TPTP Problem File: ITP215_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP215_2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Automation 00192_006252
% 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 : 0027_Automation_00192_006252 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 7053 (1743 unt;1156 typ; 0 def)
% Number of atoms : 15574 (5739 equ)
% Maximal formula atoms : 22 ( 2 avg)
% Number of connectives : 11175 (1498 ~; 188 |;1070 &)
% (1091 <=>;7328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Maximal term depth : 40 ( 2 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 1081 ( 770 >; 311 *; 0 +; 0 <<)
% Number of predicates : 243 ( 240 usr; 2 prp; 0-7 aty)
% Number of functors : 905 ( 905 usr; 32 con; 0-8 aty)
% Number of variables : 20850 (18882 !; 372 ?;20850 :)
% (1596 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 15:59:57.069
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
tff(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
tff(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
tff(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
tff(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
tff(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
tff(ty_t_Product__Type_Oprod,type,
product_prod: ( $tType * $tType ) > $tType ).
tff(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
tff(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
tff(ty_t_Assertions_Oassn,type,
assn: $tType ).
tff(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
tff(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
tff(ty_t_Option_Ooption,type,
option: $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 ).
% Explicit typings (1132)
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
tff(sy_cl_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_Oord,type,
ord:
!>[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_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[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_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_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_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_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_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_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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__af____,type,
aTP_Lamp_af:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ag____,type,
aTP_Lamp_ag:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ah____,type,
aTP_Lamp_ah:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ap____,type,
aTP_Lamp_ap:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aq____,type,
aTP_Lamp_aq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ar____,type,
aTP_Lamp_ar:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__as____,type,
aTP_Lamp_as:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__at____,type,
aTP_Lamp_at:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__au____,type,
aTP_Lamp_au:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__av____,type,
aTP_Lamp_av:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__aw____,type,
aTP_Lamp_aw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ax____,type,
aTP_Lamp_ax:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ay____,type,
aTP_Lamp_ay:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__az____,type,
aTP_Lamp_az:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ba____,type,
aTP_Lamp_ba:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cb____,type,
aTP_Lamp_cb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cc____,type,
aTP_Lamp_cc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cd____,type,
aTP_Lamp_cd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ce____,type,
aTP_Lamp_ce:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cf____,type,
aTP_Lamp_cf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cg____,type,
aTP_Lamp_cg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ch____,type,
aTP_Lamp_ch:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cu____,type,
aTP_Lamp_cu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cv____,type,
aTP_Lamp_cv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cw____,type,
aTP_Lamp_cw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cx____,type,
aTP_Lamp_cx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cy____,type,
aTP_Lamp_cy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cz____,type,
aTP_Lamp_cz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__da____,type,
aTP_Lamp_da:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__db____,type,
aTP_Lamp_db:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ds____,type,
aTP_Lamp_ds:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dt____,type,
aTP_Lamp_dt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__du____,type,
aTP_Lamp_du:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dv____,type,
aTP_Lamp_dv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dw____,type,
aTP_Lamp_dw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dx____,type,
aTP_Lamp_dx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dy____,type,
aTP_Lamp_dy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dz____,type,
aTP_Lamp_dz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ea____,type,
aTP_Lamp_ea:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eb____,type,
aTP_Lamp_eb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ec____,type,
aTP_Lamp_ec:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ed____,type,
aTP_Lamp_ed:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ee____,type,
aTP_Lamp_ee:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ef____,type,
aTP_Lamp_ef:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eg____,type,
aTP_Lamp_eg:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ej____,type,
aTP_Lamp_ej:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ek____,type,
aTP_Lamp_ek:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__el____,type,
aTP_Lamp_el:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__em____,type,
aTP_Lamp_em:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__en____,type,
aTP_Lamp_en:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eo____,type,
aTP_Lamp_eo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eq____,type,
aTP_Lamp_eq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__er____,type,
aTP_Lamp_er:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__es____,type,
aTP_Lamp_es:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__et____,type,
aTP_Lamp_et:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eu____,type,
aTP_Lamp_eu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ew____,type,
aTP_Lamp_ew:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ex____,type,
aTP_Lamp_ex:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ez____,type,
aTP_Lamp_ez:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fa____,type,
aTP_Lamp_fa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fb____,type,
aTP_Lamp_fb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fc____,type,
aTP_Lamp_fc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fd____,type,
aTP_Lamp_fd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fe____,type,
