),fun(set(A),bool),Uu),Uua))
<=> ? [B9: set(A)] :
( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B9) )
& pp(member2(set(A),Uu,pow(A,B9))) ) ) ).
% ATP.lambda_161
tff(fact_6300_ATP_Olambda__162,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_qb(set(A),fun(A,bool),Uu),Uua))
<=> ! [X4: A] :
( pp(member2(A,X4,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).
% ATP.lambda_162
tff(fact_6301_ATP_Olambda__163,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_pv(set(A),fun(A,bool),Uu),Uua))
<=> ! [X4: A] :
( pp(member2(A,X4,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).
% ATP.lambda_163
tff(fact_6302_ATP_Olambda__164,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_sv(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
<=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,aa(A,assn,Uu,X4)),Uua)) ) ).
% ATP.lambda_164
tff(fact_6303_ATP_Olambda__165,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_ol(A,fun(product_prod(A,B),bool),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_165
tff(fact_6304_ATP_Olambda__166,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_kx(fun(B,A),fun(A,bool),Uu),Uua))
<=> ? [X4: B] : Uua = aa(B,A,Uu,X4) ) ).
% ATP.lambda_166
tff(fact_6305_ATP_Olambda__167,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod(bool,A),product_prod(bool,B))] :
( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_jx(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu),Uua))
<=> ? [X4: A,Y3: B] :
( ( Uua = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fFalse),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y3)) )
& pp(member2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),Uu)) ) ) ).
% ATP.lambda_167
tff(fact_6306_ATP_Olambda__168,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_qq(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
<=> ? [X11: 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)),X11),Y8) )
& ( X11 != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member2(A,X4,Y8))
=> ? [Xa2: A] :
( pp(member2(A,Xa2,X11))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu)) ) ) ) ) ).
% ATP.lambda_168
tff(fact_6307_ATP_Olambda__169,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_ns(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).
% ATP.lambda_169
tff(fact_6308_ATP_Olambda__170,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_ow(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).
% ATP.lambda_170
tff(fact_6309_ATP_Olambda__171,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_te(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).
% ATP.lambda_171
tff(fact_6310_ATP_Olambda__172,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_tf(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_172
tff(fact_6311_ATP_Olambda__173,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_mm(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_173
tff(fact_6312_ATP_Olambda__174,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_mg(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).
% ATP.lambda_174
tff(fact_6313_ATP_Olambda__175,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_oh(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).
% ATP.lambda_175
tff(fact_6314_ATP_Olambda__176,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_mk(set(A),fun(list(A),set(list(A))),Uu),Uua) = aa(set(A),set(list(A)),lists(A),Uu) ).
% ATP.lambda_176
tff(fact_6315_ATP_Olambda__177,axiom,
! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_ai(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).
% ATP.lambda_177
tff(fact_6316_ATP_Olambda__178,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_ho(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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_178
tff(fact_6317_ATP_Olambda__179,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ma(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uu),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub))) ).
% ATP.lambda_179
tff(fact_6318_ATP_Olambda__180,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_gj(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,member2(B,Uub,Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).
% ATP.lambda_180
tff(fact_6319_ATP_Olambda__181,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_di(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,member2(B,Uub,Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).
% ATP.lambda_181
tff(fact_6320_ATP_Olambda__182,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aj(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).
% ATP.lambda_182
tff(fact_6321_ATP_Olambda__183,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ak(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).
% ATP.lambda_183
tff(fact_6322_ATP_Olambda__184,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rl(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,zero_zero(A)) ) ).
% ATP.lambda_184
tff(fact_6323_ATP_Olambda__185,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_ro(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,one_one(A)) ) ).
% ATP.lambda_185
tff(fact_6324_ATP_Olambda__186,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rk(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,zero_zero(A)) ) ).
% ATP.lambda_186
tff(fact_6325_ATP_Olambda__187,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_rp(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,one_one(A)) ) ).
% ATP.lambda_187
tff(fact_6326_ATP_Olambda__188,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_fn(fun(A,bool),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).
% ATP.lambda_188
tff(fact_6327_ATP_Olambda__189,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_ig(fun(A,bool),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = if(set(A),aa(A,bool,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).
% ATP.lambda_189
tff(fact_6328_ATP_Olambda__190,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ru(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).
% ATP.lambda_190
tff(fact_6329_ATP_Olambda__191,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_cs(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).
% ATP.lambda_191
tff(fact_6330_ATP_Olambda__192,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,bool),Uub: B] : aa(B,fun(A,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_np(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),Uu),Uua),Uub) = if(fun(A,A),aa(B,bool,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).
% ATP.lambda_192
tff(fact_6331_ATP_Olambda__193,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_os(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).
% ATP.lambda_193
tff(fact_6332_ATP_Olambda__194,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_lo(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).
% ATP.lambda_194
tff(fact_6333_ATP_Olambda__195,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ql(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_195
tff(fact_6334_ATP_Olambda__196,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_dd(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu,Uub),Uua) ).
% ATP.lambda_196
tff(fact_6335_ATP_Olambda__197,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_td(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_197
tff(fact_6336_ATP_Olambda__198,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_de(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu,Uub),Uua) ).
% ATP.lambda_198
tff(fact_6337_ATP_Olambda__199,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ay(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_199
tff(fact_6338_ATP_Olambda__200,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_gd(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_200
tff(fact_6339_ATP_Olambda__201,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_il(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
| ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& pp(member2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu))) ) ) ) ).
% ATP.lambda_201
tff(fact_6340_ATP_Olambda__202,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_qn(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
& ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).
% ATP.lambda_202
tff(fact_6341_ATP_Olambda__203,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_jk(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& ? [Xys2: list(A),X4: A,Y3: A,Xs5: 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),X4),Xs5)) )
& ( 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)) )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu)) ) ) ) ).
% ATP.lambda_203
tff(fact_6342_ATP_Olambda__204,axiom,
! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_ez(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
& pp(aa(list(A),bool,distinct(A),Uub))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).
% ATP.lambda_204
tff(fact_6343_ATP_Olambda__205,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_aw(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& pp(aa(list(A),bool,distinct(A),Uub))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).
% ATP.lambda_205
tff(fact_6344_ATP_Olambda__206,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_mt(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).
% ATP.lambda_206
tff(fact_6345_ATP_Olambda__207,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ed(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).
% ATP.lambda_207
tff(fact_6346_ATP_Olambda__208,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ei(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).
% ATP.lambda_208
tff(fact_6347_ATP_Olambda__209,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).
% ATP.lambda_209
tff(fact_6348_ATP_Olambda__210,axiom,
! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_lm(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
<=> pp(member2(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu)) ) ).
% ATP.lambda_210
tff(fact_6349_ATP_Olambda__211,axiom,
! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_dl(set(list(A)),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A))),Uu)) ) ).
% ATP.lambda_211
tff(fact_6350_ATP_Olambda__212,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_kq(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
& pp(member2(nat,Uub,Uua)) ) ) ).
% ATP.lambda_212
tff(fact_6351_ATP_Olambda__213,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_fm(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
& pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).
% ATP.lambda_213
tff(fact_6352_ATP_Olambda__214,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ox(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,aa(list(A),set(A),set2(A),Uu)))
& pp(aa(A,bool,Uua,Uub)) ) ) ) ).
% ATP.lambda_214
tff(fact_6353_ATP_Olambda__215,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_fg(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,aa(list(A),set(A),set2(A),Uua)))
& pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_215
tff(fact_6354_ATP_Olambda__216,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ko(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> pp(member2(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua)) ) ).
% ATP.lambda_216
tff(fact_6355_ATP_Olambda__217,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ue(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).
% ATP.lambda_217
tff(fact_6356_ATP_Olambda__218,axiom,
! [Uu: set(nat),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_qw(set(nat),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(member2(nat,Uub,Uu))
& pp(member2(nat,finite_card(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_qv(set(nat),fun(nat,fun(nat,bool)),Uu),Uub))),Uua)) ) ) ).
% ATP.lambda_218
tff(fact_6357_ATP_Olambda__219,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sh(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Uu),Uub))
| pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Uua),Uub)) ) ) ).
% ATP.lambda_219
tff(fact_6358_ATP_Olambda__220,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_bs(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
| pp(member2(A,Uub,Uua)) ) ) ).
% ATP.lambda_220
tff(fact_6359_ATP_Olambda__221,axiom,
! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_bm(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub = Uu )
| pp(member2(A,Uub,Uua)) ) ) ).
% ATP.lambda_221
tff(fact_6360_ATP_Olambda__222,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ra(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uua))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).
% ATP.lambda_222
tff(fact_6361_ATP_Olambda__223,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_si(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Uu),Uub))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),rep_assn,Uua),Uub)) ) ) ).
% ATP.lambda_223
tff(fact_6362_ATP_Olambda__224,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_qv(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(member2(nat,Uub,Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).
% ATP.lambda_224
tff(fact_6363_ATP_Olambda__225,axiom,
! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
( pp(aa(set(A),bool,aa(A,fun(set(A),bool),aTP_Lamp_rw(set(set(A)),fun(A,fun(set(A),bool)),Uu),Uua),Uub))
<=> ( pp(member2(set(A),Uub,Uu))
& pp(member2(A,Uua,Uub)) ) ) ).
% ATP.lambda_225
tff(fact_6364_ATP_Olambda__226,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_bv(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& pp(member2(A,Uub,Uua)) ) ) ).
% ATP.lambda_226
tff(fact_6365_ATP_Olambda__227,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_dc(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( minus_minus(set(A),Uub,Uu) = minus_minus(set(A),Uua,Uu) ) ) ).
% ATP.lambda_227
tff(fact_6366_ATP_Olambda__228,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_us(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uua))
=> pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_228
tff(fact_6367_ATP_Olambda__229,axiom,
! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_cr(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member2(B,Uub,Uu))
& pp(aa(B,bool,Uua,Uub)) ) ) ).
% ATP.lambda_229
tff(fact_6368_ATP_Olambda__230,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: set(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_kh(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& pp(aa(A,bool,Uua,Uub)) ) ) ) ).
% ATP.lambda_230
tff(fact_6369_ATP_Olambda__231,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bl(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_231
tff(fact_6370_ATP_Olambda__232,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ih(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uua))
& pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_232
tff(fact_6371_ATP_Olambda__233,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bp(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uua = Uub )
& pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_233
tff(fact_6372_ATP_Olambda__234,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bq(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub = Uua )
& pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_234
tff(fact_6373_ATP_Olambda__235,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_mo(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).
% ATP.lambda_235
tff(fact_6374_ATP_Olambda__236,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_io(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).
% ATP.lambda_236
tff(fact_6375_ATP_Olambda__237,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ch(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member2(B,Uub,Uu))
& ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_237
tff(fact_6376_ATP_Olambda__238,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_kl(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member2(B,Uub,Uua))
& ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_238
tff(fact_6377_ATP_Olambda__239,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_tt(array(A),fun(list(A),fun(nat,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))) ) ).
% ATP.lambda_239
tff(fact_6378_ATP_Olambda__240,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_tv(ref(A),fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,Uu,Uua)),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua))) ) ).
% ATP.lambda_240
tff(fact_6379_ATP_Olambda__241,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: list(A),Uub: list(A)] : aa(list(A),assn,aa(list(A),fun(list(A),assn),aTP_Lamp_tw(array(A),fun(list(A),fun(list(A),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Uub),Uua))) ) ).
% ATP.lambda_241
tff(fact_6380_ATP_Olambda__242,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_cb(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& ~ pp(member2(A,Uub,Uua)) ) ) ).
% ATP.lambda_242
tff(fact_6381_ATP_Olambda__243,axiom,
! [A: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,bool)),Uua: fun(C,A),Uub: set(A)] : aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),aTP_Lamp_um(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uua),Uub)),collect(C,domainp(C,D,Uu))) ).
% ATP.lambda_243
tff(fact_6382_ATP_Olambda__244,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_eo(A,fun(A,fun(A,A)),Uu),Uua),Uub) = minus_minus(A,divide_divide(A,Uub,Uu),Uua) ) ).
% ATP.lambda_244
tff(fact_6383_ATP_Olambda__245,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_em(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_245
tff(fact_6384_ATP_Olambda__246,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_en(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_246
tff(fact_6385_ATP_Olambda__247,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_el(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_247
tff(fact_6386_ATP_Olambda__248,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_pi(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(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_248
tff(fact_6387_ATP_Olambda__249,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_bw(set(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(member2(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_249
tff(fact_6388_ATP_Olambda__250,axiom,
! [A: $tType,Uu: set(list(A)),Uua: A,Uub: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_cg(set(list(A)),fun(A,fun(list(A),bool)),Uu),Uua),Uub))
<=> pp(member2(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uu)) ) ).
% ATP.lambda_250
tff(fact_6389_ATP_Olambda__251,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ex(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).
% ATP.lambda_251
tff(fact_6390_ATP_Olambda__252,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: B,Uub: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_cu(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_252
tff(fact_6391_ATP_Olambda__253,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_cf(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_253
tff(fact_6392_ATP_Olambda__254,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_gu(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_254
tff(fact_6393_ATP_Olambda__255,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rt(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = minus_minus(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_255
tff(fact_6394_ATP_Olambda__256,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_pu(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_256
tff(fact_6395_ATP_Olambda__257,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_st(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_257
tff(fact_6396_ATP_Olambda__258,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lx(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_258
tff(fact_6397_ATP_Olambda__259,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qa(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_259
tff(fact_6398_ATP_Olambda__260,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lw(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_260
tff(fact_6399_ATP_Olambda__261,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_pz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_261
tff(fact_6400_ATP_Olambda__262,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_mw(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_262
tff(fact_6401_ATP_Olambda__263,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rs(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_263
tff(fact_6402_ATP_Olambda__264,axiom,
! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_qk(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = union_mset(A,aa(B,multiset(A),Uu,Uub),aa(B,multiset(A),Uua,Uub)) ).
% ATP.lambda_264
tff(fact_6403_ATP_Olambda__265,axiom,
! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_qh(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = inter_mset(A,aa(B,multiset(A),Uu,Uub),aa(B,multiset(A),Uua,Uub)) ).
% ATP.lambda_265
tff(fact_6404_ATP_Olambda__266,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_na(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_266
tff(fact_6405_ATP_Olambda__267,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,Uub))
=> pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_267
tff(fact_6406_ATP_Olambda__268,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sx(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Uu,Uub))
| pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Uua,Uub)) ) ) ).
% ATP.lambda_268
tff(fact_6407_ATP_Olambda__269,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bt(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,Uub))
| pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_269
tff(fact_6408_ATP_Olambda__270,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_sw(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Uu,Uub))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Uua,Uub)) ) ) ).
% ATP.lambda_270
tff(fact_6409_ATP_Olambda__271,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bu(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,Uub))
& pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_271
tff(fact_6410_ATP_Olambda__272,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ff(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uua,Uub))
& pp(aa(A,bool,Uu,Uub)) ) ) ).
% ATP.lambda_272
tff(fact_6411_ATP_Olambda__273,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: B,Uub: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_nv(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uu,Uua) = aa(B,A,Uu,Uub) ) ) ) ).
% ATP.lambda_273
tff(fact_6412_ATP_Olambda__274,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_fk(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).
% ATP.lambda_274
tff(fact_6413_ATP_Olambda__275,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ry(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).
% ATP.lambda_275
tff(fact_6414_ATP_Olambda__276,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_iu(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).
% ATP.lambda_276
tff(fact_6415_ATP_Olambda__277,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_gb(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).
% ATP.lambda_277
tff(fact_6416_ATP_Olambda__278,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_rf(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uua,Uub))
& ! [Y3: A] :
( pp(aa(A,bool,Uua,Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y3)) ) ) ) ).
% ATP.lambda_278
tff(fact_6417_ATP_Olambda__279,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_ss(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_279
tff(fact_6418_ATP_Olambda__280,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_oo(fun(A,option(B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).
% ATP.lambda_280
tff(fact_6419_ATP_Olambda__281,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).
% ATP.lambda_281
tff(fact_6420_ATP_Olambda__282,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_gg(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_282
tff(fact_6421_ATP_Olambda__283,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_sm(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_283
tff(fact_6422_ATP_Olambda__284,axiom,
! [E4: $tType,F6: $tType,Uu: fun(E4,set(F6)),Uua: set(F6),Uub: E4] : aa(E4,set(F6),aa(set(F6),fun(E4,set(F6)),aTP_Lamp_lv(fun(E4,set(F6)),fun(set(F6),fun(E4,set(F6))),Uu),Uua),Uub) = minus_minus(set(F6),aa(E4,set(F6),Uu,Uub),Uua) ).
% ATP.lambda_284
tff(fact_6423_ATP_Olambda__285,axiom,
! [E4: $tType,Uu: fun(E4,assn),Uua: assn,Uub: E4] : aa(E4,assn,aa(assn,fun(E4,assn),aTP_Lamp_so(fun(E4,assn),fun(assn,fun(E4,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(E4,assn,Uu,Uub)),Uua) ).
% ATP.lambda_285
tff(fact_6424_ATP_Olambda__286,axiom,
! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: set(D),Uub: C] : aa(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_as(fun(C,set(D)),fun(set(D),fun(C,set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(C,set(D),Uu,Uub)),Uua) ).
% ATP.lambda_286
tff(fact_6425_ATP_Olambda__287,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_su(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_287
tff(fact_6426_ATP_Olambda__288,axiom,
! [G4: $tType,H2: $tType,Uu: fun(G4,set(H2)),Uua: set(H2),Uub: G4] : aa(G4,set(H2),aa(set(H2),fun(G4,set(H2)),aTP_Lamp_cw(fun(G4,set(H2)),fun(set(H2),fun(G4,set(H2))),Uu),Uua),Uub) = aa(set(H2),set(H2),aa(set(H2),fun(set(H2),set(H2)),inf_inf(set(H2)),aa(G4,set(H2),Uu,Uub)),Uua) ).
% ATP.lambda_288
tff(fact_6427_ATP_Olambda__289,axiom,
! [C: $tType,Uu: fun(C,assn),Uua: assn,Uub: C] : aa(C,assn,aa(assn,fun(C,assn),aTP_Lamp_sp(fun(C,assn),fun(assn,fun(C,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(C,assn,Uu,Uub)),Uua) ).
% ATP.lambda_289
tff(fact_6428_ATP_Olambda__290,axiom,
! [B: $tType,A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_290
tff(fact_6429_ATP_Olambda__291,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_lz(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_291
tff(fact_6430_ATP_Olambda__292,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_sr(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_292
tff(fact_6431_ATP_Olambda__293,axiom,
! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lu(fun(A,set(B)),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),Uua) ).
% ATP.lambda_293
tff(fact_6432_ATP_Olambda__294,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_du(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_294
tff(fact_6433_ATP_Olambda__295,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_uz(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).
% ATP.lambda_295
tff(fact_6434_ATP_Olambda__296,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ne(fun(A,B),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(B,aa(A,B,Uu,Uub),Uua)) ) ).
% ATP.lambda_296
tff(fact_6435_ATP_Olambda__297,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ky(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(member2(A,aa(B,A,Uua,Uub),Uu)) ) ).
% ATP.lambda_297
tff(fact_6436_ATP_Olambda__298,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,set(C)),Uua: fun(C,set(B)),Uub: A] : aa(A,set(B),aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_qr(fun(A,set(C)),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua),Uub) = bind2(C,B,aa(A,set(C),Uu,Uub),Uua) ).
% ATP.lambda_298
tff(fact_6437_ATP_Olambda__299,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_od(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_299
tff(fact_6438_ATP_Olambda__300,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ix(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_300
tff(fact_6439_ATP_Olambda__301,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_nm(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ).