aTP_Lamp_fe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ff____,type,
aTP_Lamp_ff:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fg____,type,
aTP_Lamp_fg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fh____,type,
aTP_Lamp_fh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fi____,type,
aTP_Lamp_fi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fj____,type,
aTP_Lamp_fj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fk____,type,
aTP_Lamp_fk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fl____,type,
aTP_Lamp_fl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fm____,type,
aTP_Lamp_fm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fn____,type,
aTP_Lamp_fn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fo____,type,
aTP_Lamp_fo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fp____,type,
aTP_Lamp_fp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fq____,type,
aTP_Lamp_fq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fr____,type,
aTP_Lamp_fr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fs____,type,
aTP_Lamp_fs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ft____,type,
aTP_Lamp_ft:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fu____,type,
aTP_Lamp_fu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fv____,type,
aTP_Lamp_fv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fw____,type,
aTP_Lamp_fw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fx____,type,
aTP_Lamp_fx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fy____,type,
aTP_Lamp_fy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fz____,type,
aTP_Lamp_fz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ga____,type,
aTP_Lamp_ga:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gi____,type,
aTP_Lamp_gi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gj____,type,
aTP_Lamp_gj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gk____,type,
aTP_Lamp_gk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gl____,type,
aTP_Lamp_gl:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gp____,type,
aTP_Lamp_gp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gq____,type,
aTP_Lamp_gq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gr____,type,
aTP_Lamp_gr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gs____,type,
aTP_Lamp_gs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gt____,type,
aTP_Lamp_gt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gu____,type,
aTP_Lamp_gu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gv____,type,
aTP_Lamp_gv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gw____,type,
aTP_Lamp_gw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gx____,type,
aTP_Lamp_gx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gy____,type,
aTP_Lamp_gy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ho____,type,
aTP_Lamp_ho:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hp____,type,
aTP_Lamp_hp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hq____,type,
aTP_Lamp_hq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hr____,type,
aTP_Lamp_hr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hs____,type,
aTP_Lamp_hs:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ht____,type,
aTP_Lamp_ht:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hu____,type,
aTP_Lamp_hu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__if____,type,
aTP_Lamp_if:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ig____,type,
aTP_Lamp_ig:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iw____,type,
aTP_Lamp_iw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ix____,type,
aTP_Lamp_ix:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iy____,type,
aTP_Lamp_iy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iz____,type,
aTP_Lamp_iz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__je____,type,
aTP_Lamp_je:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jo____,type,
aTP_Lamp_jo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jq____,type,
aTP_Lamp_jq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jr____,type,
aTP_Lamp_jr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__js____,type,
aTP_Lamp_js:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jt____,type,
aTP_Lamp_jt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ju____,type,
aTP_Lamp_ju:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jv____,type,
aTP_Lamp_jv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jw____,type,
aTP_Lamp_jw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jx____,type,
aTP_Lamp_jx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jy____,type,
aTP_Lamp_jy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ka____,type,
aTP_Lamp_ka:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kb____,type,
aTP_Lamp_kb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kc____,type,
aTP_Lamp_kc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kd____,type,
aTP_Lamp_kd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ke____,type,
aTP_Lamp_ke:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ko____,type,
aTP_Lamp_ko:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kp____,type,
aTP_Lamp_kp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kq____,type,
aTP_Lamp_kq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kr____,type,
aTP_Lamp_kr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kt____,type,
aTP_Lamp_kt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ku____,type,
aTP_Lamp_ku:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kv____,type,
aTP_Lamp_kv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kw____,type,
aTP_Lamp_kw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mc____,type,
aTP_Lamp_mc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__md____,type,
aTP_Lamp_md:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__me____,type,
aTP_Lamp_me:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mf____,type,
aTP_Lamp_mf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mi____,type,
aTP_Lamp_mi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mj____,type,
aTP_Lamp_mj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mk____,type,
aTP_Lamp_mk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mm____,type,
aTP_Lamp_mm:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nz____,type,
aTP_Lamp_nz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oa____,type,
aTP_Lamp_oa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ob____,type,
aTP_Lamp_ob:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oc____,type,
aTP_Lamp_oc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oe____,type,
aTP_Lamp_oe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__of____,type,
aTP_Lamp_of:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__og____,type,
aTP_Lamp_og:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oh____,type,
aTP_Lamp_oh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oi____,type,
aTP_Lamp_oi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oj____,type,
aTP_Lamp_oj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__or____,type,
aTP_Lamp_or:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : fun(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_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).