% ATP.lambda_301
tff(fact_6440_ATP_Olambda__302,axiom,
! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bn(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub != Uu )
=> pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_302
tff(fact_6441_ATP_Olambda__303,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_jg(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uua != Uu )
& ( Uub != Uu ) ) ) ).
% ATP.lambda_303
tff(fact_6442_ATP_Olambda__304,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rx(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ~ pp(aa(A,bool,Uu,Uub))
| ( Uua = Uub ) ) ) ).
% ATP.lambda_304
tff(fact_6443_ATP_Olambda__305,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gc(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_305
tff(fact_6444_ATP_Olambda__306,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_ka(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
<=> ( ? [Y3: A,Ys3: list(A)] :
( ( Uua = nil(A) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys3)) ) ) ) ) ).
% ATP.lambda_306
tff(fact_6445_ATP_Olambda__307,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_rc(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
& pp(aa(set(A),bool,finite_finite2(A),Uub))
& ( Uub != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member2(A,X4,Uua))
=> ? [Xa2: A] :
( pp(member2(A,Xa2,Uub))
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu)) ) ) ) ) ).
% ATP.lambda_307
tff(fact_6446_ATP_Olambda__308,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_ub(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(product_prod(A,A)),list(A),aa(fun(product_prod(A,A),A),fun(list(product_prod(A,A)),list(A)),map(product_prod(A,A),A),product_case_prod(A,A,A,Uu)),aa(list(A),list(product_prod(A,A)),aa(list(A),fun(list(A),list(product_prod(A,A))),zip(A,A),Uua),Uub)) ).
% ATP.lambda_308
tff(fact_6447_ATP_Olambda__309,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_rb(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_309
tff(fact_6448_ATP_Olambda__310,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: list(B),Uub: B] : aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_it(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub) = aa(fun(B,fun(list(B),list(B))),fun(list(B),list(B)),aa(list(B),fun(fun(B,fun(list(B),list(B))),fun(list(B),list(B))),case_list(list(B),B),Uua),aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_is(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub)) ) ).
% ATP.lambda_310
tff(fact_6449_ATP_Olambda__311,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: list(B),Uub: A] : aa(A,list(B),aa(list(B),fun(A,list(B)),aTP_Lamp_ef(fun(A,B),fun(list(B),fun(A,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uua),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,Uu,Uub)),nil(B))) ).
% ATP.lambda_311
tff(fact_6450_ATP_Olambda__312,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hl(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_312
tff(fact_6451_ATP_Olambda__313,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(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_313
tff(fact_6452_ATP_Olambda__314,axiom,
! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pp(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(A,Uub,minus_minus(set(A),Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))))) ) ).
% ATP.lambda_314
tff(fact_6453_ATP_Olambda__315,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: array(A)] : aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_ts(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),Uua),upt(zero_zero(nat),Uu))) ) ).
% ATP.lambda_315
tff(fact_6454_ATP_Olambda__316,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_ha(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_316
tff(fact_6455_ATP_Olambda__317,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_pk(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(A,Uub,minus_minus(set(A),Uua,dom(A,B,Uu)))) ) ).
% ATP.lambda_317
tff(fact_6456_ATP_Olambda__318,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_pl(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu)))) ) ).
% ATP.lambda_318
tff(fact_6457_ATP_Olambda__319,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_gk(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_319
tff(fact_6458_ATP_Olambda__320,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_tu(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Uu),Uua)) ) ).
% ATP.lambda_320
tff(fact_6459_ATP_Olambda__321,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_ii(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_321
tff(fact_6460_ATP_Olambda__322,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_hr(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_322
tff(fact_6461_ATP_Olambda__323,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_qd(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Uub),Uua)) ).
% ATP.lambda_323
tff(fact_6462_ATP_Olambda__324,axiom,
! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_ph(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).
% ATP.lambda_324
tff(fact_6463_ATP_Olambda__325,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_mp(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(member2(B,Uub,aa(set(A),set(B),image2(A,B,Uu),Uua))) ) ) ).
% ATP.lambda_325
tff(fact_6464_ATP_Olambda__326,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_eh(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_326
tff(fact_6465_ATP_Olambda__327,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_gs(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_327
tff(fact_6466_ATP_Olambda__328,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_im(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_328
tff(fact_6467_ATP_Olambda__329,axiom,
! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_sl(assn,fun(fun(B,assn),fun(B,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(B,assn,Uua,Uub)) ).
% ATP.lambda_329
tff(fact_6468_ATP_Olambda__330,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_gf(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_330
tff(fact_6469_ATP_Olambda__331,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_tp(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu,Uub)) ).
% ATP.lambda_331
tff(fact_6470_ATP_Olambda__332,axiom,
! [G4: $tType,H2: $tType,Uu: set(G4),Uua: fun(H2,set(G4)),Uub: H2] : aa(H2,set(G4),aa(fun(H2,set(G4)),fun(H2,set(G4)),aTP_Lamp_lr(set(G4),fun(fun(H2,set(G4)),fun(H2,set(G4))),Uu),Uua),Uub) = minus_minus(set(G4),Uu,aa(H2,set(G4),Uua,Uub)) ).
% ATP.lambda_332
tff(fact_6471_ATP_Olambda__333,axiom,
! [F6: $tType,Uu: assn,Uua: fun(F6,assn),Uub: F6] : aa(F6,assn,aa(fun(F6,assn),fun(F6,assn),aTP_Lamp_sn(assn,fun(fun(F6,assn),fun(F6,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Uu),aa(F6,assn,Uua,Uub)) ).
% ATP.lambda_333
tff(fact_6472_ATP_Olambda__334,axiom,
! [E4: $tType,F6: $tType,Uu: set(E4),Uua: fun(F6,set(E4)),Uub: F6] : aa(F6,set(E4),aa(fun(F6,set(E4)),fun(F6,set(E4)),aTP_Lamp_ar(set(E4),fun(fun(F6,set(E4)),fun(F6,set(E4))),Uu),Uua),Uub) = aa(set(E4),set(E4),aa(set(E4),fun(set(E4),set(E4)),sup_sup(set(E4)),Uu),aa(F6,set(E4),Uua,Uub)) ).
% ATP.lambda_334
tff(fact_6473_ATP_Olambda__335,axiom,
! [D: $tType,Uu: assn,Uua: fun(D,assn),Uub: D] : aa(D,assn,aa(fun(D,assn),fun(D,assn),aTP_Lamp_sq(assn,fun(fun(D,assn),fun(D,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Uu),aa(D,assn,Uua,Uub)) ).
% ATP.lambda_335
tff(fact_6474_ATP_Olambda__336,axiom,
! [I7: $tType,J5: $tType,Uu: set(I7),Uua: fun(J5,set(I7)),Uub: J5] : aa(J5,set(I7),aa(fun(J5,set(I7)),fun(J5,set(I7)),aTP_Lamp_cv(set(I7),fun(fun(J5,set(I7)),fun(J5,set(I7))),Uu),Uua),Uub) = aa(set(I7),set(I7),aa(set(I7),fun(set(I7),set(I7)),inf_inf(set(I7)),Uu),aa(J5,set(I7),Uua,Uub)) ).
% ATP.lambda_336
tff(fact_6475_ATP_Olambda__337,axiom,
! [C: $tType,D: $tType,Uu: set(C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lt(set(C),fun(fun(D,set(C)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),aa(D,set(C),Uua,Uub)) ).
% ATP.lambda_337
tff(fact_6476_ATP_Olambda__338,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_cx(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_338
tff(fact_6477_ATP_Olambda__339,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_co(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_339
tff(fact_6478_ATP_Olambda__340,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ly(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_340
tff(fact_6479_ATP_Olambda__341,axiom,
! [A: $tType,B: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_gq(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_341
tff(fact_6480_ATP_Olambda__342,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_va(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).
% ATP.lambda_342
tff(fact_6481_ATP_Olambda__343,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_hw(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_343
tff(fact_6482_ATP_Olambda__344,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_qm(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_344
tff(fact_6483_ATP_Olambda__345,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_aq(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_345
tff(fact_6484_ATP_Olambda__346,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ps(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub)) ) ).
% ATP.lambda_346
tff(fact_6485_ATP_Olambda__347,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_fi(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).
% ATP.lambda_347
tff(fact_6486_ATP_Olambda__348,axiom,
! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_tx(assn,fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua))) ).
% ATP.lambda_348
tff(fact_6487_ATP_Olambda__349,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_kp(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
<=> pp(member2(A,Uub,aa(list(A),set(A),set2(A),aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Uu),Uua)))) ) ).
% ATP.lambda_349
tff(fact_6488_ATP_Olambda__350,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: A] : aa(A,set(A),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_qo(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uua),field2(B,Uu)) ).
% ATP.lambda_350
tff(fact_6489_ATP_Olambda__351,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_nr(set(A),fun(set(A),fun(A,set(A))),Uu),Uua),Uub) = minus_minus(set(A),Uu,Uua) ).
% ATP.lambda_351
tff(fact_6490_ATP_Olambda__352,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_mx(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_352
tff(fact_6491_ATP_Olambda__353,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_qp(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).
% ATP.lambda_353
tff(fact_6492_ATP_Olambda__354,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_oc(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).
% ATP.lambda_354
tff(fact_6493_ATP_Olambda__355,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: product_unit] : aa(product_unit,assn,aa(A,fun(product_unit,assn),aTP_Lamp_ua(ref(A),fun(A,fun(product_unit,assn)),Uu),Uua),Uub) = sngr_assn(A,Uu,Uua) ) ).
% ATP.lambda_355
tff(fact_6494_ATP_Olambda__356,axiom,
! [A: $tType,Uu: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ba(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).
% ATP.lambda_356
tff(fact_6495_ATP_Olambda__357,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_mi(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_357
tff(fact_6496_ATP_Olambda__358,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_gm(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ).
% ATP.lambda_358
tff(fact_6497_ATP_Olambda__359,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_qg(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_359
tff(fact_6498_ATP_Olambda__360,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ny(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).
% ATP.lambda_360
tff(fact_6499_ATP_Olambda__361,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_kw(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).
% ATP.lambda_361
tff(fact_6500_ATP_Olambda__362,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_bc(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).
% ATP.lambda_362
tff(fact_6501_ATP_Olambda__363,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_pw(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_363
tff(fact_6502_ATP_Olambda__364,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_nx(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).
% ATP.lambda_364
tff(fact_6503_ATP_Olambda__365,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ri(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).
% ATP.lambda_365
tff(fact_6504_ATP_Olambda__366,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_nc(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).
% ATP.lambda_366
tff(fact_6505_ATP_Olambda__367,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_gv(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_367
tff(fact_6506_ATP_Olambda__368,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_ib(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).
% ATP.lambda_368
tff(fact_6507_ATP_Olambda__369,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ti(fun(A,bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_369
tff(fact_6508_ATP_Olambda__370,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_lk(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).
% ATP.lambda_370
tff(fact_6509_ATP_Olambda__371,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_qz(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = product_case_prod(A,A,bool,aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_qy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).
% ATP.lambda_371
tff(fact_6510_ATP_Olambda__372,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tj(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ? [X_12: C] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,Uua),Uub),X_12)) ) ).
% ATP.lambda_372
tff(fact_6511_ATP_Olambda__373,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_pt(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).
% ATP.lambda_373
tff(fact_6512_ATP_Olambda__374,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_py(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).
% ATP.lambda_374
tff(fact_6513_ATP_Olambda__375,axiom,
! [A: $tType,B: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_gr(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ~ pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_375
tff(fact_6514_ATP_Olambda__376,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: list(B)] : aa(list(B),fun(list(A),list(A)),aa(B,fun(list(B),fun(list(A),list(A))),aTP_Lamp_og(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),Uu),Uua),Uub) = aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)) ).
% ATP.lambda_376
tff(fact_6515_ATP_Olambda__377,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: A,Uub: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aTP_Lamp_tk(fun(A,fun(B,fun(C,bool))),fun(A,fun(C,bool)),Uu),Uua),Uub))
<=> ? [B5: B] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,Uua),B5),Uub)) ) ).
% ATP.lambda_377
tff(fact_6516_ATP_Olambda__378,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: B,Uub: C] :
( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_tl(fun(A,fun(B,fun(C,bool))),fun(B,fun(C,bool)),Uu),Uua),Uub))
<=> ? [A6: A] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,A6),Uua),Uub)) ) ).
% ATP.lambda_378
tff(fact_6517_ATP_Olambda__379,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
( pp(aa(A,bool,aa(set(C),fun(A,bool),aTP_Lamp_le(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),Uu),Uua),Uub))
<=> ? [B5: B] : pp(member2(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B5),Uua)) ) ).
% ATP.lambda_379
tff(fact_6518_ATP_Olambda__380,axiom,
! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_jb(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [I2: 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),I2)),aa(nat,B,nth(B,Uua),I2)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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_380
tff(fact_6519_ATP_Olambda__381,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_kk(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
<=> ? [I2: nat] :
( ( Uub = aa(nat,A,nth(A,Uu),I2) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uu)))
& pp(member2(nat,I2,Uua)) ) ) ).
% ATP.lambda_381
tff(fact_6520_ATP_Olambda__382,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_jt(nat,fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [I2: nat] :
( ( Uub = aa(nat,A,nth(A,Uua),I2) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uu),I2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Uua))) ) ) ).
% ATP.lambda_382
tff(fact_6521_ATP_Olambda__383,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: B] :
( pp(aa(B,bool,aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_qt(set(A),fun(fun(A,set(B)),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X4: set(B)] :
( pp(member2(set(B),X4,aa(set(A),set(set(B)),image2(A,set(B),Uua),Uu)))
& pp(member2(B,Uub,X4)) ) ) ).
% ATP.lambda_383
tff(fact_6522_ATP_Olambda__384,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_jq(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [A6: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A6) )
& pp(member2(A,A6,Uu)) ) ) ) ).
% ATP.lambda_384
tff(fact_6523_ATP_Olambda__385,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [Uu: A,Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_jv(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [B5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B5) )
& pp(member2(A,B5,Uua)) ) ) ) ).
% ATP.lambda_385
tff(fact_6524_ATP_Olambda__386,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_jn(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [A6: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A6) )
& pp(member2(A,A6,Uu)) ) ) ) ).
% ATP.lambda_386
tff(fact_6525_ATP_Olambda__387,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_js(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
<=> ? [I2: nat] :
( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I2)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).
% ATP.lambda_387
tff(fact_6526_ATP_Olambda__388,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_jl(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
<=> ? [A6: B] :
( ( Uub = aa(B,set(A),Uu,A6) )
& pp(member2(B,A6,aa(list(B),set(B),set2(B),Uua))) ) ) ).
% ATP.lambda_388
tff(fact_6527_ATP_Olambda__389,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [L4: B] :
( ( Uub = aa(B,A,Uu,L4) )
& pp(member2(B,L4,Uua)) ) ) ).
% ATP.lambda_389
tff(fact_6528_ATP_Olambda__390,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_jf(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X4: B] :
( ( Uub = aa(B,A,Uu,X4) )
& pp(aa(B,bool,Uua,X4)) ) ) ).
% ATP.lambda_390
tff(fact_6529_ATP_Olambda__391,axiom,
! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
( pp(aa(A,bool,aa(list(B),fun(A,bool),aTP_Lamp_ot(fun(B,option(A)),fun(list(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),Uua)))
& ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).
% ATP.lambda_391
tff(fact_6530_ATP_Olambda__392,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_ld(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [P5: product_prod(A,B)] :
( ( Uub = P5 )
& pp(member2(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P5)),aa(product_prod(A,B),B,product_snd(A,B),P5)),Uua)) ) ) ).
% ATP.lambda_392
tff(fact_6531_ATP_Olambda__393,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_qu(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X4: A] :
( pp(member2(A,X4,Uua))
& ( aa(A,B,Uu,X4) = aa(A,B,Uu,Uub) ) ) ) ).
% ATP.lambda_393
tff(fact_6532_ATP_Olambda__394,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_qx(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X4: A] :
( pp(member2(A,X4,Uua))
& ( aa(A,option(B),Uu,X4) = aa(B,option(B),some(B),Uub) ) ) ) ).
% ATP.lambda_394
tff(fact_6533_ATP_Olambda__395,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_qs(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X4: A] :
( pp(member2(A,X4,Uua))
& ( Uub = aa(A,B,Uu,X4) ) ) ) ).
% ATP.lambda_395
tff(fact_6534_ATP_Olambda__396,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,bool)),Uub: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_jw(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
<=> ? [A6: B,B5: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A6)),aa(B,A,Uu,B5)) )
& pp(aa(B,bool,aa(B,fun(B,bool),Uua,A6),B5)) ) ) ).
% ATP.lambda_396
tff(fact_6535_ATP_Olambda__397,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ji(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [A6: 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),A6),V4)) )
| ? [U4: list(A),Aa3: A,B5: A,Va3: list(A),W4: list(A)] :
( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B5),Uu))
& ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),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),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),W4)) ) ) ) ) ).
% ATP.lambda_397
tff(fact_6536_ATP_Olambda__398,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_jp(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [A6: A,B5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B5) )
& pp(member2(A,A6,Uu))
& pp(member2(A,B5,Uua)) ) ) ) ).
% ATP.lambda_398
tff(fact_6537_ATP_Olambda__399,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_jo(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [A6: A,B5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B5) )
& pp(member2(A,A6,Uu))
& pp(member2(A,B5,Uua)) ) ) ) ).
% ATP.lambda_399
tff(fact_6538_ATP_Olambda__400,axiom,
! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_je(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [X4: A,Xs3: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& pp(member2(A,X4,Uu))
& pp(member2(list(A),Xs3,Uua)) ) ) ).
% ATP.lambda_400
tff(fact_6539_ATP_Olambda__401,axiom,
! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_uv(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
<=> ? [Q7: fun(A,bool),R8: fun(A,bool)] :
( eventually(A,Q7,Uu)
& eventually(A,R8,Uua)
& ! [X4: A] :
( ( pp(aa(A,bool,Q7,X4))
& pp(aa(A,bool,R8,X4)) )
=> pp(aa(A,bool,Uub,X4)) ) ) ) ).
% ATP.lambda_401
tff(fact_6540_ATP_Olambda__402,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_jh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [Us: list(A),Z3: A,Z8: A,Vs2: list(A)] :
( ( Uua = 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),Z3),Vs2)) )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Z8),Uu))
& ( Uub = 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),Z8),Vs2)) ) ) ) ).
% ATP.lambda_402
tff(fact_6541_ATP_Olambda__403,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_hq(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_403
tff(fact_6542_ATP_Olambda__404,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_rz(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,member2(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).
% ATP.lambda_404
tff(fact_6543_ATP_Olambda__405,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_pm(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = if(set(B),member2(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).
% ATP.lambda_405
tff(fact_6544_ATP_Olambda__406,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_ev(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,minus_minus(nat,Uuc,aa(nat,nat,suc,zero_zero(nat)))))) ) ).
% ATP.lambda_406
tff(fact_6545_ATP_Olambda__407,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_ew(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).
% ATP.lambda_407
tff(fact_6546_ATP_Olambda__408,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_dt(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,member2(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_408
tff(fact_6547_ATP_Olambda__409,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_eg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_409
tff(fact_6548_ATP_Olambda__410,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_go(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_410
tff(fact_6549_ATP_Olambda__411,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_sj(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_411
tff(fact_6550_ATP_Olambda__412,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: fun(B,A),Uub: set(B),Uuc: B] : aa(B,A,aa(set(B),fun(B,A),aa(fun(B,A),fun(set(B),fun(B,A)),aTP_Lamp_sg(A,fun(fun(B,A),fun(set(B),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,member2(B,Uuc,Uub),aa(B,A,Uua,Uuc),Uu) ).