tff(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).
tff(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).
tff(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).
tff(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > nat ) ).
tff(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Opresent,type,
array_present:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > $o ) ).
tff(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: fun(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__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__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,bool)) ) ).
tff(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( fun(A,bool) > fun(A,fun(A,bool)) ) ).
tff(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).
tff(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(set(A),fun(set(B),bool)) ) ).
tff(sy_c_BNF__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__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_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(product_prod(A,B),bool) ) ).
tff(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).
tff(sy_c_Binomial_Obinomial,type,
binomial: nat > fun(nat,nat) ).
tff(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > fun(nat,A) ) ).
tff(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: ( num * num ) > num ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( ( nat * 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__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).
tff(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
tff(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
tff(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).
tff(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
tff(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
tff(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
tff(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
tff(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: ( int * code_integer ) > $o ).
tff(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: ( nat * code_natural ) > $o ).
tff(sy_c_Code__Numeral_Osub,type,
code_sub: ( num * num ) > code_integer ).
tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( set(A) > A ) ).
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_Oadjust__div,type,
adjust_div: product_prod(int,int) > int ).
tff(sy_c_Divides_Oadjust__mod,type,
adjust_mod: ( int * int ) > int ).
tff(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: ( int * int * product_prod(int,int) ) > $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_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).
tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_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,type,
finite_comp_fun_idem:
!>[A: $tType,B: $tType] : ( 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_finite:
!>[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_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_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).
tff(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).
tff(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_Obezw,type,
bezw: ( nat * nat ) > product_prod(int,int) ).
tff(sy_c_GCD_Obezw__rel,type,
bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).
tff(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),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_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > array(A) ) ).
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_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( fun(nat,nat) * heap_ext(Z) ) > heap_ext(Z) ) ).
tff(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > nat ) ).
tff(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ref(A) ) ).
tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( fun(A,bool) * A ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).
tff(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).
tff(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( fun(heap_ext(product_unit),bool) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : fun(A,heap_Time_Heap(A)) ).
tff(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > heap_Time_Heap(product_unit) ).
tff(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).
tff(sy_c_Hoare__Triple_Onew__addrs,type,
hoare_new_addrs: ( heap_ext(product_unit) * set(nat) * heap_ext(product_unit) ) > set(nat) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set(A) * nat ) > A ) ).
tff(sy_c_Int_OAbs__Integ,type,
abs_Integ: product_prod(nat,nat) > int ).
tff(sy_c_Int_OPos,type,
pos: num > int ).
tff(sy_c_Int_Odup,type,
dup: int > int ).
tff(sy_c_Int_Ointrel,type,
intrel: ( product_prod(nat,nat) * product_prod(nat,nat) ) > $o ).
tff(sy_c_Int_Onat,type,
nat2: int > nat ).
tff(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod(nat,nat) * int ) > $o ).
tff(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( ( A * int ) > A ) ).
tff(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : set(A) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : fun(int,A) ).
tff(sy_c_Int_Osub,type,
sub: ( num * num ) > int ).