% ATP.lambda_412
tff(fact_6551_ATP_Olambda__413,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ep(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).
% ATP.lambda_413
tff(fact_6552_ATP_Olambda__414,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_pe(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uu),aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).
% ATP.lambda_414
tff(fact_6553_ATP_Olambda__415,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B,Uuc: A] : aa(A,option(B),aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_om(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uua),aa(B,option(B),some(B),Uub),aa(A,option(B),Uu,Uuc)) ).
% ATP.lambda_415
tff(fact_6554_ATP_Olambda__416,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_nd(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,member2(A,Uuc,Uub),Uu,Uua) ).
% ATP.lambda_416
tff(fact_6555_ATP_Olambda__417,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_nh(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).
% ATP.lambda_417
tff(fact_6556_ATP_Olambda__418,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_dw(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_418
tff(fact_6557_ATP_Olambda__419,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_gn(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_419
tff(fact_6558_ATP_Olambda__420,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_al(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_420
tff(fact_6559_ATP_Olambda__421,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,option(B)),Uub: fun(A,option(B)),Uuc: A] : aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_pg(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,Uu,Uuc),aa(A,option(B),Uua,Uuc),aa(A,option(B),Uub,Uuc)) ).
% ATP.lambda_421
tff(fact_6560_ATP_Olambda__422,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_uw(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_422
tff(fact_6561_ATP_Olambda__423,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_kb(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_423
tff(fact_6562_ATP_Olambda__424,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: list(A),Uuc: bool] :
( pp(aa(bool,bool,aa(list(A),fun(bool,bool),aa(A,fun(list(A),fun(bool,bool)),aTP_Lamp_rh(fun(A,fun(A,bool)),fun(A,fun(list(A),fun(bool,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(list(A),bool,aa(fun(A,fun(list(A),bool)),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),bool)),fun(list(A),bool)),case_list(bool,A),fTrue),aa(bool,fun(A,fun(list(A),bool)),aa(A,fun(bool,fun(A,fun(list(A),bool))),aTP_Lamp_rg(fun(A,fun(A,bool)),fun(A,fun(bool,fun(A,fun(list(A),bool)))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_424
tff(fact_6563_ATP_Olambda__425,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,fun(list(B),fun(C,C))),Uua: fun(A,B),Uub: A,Uuc: list(A)] : aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_oi(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),Uu,aa(A,B,Uua,Uub)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Uua),Uuc)) ).
% ATP.lambda_425
tff(fact_6564_ATP_Olambda__426,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_bb(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ).
% ATP.lambda_426
tff(fact_6565_ATP_Olambda__427,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_hu(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).
% ATP.lambda_427
tff(fact_6566_ATP_Olambda__428,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_my(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_428
tff(fact_6567_ATP_Olambda__429,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_nz(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).
% ATP.lambda_429
tff(fact_6568_ATP_Olambda__430,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_nw(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).
% ATP.lambda_430
tff(fact_6569_ATP_Olambda__431,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_bj(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(C,B,Uua,Uuc)) ).
% ATP.lambda_431
tff(fact_6570_ATP_Olambda__432,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_gi(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_gh(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_432
tff(fact_6571_ATP_Olambda__433,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: B] : aa(B,fun(list(B),list(B)),aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_is(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub),Uuc) = aa(fun(B,fun(list(B),list(B))),fun(list(B),list(B)),aa(list(B),fun(fun(B,fun(list(B),list(B))),fun(list(B),list(B))),case_list(list(B),B),if(list(B),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uu,Uuc)),Uua,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),nil(B))))),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_ir(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua)) ) ).
% ATP.lambda_433
tff(fact_6572_ATP_Olambda__434,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_tg(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uuc))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_434
tff(fact_6573_ATP_Olambda__435,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: bool,Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(bool,fun(A,fun(B,assn)),aTP_Lamp_ki(fun(A,fun(B,assn)),fun(bool,fun(A,fun(B,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uuc)),pure_assn(Uua)) ).
% ATP.lambda_435
tff(fact_6574_ATP_Olambda__436,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_nq(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uu))
& ~ pp(member2(A,Uub,Uua))
& ~ pp(member2(A,Uuc,Uua)) ) ) ).
% ATP.lambda_436
tff(fact_6575_ATP_Olambda__437,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),bool),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_oq(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
& pp(aa(product_prod(A,B),bool,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc))) ) ) ).
% ATP.lambda_437
tff(fact_6576_ATP_Olambda__438,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_hd(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),binomial(Uub,Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,minus_minus(nat,Uub,Uuc))) ) ).
% ATP.lambda_438
tff(fact_6577_ATP_Olambda__439,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_hc(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),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_439
tff(fact_6578_ATP_Olambda__440,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_ju(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
& ? [Xys2: list(A),X4: A,Y3: A,Xs5: 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),X4),Xs5)) )
& ( 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)) )
& pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu)) ) ) ) ).
% ATP.lambda_440
tff(fact_6579_ATP_Olambda__441,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_gy(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_441
tff(fact_6580_ATP_Olambda__442,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_bd(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(A,Uuc,aa(set(B),set(A),image2(B,A,Uu),Uua)))
& pp(aa(A,bool,Uub,Uuc)) ) ) ).
% ATP.lambda_442
tff(fact_6581_ATP_Olambda__443,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_gw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(A,Uuc,Uu))
& ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_443
tff(fact_6582_ATP_Olambda__444,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nk(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(A,Uuc,Uua))
& ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).
% ATP.lambda_444
tff(fact_6583_ATP_Olambda__445,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_gz(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_445
tff(fact_6584_ATP_Olambda__446,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: list(B),Uub: list(B),Uuc: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_ip(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),Uu),Uua)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uub),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),Uu),Uuc))) ) ).
% ATP.lambda_446
tff(fact_6585_ATP_Olambda__447,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_eb(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Uua),Uuc)) ).
% ATP.lambda_447
tff(fact_6586_ATP_Olambda__448,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_ec(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)),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Uuc),Uua)) ).
% ATP.lambda_448
tff(fact_6587_ATP_Olambda__449,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_be(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(B,Uuc,Uua))
& pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).
% ATP.lambda_449
tff(fact_6588_ATP_Olambda__450,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ci(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member2(B,Uuc,Uu))
& ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).
% ATP.lambda_450
tff(fact_6589_ATP_Olambda__451,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A] : aa(A,assn,aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_to(nat,fun(list(A),fun(array(A),fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uub,Uua)),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),aa(nat,A,nth(A,Uua),Uu)))) ) ).
% ATP.lambda_451
tff(fact_6590_ATP_Olambda__452,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_he(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_452
tff(fact_6591_ATP_Olambda__453,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_dz(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_453
tff(fact_6592_ATP_Olambda__454,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( semiring_0(B)
& comm_monoid_add(A)
& times(A) )
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_rm(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).
% ATP.lambda_454
tff(fact_6593_ATP_Olambda__455,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_gh(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).
% ATP.lambda_455
tff(fact_6594_ATP_Olambda__456,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_cy(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).
% ATP.lambda_456
tff(fact_6595_ATP_Olambda__457,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_cm(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_457
tff(fact_6596_ATP_Olambda__458,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_ux(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).
% ATP.lambda_458
tff(fact_6597_ATP_Olambda__459,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_hx(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_459
tff(fact_6598_ATP_Olambda__460,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_fl(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).
% ATP.lambda_460
tff(fact_6599_ATP_Olambda__461,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_ir(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(B),product_prod(list(B),list(B))),list(B),product_case_prod(list(B),product_prod(list(B),list(B)),list(B),aTP_Lamp_iq(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu)),linorder_part(B,A,Uu,aa(B,A,Uu,aa(nat,B,nth(B,Uua),divide_divide(nat,aa(list(B),nat,size_size(list(B)),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Uua)) ) ).
% ATP.lambda_461
tff(fact_6600_ATP_Olambda__462,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_rv(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ) ).
% ATP.lambda_462
tff(fact_6601_ATP_Olambda__463,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_gx(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,collect(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_gw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_463
tff(fact_6602_ATP_Olambda__464,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(list(A),fun(list(A),bool)),Uub: list(A),Uuc: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_kc(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
<=> ( ? [Y3: A,Ys3: list(A)] :
( ( Uub = nil(A) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) ) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y3)) )
| ? [X4: A,Y3: A,Xs3: list(A),Ys3: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y3),X4))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs3),Ys3)) ) ) ) ).
% ATP.lambda_464
tff(fact_6603_ATP_Olambda__465,axiom,
! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,assn)),Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_kj(bool,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uu)),aa(B,assn,aa(A,fun(B,assn),Uua,Uub),Uuc)) ).
% ATP.lambda_465
tff(fact_6604_ATP_Olambda__466,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_qf(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Uua),Uuc),Uub)) ).
% ATP.lambda_466
tff(fact_6605_ATP_Olambda__467,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_nb(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_467
tff(fact_6606_ATP_Olambda__468,axiom,
! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(bool),aa(A,fun(list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_pb(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),Uuc)) ).
% ATP.lambda_468
tff(fact_6607_ATP_Olambda__469,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_dq(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu),min_list(A,Uua)) ) ).
% ATP.lambda_469
tff(fact_6608_ATP_Olambda__470,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_br(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).
% ATP.lambda_470
tff(fact_6609_ATP_Olambda__471,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: nat,Uub: list(A),Uuc: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_gl(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Uua),Uub)),Uuc))) ) ).
% ATP.lambda_471
tff(fact_6610_ATP_Olambda__472,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_hy(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_472
tff(fact_6611_ATP_Olambda__473,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_hz(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_473
tff(fact_6612_ATP_Olambda__474,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: fun(B,A)] : aa(fun(B,A),fun(B,A),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_uq(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uuc))) ).
% ATP.lambda_474
tff(fact_6613_ATP_Olambda__475,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_iz(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),Uu),Uua),Uub),Uuc) = product_case_prod(list(B),list(B),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_iy(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc)) ) ).
% ATP.lambda_475
tff(fact_6614_ATP_Olambda__476,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( semiring_0(B)
& comm_monoid_add(A)
& times(A) )
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: multiset(C),Uuc: A] : aa(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_rn(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(B,image_mset(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_rm(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_476
tff(fact_6615_ATP_Olambda__477,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_cz(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_cy(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).
% ATP.lambda_477
tff(fact_6616_ATP_Olambda__478,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cn(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_cm(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_478
tff(fact_6617_ATP_Olambda__479,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_uy(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_ux(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).
% ATP.lambda_479
tff(fact_6618_ATP_Olambda__480,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_or(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),comp(bool,bool,A,fNot,Uu)),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_480
tff(fact_6619_ATP_Olambda__481,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_ou(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_481
tff(fact_6620_ATP_Olambda__482,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_px(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_482
tff(fact_6621_ATP_Olambda__483,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_jc(fun(B,A),fun(B,fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ? [L4: B] :
( ( Uuc = aa(B,A,Uu,L4) )
& ( ( L4 = Uua )
| pp(member2(B,L4,Uub)) ) ) ) ).
% ATP.lambda_483
tff(fact_6622_ATP_Olambda__484,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_jj(A,fun(A,fun(fun(A,B),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ? [I2: A] :
( ( Uuc = aa(A,B,Uub,I2) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),I2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),Uua)) ) ) ) ).
% ATP.lambda_484
tff(fact_6623_ATP_Olambda__485,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_se(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_485
tff(fact_6624_ATP_Olambda__486,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_nj(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),quicksort_by_rel(A,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),aa(list(A),list(A),quicksort_by_rel(A,Uu,Uua),Uud))),Uuc) ).
% ATP.lambda_486
tff(fact_6625_ATP_Olambda__487,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: B,Uud: fun(B,A)] : aa(fun(B,A),A,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_up(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,A,aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uud))),Uuc) ).
% ATP.lambda_487
tff(fact_6626_ATP_Olambda__488,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_ln(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_488
tff(fact_6627_ATP_Olambda__489,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: bool,Uuc: A,Uud: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aa(bool,fun(A,fun(list(A),bool)),aa(A,fun(bool,fun(A,fun(list(A),bool))),aTP_Lamp_rg(fun(A,fun(A,bool)),fun(A,fun(bool,fun(A,fun(list(A),bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uua),Uuc))
& pp(Uub) ) ) ).
% ATP.lambda_489
tff(fact_6628_ATP_Olambda__490,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_hj(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_490
tff(fact_6629_ATP_Olambda__491,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_hg(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_491
tff(fact_6630_ATP_Olambda__492,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_hh(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_492
tff(fact_6631_ATP_Olambda__493,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_qy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),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))))))),field2(A,Uu)))
& ( ( ( Uua = Uuc )
& ( Uub = Uud ) )
| pp(member2(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) )
& pp(member2(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 )
& pp(member2(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_493
tff(fact_6632_ATP_Olambda__494,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_tr(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uub,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Uua),Uu),Uuc))),pure_assn(aa(array(A),bool,aa(array(A),fun(array(A),bool),fequal(array(A)),Uud),Uub))) ) ).
% ATP.lambda_494
tff(fact_6633_ATP_Olambda__495,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_no(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ? [A6: A,B5: C] :
( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A6)),aa(C,B,Uua,B5)) )
& pp(member2(A,A6,Uub))
& pp(member2(C,B5,Uuc)) ) ) ) ).
% ATP.lambda_495
tff(fact_6634_ATP_Olambda__496,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B),Uud: list(B),Uue: list(B)] : aa(list(B),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_iy(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uuc)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),Uue)),if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub)),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uue))),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uud)),Uue)))) ) ).
% ATP.lambda_496
tff(fact_6635_ATP_Olambda__497,axiom,
! [B: $tType,A: $tType,Uu: multiset(A),Uua: B] : aa(B,multiset(A),aTP_Lamp_ac(multiset(A),fun(B,multiset(A)),Uu),Uua) = Uu ).
% ATP.lambda_497
tff(fact_6636_ATP_Olambda__498,axiom,
! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_sk(assn,fun(A,assn),Uu),Uua) = Uu ).
% ATP.lambda_498
tff(fact_6637_ATP_Olambda__499,axiom,
! [A: $tType,Uu: bool,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ku(bool,fun(A,bool),Uu),Uua))
<=> pp(Uu) ) ).
% ATP.lambda_499
tff(fact_6638_ATP_Olambda__500,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_nf(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_500
tff(fact_6639_ATP_Olambda__501,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_nn(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_501
tff(fact_6640_ATP_Olambda__502,axiom,
! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_mh(set(B),fun(B,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_502
tff(fact_6641_ATP_Olambda__503,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_me(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_503
tff(fact_6642_ATP_Olambda__504,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_mu(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).
% ATP.lambda_504
tff(fact_6643_ATP_Olambda__505,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_an(set(A),fun(B,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_505
tff(fact_6644_ATP_Olambda__506,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_mj(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_506
tff(fact_6645_ATP_Olambda__507,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_th(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).
% ATP.lambda_507
tff(fact_6646_ATP_Olambda__508,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_li(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).
% ATP.lambda_508
tff(fact_6647_ATP_Olambda__509,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_ik(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).
% ATP.lambda_509
tff(fact_6648_ATP_Olambda__510,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_iw(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_510
tff(fact_6649_ATP_Olambda__511,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_rq(B,fun(C,B),Uu),Uua) = Uu ) ).
% ATP.lambda_511
tff(fact_6650_ATP_Olambda__512,axiom,
! [A: $tType,B: $tType] :
( semiring_1(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_rj(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_512
tff(fact_6651_ATP_Olambda__513,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ck(B,fun(A,B),Uu),Uua) = Uu ).
% ATP.lambda_513
tff(fact_6652_ATP_Olambda__514,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ag(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_514
tff(fact_6653_ATP_Olambda__515,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ah(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_515
tff(fact_6654_ATP_Olambda__516,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_ja(A,fun(list(A),A)),Uu),Uua) = Uu ) ).
% ATP.lambda_516
tff(fact_6655_ATP_Olambda__517,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_at(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_517
tff(fact_6656_ATP_Olambda__518,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_fz(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_518
tff(fact_6657_ATP_Olambda__519,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_kg(A,fun(list(A),A)),Uu),Uua) = Uu ).
% ATP.lambda_519
tff(fact_6658_ATP_Olambda__520,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kz(A,fun(nat,A),Uu),Uua) = Uu ).
% ATP.lambda_520
tff(fact_6659_ATP_Olambda__521,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_cj(A,fun(B,A),Uu),Uua) = Uu ).
% ATP.lambda_521
tff(fact_6660_ATP_Olambda__522,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ce(A,fun(list(A),list(A))),Uu),Uua) = Uua ).
% ATP.lambda_522
tff(fact_6661_ATP_Olambda__523,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_cc(A,fun(list(A),bool)),Uu),Uua))
<=> $false ) ).
% ATP.lambda_523
tff(fact_6662_ATP_Olambda__524,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_cd(A,fun(list(A),bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_524
tff(fact_6663_ATP_Olambda__525,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ax(A,fun(A,bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_525
tff(fact_6664_ATP_Olambda__526,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_fa(nat,nat),Uu) = Uu ).
% ATP.lambda_526
tff(fact_6665_ATP_Olambda__527,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_iv(int,int),Uu) = Uu ).
% ATP.lambda_527
tff(fact_6666_ATP_Olambda__528,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_az(A,A),Uu) = Uu ) ).
% ATP.lambda_528
tff(fact_6667_ATP_Olambda__529,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_bz(A,A),Uu) = Uu ) ).
% ATP.lambda_529
tff(fact_6668_ATP_Olambda__530,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_sc(A,A),Uu) = Uu ) ).
% ATP.lambda_530
tff(fact_6669_ATP_Olambda__531,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu) = Uu ).
% ATP.lambda_531
tff(fact_6670_ATP_Olambda__532,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_mf(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_532
tff(fact_6671_ATP_Olambda__533,axiom,
! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_mb(C,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_533
tff(fact_6672_ATP_Olambda__534,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_ct(B,set(A)),Uu) = bot_bot(set(A)) ).
% ATP.lambda_534
tff(fact_6673_ATP_Olambda__535,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_af(B,A),Uu) = bot_bot(A) ) ).
% ATP.lambda_535
tff(fact_6674_ATP_Olambda__536,axiom,
! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_mc(A,set(D)),Uu) = bot_bot(set(D)) ).
% ATP.lambda_536
tff(fact_6675_ATP_Olambda__537,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_mv(A,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_537
tff(fact_6676_ATP_Olambda__538,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_rr(B,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_538
tff(fact_6677_ATP_Olambda__539,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_ca(A,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_539
tff(fact_6678_ATP_Olambda__540,axiom,
! [Uu: product_unit] : aa(product_unit,assn,aTP_Lamp_tz(product_unit,assn),Uu) = one_one(assn) ).
% ATP.lambda_540
tff(fact_6679_ATP_Olambda__541,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_ae(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_541
tff(fact_6680_ATP_Olambda__542,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_ok(B,option(A)),Uu) = none(A) ).
% ATP.lambda_542
tff(fact_6681_ATP_Olambda__543,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_oj(A,option(B)),Uu) = none(B) ).
% ATP.lambda_543
tff(fact_6682_ATP_Olambda__544,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_sa(A,B),Uu) = undefined(B) ).
% ATP.lambda_544
tff(fact_6683_ATP_Olambda__545,axiom,
! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_sf(product_prod(heap_ext(product_unit),set(nat)),bool),Uu))
<=> $false ) ).