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_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : fun(set(A),A) ).
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_OListMem,type,
listMem:
!>[A: $tType] : ( ( A * list(A) ) > $o ) ).
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,
filter:
!>[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__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_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_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_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__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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] : ( list(A) > list(A) ) ).
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__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,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_Oslice,type,
slice:
!>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).
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_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( fun(C,fun(B,bool)) > fun(fun(C,nat),fun(multiset(B),bool)) ) ).
tff(sy_c_Multiset_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_Onat__of__num,type,
nat_of_num: num > nat ).
tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( ( num * num ) > A ) ).
tff(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
tff(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
tff(sy_c_Num_Onum_OOne,type,
one2: num ).
tff(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
tff(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : fun(num,A) ).
tff(sy_c_Num_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_Othe,type,
the2:
!>[A: $tType] : fun(option(A),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__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(A,B) ) > $o ) ).
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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : 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_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : fun(A,fun(nat,A)) ).
tff(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( fun(product_unit,seq(A)) > pred(A) ) ).
tff(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > pred(A) ) ).
tff(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool)) ).
tff(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : seq(A) ).
tff(sy_c_Predicate_Oseq_OInsert,type,
insert2:
!>[A: $tType] : ( ( A * pred(A) ) > seq(A) ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
tff(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).
tff(sy_c_Product__Type_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_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_Random_Oinc__shift,type,
inc_shift: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).
tff(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).
tff(sy_c_Random_Olog,type,
log: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Olog__rel,type,
log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),bool)) ).
tff(sy_c_Random_Ominus__shift,type,
minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Onext,type,
next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).
tff(sy_c_Random_Orange,type,
range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Rat_OAbs__Rat,type,
abs_Rat: fun(product_prod(int,int),rat) ).
tff(sy_c_Rat_OFract,type,
fract: fun(int,fun(int,rat)) ).
tff(sy_c_Rat_OFrct,type,
frct: product_prod(int,int) > rat ).
tff(sy_c_Rat_ORep__Rat,type,
rep_Rat: fun(rat,product_prod(int,int)) ).
tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
tff(sy_c_Rat_Onormalize,type,
normalize: product_prod(int,int) > product_prod(int,int) ).
tff(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
tff(sy_c_Rat_Opcr__rat,type,
pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).
tff(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod(int,int) ).
tff(sy_c_Rat_Oratrel,type,
ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( ( A * heap_ext(product_unit) ) > product_prod(ref(A),heap_ext(product_unit)) ) ).
tff(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).
tff(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > $o ) ).
tff(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).
tff(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).
tff(sy_c_Refine__Imp__Hol_Oassert_H,type,
refine_Imp_assert: bool > 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,
range2:
!>[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_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).
tff(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) ) > fun(A,fun(C,bool)) ) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Relation_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_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).
tff(sy_c_Set_Ofilter,type,
filter2:
!>[A: $tType] : ( ( fun(A,bool) * set(A) ) > set(A) ) ).
tff(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).
tff(sy_c_Set_Oinsert,type,
insert3:
!>[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_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_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool)) ).
tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( 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_String_Ochar_OChar,type,
char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).
tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
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_Syntax__Match_Osyntax__nomatch,type,
syntax2379306206330768139omatch:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_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_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_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_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
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_F,type,
f: assn ).
tff(sy_v_P,type,
p: assn ).
tff(sy_v_Q,type,
q: assn ).
% Relevant facts (5278)
tff(fact_0_assms,axiom,
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,f) ).
% assms
tff(fact_1_FI__QUERY__def,axiom,
! [P: assn,Q: assn,F: assn] :
( fI_QUERY(P,Q,F)
<=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F))) ) ).
% FI_QUERY_def
tff(fact_2_assn__aci_I10_J,axiom,
! [A3: assn,B2: assn,C2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C2)),B2) ).
% assn_aci(10)
tff(fact_3_star__aci_I3_J,axiom,
! [A3: assn,B2: assn,C2: 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),C2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C2)) ).