% ATP.lambda_545
tff(fact_6684_ATP_Olambda__546,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_bo(A,bool),Uu))
<=> $false ) ).
% ATP.lambda_546
tff(fact_6685_ATP_Olambda__547,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_kv(A,bool),Uu))
<=> $true ) ).
% ATP.lambda_547
tff(fact_6686_ATP_Olambda__548,axiom,
! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_au(A,fun(nat,nat)),Uu) = suc ).
% ATP.lambda_548
% Type constructors (565)
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___Lattices_Obounded__semilattice__inf__top,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounde4346867609351753570nf_top(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__top,axiom,
! [A18: $tType,A19: $tType] :
( bounded_lattice(A19)
=> bounded_lattice_top(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_fun___Lattices_Oinf,axiom,
! [A18: $tType,A19: $tType] :
( semilattice_inf(A19)
=> inf(fun(A18,A19)) ) ).
tff(tcon_fun___Groups_Ominus,axiom,
! [A18: $tType,A19: $tType] :
( minus(A19)
=> minus(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_1,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_2,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Lattices_Odistrib__lattice_4,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_5,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_6,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_7,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_8,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_9,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Lattices_Oinf_10,axiom,
inf(int) ).
tff(tcon_Int_Oint___Groups_Otimes,axiom,
times(int) ).
tff(tcon_Int_Oint___Groups_Ominus_11,axiom,
minus(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_12,axiom,
condit6923001295902523014norder(nat) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_20,axiom,
normal6328177297339901930cative(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_21,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_22,axiom,
semiri6575147826004484403cancel(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_23,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_24,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_25,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_26,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_27,axiom,
linord2810124833399127020strict(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_28,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_29,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_30,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_31,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_32,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_33,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_34,axiom,
linord8928482502909563296strict(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_35,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_36,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_37,axiom,
semiri6843258321239162965malize(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_38,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_39,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_40,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_41,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_42,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_43,axiom,
ordere2520102378445227354miring(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_44,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_45,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring_46,axiom,
linordered_semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_47,axiom,
ordered_semiring_0(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_48,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_49,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_50,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_51,axiom,
distrib_lattice(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_52,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_53,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_54,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_55,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring_56,axiom,
ordered_semiring(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_57,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_58,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_59,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_60,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_61,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_62,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_63,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_64,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_65,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_66,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_67,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_68,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder__bot_69,axiom,
order_bot(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_70,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_71,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_72,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_73,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_74,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_75,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_76,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_77,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Orderings_Ono__top_78,axiom,
no_top(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_79,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_80,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_81,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Rings_Omult__zero_82,axiom,
mult_zero(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_83,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_84,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_85,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Orderings_Obot_86,axiom,
bot(nat) ).
tff(tcon_Nat_Onat___Lattices_Oinf_87,axiom,
inf(nat) ).
tff(tcon_Nat_Onat___Groups_Otimes_88,axiom,
times(nat) ).
tff(tcon_Nat_Onat___Groups_Ominus_89,axiom,
minus(nat) ).
tff(tcon_Nat_Onat___Power_Opower_90,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_91,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_92,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_93,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_94,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_95,axiom,
heap(nat) ).
tff(tcon_Num_Onum___Orderings_Opreorder_96,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_97,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_98,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_99,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Otimes_100,axiom,
times(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_101,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_102,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_103,axiom,
semiri6575147826004484403cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_104,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_105,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_106,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_107,axiom,
linord2810124833399127020strict(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_108,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_109,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_110,axiom,
linord715952674999750819strict(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_111,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_112,axiom,
linord181362715937106298miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_113,axiom,
linord8928482502909563296strict(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_114,axiom,
semiri3467727345109120633visors(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_115,axiom,
ordere6658533253407199908up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_116,axiom,
ordere166539214618696060dd_abs(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_117,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_118,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_119,axiom,
cancel2418104881723323429up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_120,axiom,
ring_15535105094025558882visors(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_121,axiom,
cancel1802427076303600483id_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_122,axiom,
linord4710134922213307826strict(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_123,axiom,
comm_s4317794764714335236cancel(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_124,axiom,
ordere2520102378445227354miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_125,axiom,
linord6961819062388156250ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_126,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_127,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring_128,axiom,
linordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_129,axiom,
ordered_semiring_0(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_130,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_131,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_132,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_133,axiom,
distrib_lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_134,axiom,
ab_semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_135,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_136,axiom,
ab_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring_137,axiom,
ordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_138,axiom,
ordered_ring_abs(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_139,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_140,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_141,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_142,axiom,
comm_semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_143,axiom,
comm_semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__mult_144,axiom,
semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom__divide_145,axiom,
semidom_divide(rat) ).
tff(tcon_Rat_Orat___Num_Osemiring__numeral_146,axiom,
semiring_numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__add_147,axiom,
semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__less__one_148,axiom,
zero_less_one(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring_149,axiom,
comm_semiring(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_150,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_151,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_152,axiom,
zero_neq_one(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring_153,axiom,
ordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_154,axiom,
idom_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_155,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_156,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__mult_157,axiom,
monoid_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_158,axiom,
comm_ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__add_159,axiom,
monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_160,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_161,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__top_162,axiom,
no_top(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__bot_163,axiom,
no_bot(rat) ).
tff(tcon_Rat_Orat___Lattices_Olattice_164,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_165,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Rings_Omult__zero_166,axiom,
mult_zero(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_167,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_168,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_169,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_170,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_171,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_172,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_173,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_174,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Lattices_Oinf_175,axiom,
inf(rat) ).
tff(tcon_Rat_Orat___Groups_Otimes_176,axiom,
times(rat) ).
tff(tcon_Rat_Orat___Groups_Ominus_177,axiom,
minus(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___Lattices_Obounded__semilattice__inf__top_188,axiom,
! [A18: $tType] : bounde4346867609351753570nf_top(set(A18)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_189,axiom,
! [A18: $tType] : comple6319245703460814977attice(set(A18)) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_190,axiom,
! [A18: $tType] : boolea8198339166811842893lgebra(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_191,axiom,
! [A18: $tType] : bounded_lattice_top(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_192,axiom,
! [A18: $tType] : bounded_lattice_bot(set(A18)) ).
tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_193,axiom,
! [A18: $tType] : comple9053668089753744459l_ccpo(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_194,axiom,
! [A18: $tType] : semilattice_sup(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_195,axiom,
! [A18: $tType] : semilattice_inf(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Odistrib__lattice_196,axiom,
! [A18: $tType] : distrib_lattice(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_197,axiom,
! [A18: $tType] : bounded_lattice(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder__top_198,axiom,
! [A18: $tType] : order_top(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder__bot_199,axiom,
! [A18: $tType] : order_bot(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Opreorder_200,axiom,
! [A18: $tType] : preorder(set(A18)) ).
tff(tcon_Set_Oset___Finite__Set_Ofinite_201,axiom,
! [A18: $tType] :
( finite_finite(A18)
=> finite_finite(set(A18)) ) ).
tff(tcon_Set_Oset___Lattices_Olattice_202,axiom,
! [A18: $tType] : lattice(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oorder_203,axiom,
! [A18: $tType] : order(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Otop_204,axiom,
! [A18: $tType] : top(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Oord_205,axiom,
! [A18: $tType] : ord(set(A18)) ).
tff(tcon_Set_Oset___Orderings_Obot_206,axiom,
! [A18: $tType] : bot(set(A18)) ).
tff(tcon_Set_Oset___Groups_Ouminus_207,axiom,
! [A18: $tType] : uminus(set(A18)) ).
tff(tcon_Set_Oset___Lattices_Oinf_208,axiom,
! [A18: $tType] : inf(set(A18)) ).
tff(tcon_Set_Oset___Groups_Ominus_209,axiom,
! [A18: $tType] : minus(set(A18)) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_210,axiom,
condit1219197933456340205attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_211,axiom,
comple592849572758109894attice(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_212,axiom,
bounde4967611905675639751up_bot(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_213,axiom,
bounde4346867609351753570nf_top(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_214,axiom,
comple6319245703460814977attice(bool) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_215,axiom,
boolea8198339166811842893lgebra(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_216,axiom,
bounded_lattice_top(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_217,axiom,
bounded_lattice_bot(bool) ).
tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_218,axiom,
comple9053668089753744459l_ccpo(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_219,axiom,
semilattice_sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_220,axiom,
semilattice_inf(bool) ).
tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_221,axiom,
distrib_lattice(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_222,axiom,
bounded_lattice(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__top_223,axiom,
order_top(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__bot_224,axiom,
order_bot(bool) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_225,axiom,
preorder(bool) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_226,axiom,
linorder(bool) ).
tff(tcon_HOL_Obool___Finite__Set_Ofinite_227,axiom,
finite_finite(bool) ).
tff(tcon_HOL_Obool___Lattices_Olattice_228,axiom,
lattice(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder_229,axiom,
order(bool) ).
tff(tcon_HOL_Obool___Orderings_Otop_230,axiom,
top(bool) ).
tff(tcon_HOL_Obool___Orderings_Oord_231,axiom,
ord(bool) ).
tff(tcon_HOL_Obool___Orderings_Obot_232,axiom,
bot(bool) ).
tff(tcon_HOL_Obool___Groups_Ouminus_233,axiom,
uminus(bool) ).
tff(tcon_HOL_Obool___Lattices_Oinf_234,axiom,
inf(bool) ).
tff(tcon_HOL_Obool___Groups_Ominus_235,axiom,
minus(bool) ).
tff(tcon_HOL_Obool___Heap_Oheap_236,axiom,
heap(bool) ).
tff(tcon_Heap_Oref___Heap_Oheap_237,axiom,
! [A18: $tType] : heap(ref(A18)) ).
tff(tcon_List_Olist___Heap_Oheap_238,axiom,
! [A18: $tType] :
( heap(A18)
=> heap(list(A18)) ) ).
tff(tcon_Heap_Oarray___Heap_Oheap_239,axiom,
! [A18: $tType] : heap(array(A18)) ).
tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_240,axiom,
! [A18: $tType,A19: $tType] :
( ( finite_finite(A18)
& finite_finite(A19) )
=> finite_finite(sum_sum(A18,A19)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_241,axiom,
! [A18: $tType,A19: $tType] :
( ( heap(A18)
& heap(A19) )
=> heap(sum_sum(A18,A19)) ) ).
tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_242,axiom,
! [A18: $tType] : condit1219197933456340205attice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_243,axiom,
! [A18: $tType] : bounde4967611905675639751up_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_244,axiom,
! [A18: $tType] : bounde4346867609351753570nf_top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_245,axiom,
! [A18: $tType] : comple6319245703460814977attice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_246,axiom,
! [A18: $tType] : bounded_lattice_top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_247,axiom,
! [A18: $tType] : bounded_lattice_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_248,axiom,
! [A18: $tType] : comple9053668089753744459l_ccpo(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_249,axiom,
! [A18: $tType] : semilattice_sup(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_250,axiom,
! [A18: $tType] : semilattice_inf(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_251,axiom,
! [A18: $tType] : distrib_lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_252,axiom,
! [A18: $tType] : bounded_lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__top_253,axiom,
! [A18: $tType] : order_top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_254,axiom,
! [A18: $tType] : order_bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Opreorder_255,axiom,
! [A18: $tType] : preorder(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Olattice_256,axiom,
! [A18: $tType] : lattice(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder_257,axiom,
! [A18: $tType] : order(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Otop_258,axiom,
! [A18: $tType] : top(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Oord_259,axiom,
! [A18: $tType] : ord(filter(A18)) ).
tff(tcon_Filter_Ofilter___Orderings_Obot_260,axiom,
! [A18: $tType] : bot(filter(A18)) ).
tff(tcon_Filter_Ofilter___Lattices_Oinf_261,axiom,
! [A18: $tType] : inf(filter(A18)) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_262,axiom,
! [A18: $tType] :
( comple5582772986160207858norder(A18)
=> condit6923001295902523014norder(option(A18)) ) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_263,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> condit1219197933456340205attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_264,axiom,
! [A18: $tType] :
( comple592849572758109894attice(A18)
=> comple592849572758109894attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_265,axiom,
! [A18: $tType] :
( lattice(A18)
=> bounde4967611905675639751up_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_266,axiom,
! [A18: $tType] :
( bounded_lattice_top(A18)
=> bounde4346867609351753570nf_top(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_267,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> comple6319245703460814977attice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_268,axiom,
! [A18: $tType] :
( bounded_lattice_top(A18)
=> bounded_lattice_top(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_269,axiom,
! [A18: $tType] :
( lattice(A18)
=> bounded_lattice_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_270,axiom,
! [A18: $tType] :
( comple6319245703460814977attice(A18)
=> comple9053668089753744459l_ccpo(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_271,axiom,
! [A18: $tType] :
( semilattice_sup(A18)
=> semilattice_sup(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_272,axiom,
! [A18: $tType] :
( semilattice_inf(A18)
=> semilattice_inf(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_273,axiom,
! [A18: $tType] :
( distrib_lattice(A18)
=> distrib_lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice_274,axiom,
! [A18: $tType] :
( bounded_lattice_top(A18)
=> bounded_lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Owellorder_275,axiom,
! [A18: $tType] :
( wellorder(A18)
=> wellorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__top_276,axiom,
! [A18: $tType] :
( order_top(A18)
=> order_top(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__bot_277,axiom,
! [A18: $tType] :
( order(A18)
=> order_bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Opreorder_278,axiom,
! [A18: $tType] :
( preorder(A18)
=> preorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Olinorder_279,axiom,
! [A18: $tType] :
( linorder(A18)
=> linorder(option(A18)) ) ).
tff(tcon_Option_Ooption___Finite__Set_Ofinite_280,axiom,
! [A18: $tType] :
( finite_finite(A18)
=> finite_finite(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Olattice_281,axiom,
! [A18: $tType] :
( lattice(A18)
=> lattice(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder_282,axiom,
! [A18: $tType] :
( order(A18)
=> order(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Otop_283,axiom,
! [A18: $tType] :
( order_top(A18)
=> top(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Oord_284,axiom,
! [A18: $tType] :
( preorder(A18)
=> ord(option(A18)) ) ).
tff(tcon_Option_Ooption___Orderings_Obot_285,axiom,
! [A18: $tType] :
( order(A18)
=> bot(option(A18)) ) ).
tff(tcon_Option_Ooption___Lattices_Oinf_286,axiom,
! [A18: $tType] :
( inf(A18)
=> inf(option(A18)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_287,axiom,
! [A18: $tType] :
( heap(A18)
=> heap(option(A18)) ) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_288,axiom,
bounde4967611905675639751up_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_289,axiom,
bounde4346867609351753570nf_top(assn) ).
tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_290,axiom,
boolea8198339166811842893lgebra(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_291,axiom,
bounded_lattice_top(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_292,axiom,
bounded_lattice_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_293,axiom,
semilattice_sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_294,axiom,
semilattice_inf(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_295,axiom,
distrib_lattice(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_296,axiom,
bounded_lattice(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_297,axiom,
ab_semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_298,axiom,
comm_monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_299,axiom,
semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__top_300,axiom,
order_top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_301,axiom,
order_bot(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Opreorder_302,axiom,
preorder(assn) ).
tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_303,axiom,
monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Olattice_304,axiom,
lattice(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder_305,axiom,
order(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Otop_306,axiom,
top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oord_307,axiom,
ord(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Obot_308,axiom,
bot(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ouminus_309,axiom,
uminus(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Oinf_310,axiom,
inf(assn) ).
tff(tcon_Assertions_Oassn___Groups_Otimes_311,axiom,
times(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ominus_312,axiom,
minus(assn) ).
tff(tcon_Assertions_Oassn___Power_Opower_313,axiom,
power(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oone_314,axiom,
one(assn) ).
tff(tcon_Assertions_Oassn___Rings_Odvd_315,axiom,
dvd(assn) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_316,axiom,
heap(typerep) ).
tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_317,axiom,
! [A18: $tType] :
( preorder(A18)
=> ordere6658533253407199908up_add(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_318,axiom,
! [A18: $tType] : cancel2418104881723323429up_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_319,axiom,
! [A18: $tType] : cancel1802427076303600483id_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_320,axiom,
! [A18: $tType] : cancel_semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_321,axiom,
! [A18: $tType] : comm_monoid_diff(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_322,axiom,
! [A18: $tType] : ab_semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_323,axiom,
! [A18: $tType] : comm_monoid_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_324,axiom,
! [A18: $tType] : semigroup_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Orderings_Opreorder_325,axiom,
! [A18: $tType] :
( preorder(A18)
=> preorder(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_326,axiom,
! [A18: $tType] : monoid_add(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Orderings_Oorder_327,axiom,
! [A18: $tType] :
( preorder(A18)
=> order(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Orderings_Oord_328,axiom,
! [A18: $tType] :
( preorder(A18)
=> ord(multiset(A18)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ominus_329,axiom,
! [A18: $tType] : minus(multiset(A18)) ).
tff(tcon_Multiset_Omultiset___Groups_Ozero_330,axiom,
! [A18: $tType] : zero(multiset(A18)) ).
tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_331,axiom,
! [A18: $tType,A19: $tType] :
( ( finite_finite(A18)
& finite_finite(A19) )
=> finite_finite(product_prod(A18,A19)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_332,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_333,axiom,
condit6923001295902523014norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_334,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_335,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_336,axiom,
bounde4967611905675639751up_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_337,axiom,
bounde4346867609351753570nf_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_338,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_339,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_340,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_341,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_342,axiom,
bounded_lattice_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_343,axiom,
comple9053668089753744459l_ccpo(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_344,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_345,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_346,axiom,
distrib_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_347,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_348,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_349,axiom,
order_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_350,axiom,
order_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_351,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_352,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_353,axiom,
finite_finite(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_354,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_355,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Otop_356,axiom,
top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_357,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Obot_358,axiom,
bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_359,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Oinf_360,axiom,
inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ominus_361,axiom,
minus(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_362,axiom,
heap(product_unit) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_363,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_364,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_365,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_366,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_367,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_368,axiom,
semiri6575147826004484403cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_369,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_370,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_371,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_372,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_373,axiom,
linord2810124833399127020strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_374,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_375,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_376,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_377,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_378,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_379,axiom,
linord8928482502909563296strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_380,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_381,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_382,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_383,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_384,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_385,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_386,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_387,axiom,
ordere2520102378445227354miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_388,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_389,axiom,
linordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_390,axiom,
ordered_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_391,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_392,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_393,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_394,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_395,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_396,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_397,axiom,
ordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_398,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_399,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_400,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_401,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_402,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_403,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_404,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_405,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_406,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_407,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_408,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_409,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_410,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_411,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_412,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_413,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_414,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_415,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_416,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_417,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_418,axiom,
mult_zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_419,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_420,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_421,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Otimes_422,axiom,
times(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ominus_423,axiom,
minus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_424,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_425,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_426,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_427,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_428,axiom,
dvd(code_natural) ).
% Helper facts (16)
tff(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(Q) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(P) ) ).
tff(help_fconj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(fconj(P,Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fdisj(P,Q))
| pp(P)
| pp(Q) ) ).
tff(help_fdisj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(Q)
| pp(fdisj(P,Q)) ) ).
tff(help_fdisj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| pp(fdisj(P,Q)) ) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fChoice_1_1_T,axiom,
! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).
% Conjectures (1)
tff(conj_0,conjecture,
vEBT_List_list_assn(a,b,p,nil(a),l) = pure_assn(aa(list(b),bool,aa(list(b),fun(list(b),bool),fequal(list(b)),l),nil(b))) ).
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