% star_aci(3)
tff(fact_4_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_5_star__assoc,axiom,
! [A3: assn,B2: assn,C2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C2)) ).
% star_assoc
tff(fact_6_SLN__left,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sln),P) = P ).
% SLN_left
tff(fact_7_SLN__right,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),sln) = P ).
% SLN_right
tff(fact_8_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_9_ent__star__mono,axiom,
! [P: assn,P2: assn,Q: assn,Q2: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P2))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,Q),Q2))
=> 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),P2),Q2))) ) ) ).
% ent_star_mono
tff(fact_10_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_11_ent__refl,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P)) ).
% ent_refl
tff(fact_12_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_13_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_14_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_15_mult_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% mult.right_assoc
tff(fact_16_mult_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ).
% mult.right_commute
tff(fact_17_mult_Oassoc,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% mult.assoc
tff(fact_18_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_19_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_20_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_21_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_22_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : ~ list_ex1(A,P,nil(A)) ).
% list_ex1_simps(1)
tff(fact_23_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_24_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_25_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,list(A))] : bind(B,A,nil(B),F2) = nil(A) ).
% bind_simps(1)
tff(fact_26_member__rec_I2_J,axiom,
! [A: $tType,Y: A] : ~ member(A,nil(A),Y) ).
% member_rec(2)
tff(fact_27_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_28_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_29_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_30_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_31_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,list(A))] : maps(B,A,F2,nil(B)) = nil(A) ).
% maps_simps(2)
tff(fact_32_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_33_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_34_is__pure__assn__pure,axiom,
! [P: bool] : is_pure_assn(pure_assn(P)) ).
% is_pure_assn_pure
tff(fact_35_is__pure__assn__def,axiom,
! [A3: assn] :
( is_pure_assn(A3)
<=> ? [P3: bool] : A3 = pure_assn(P3) ) ).
% is_pure_assn_def
tff(fact_36_is__pure__assnE,axiom,
! [A3: assn] :
( is_pure_assn(A3)
=> ~ ! [P4: bool] : A3 != pure_assn(P4) ) ).
% is_pure_assnE
tff(fact_37_syntax__fo__nomatch__def,axiom,
! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax7388354845996824322omatch(A,B,Pat,Obj) ).
% syntax_fo_nomatch_def
tff(fact_38_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))))
<=> ( ! [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(B2) )
& pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q)) ) ) ).
% ent_pure_post_iff
tff(fact_39_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_40_ac__operator_Osafe__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),X: A,Y: A,A3: A,B2: A] :
( syntax_ac_operator(A,F2)
=> ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),F2,X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = aa(A,A,aa(A,fun(A,A),F2,B2),A3) ) ) ) ).
% ac_operator.safe_commute
tff(fact_41_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_42_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_43_null__rec_I2_J,axiom,
! [B: $tType] : pp(aa(list(B),bool,null(B),nil(B))) ).
% null_rec(2)
tff(fact_44_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_45_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_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
( ! [X2: A] : aa(A,B,F2,X2) = aa(A,B,G,X2)
=> ( F2 = G ) ) ).
% ext
tff(fact_48_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_49_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_50_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_51_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_52_mod__pure__star__dist,axiom,
! [P: assn,B2: bool,H2: 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))),H2))
<=> ( 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),H2))
& pp(B2) ) ) ).
% mod_pure_star_dist
tff(fact_53_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_54_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_55_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_56_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_57_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_58_pure__true,axiom,
pure_assn(fTrue) = one_one(assn) ).
% pure_true
tff(fact_59_pure__assn__eq__emp__iff,axiom,
! [P: bool] :
( ( pure_assn(P) = one_one(assn) )
<=> pp(P) ) ).
% pure_assn_eq_emp_iff
tff(fact_60_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: bool] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),pure_assn(B2)))
<=> ( ! [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(B2) )
& pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),one_one(assn))) ) ) ).
% ent_pure_post_iff_sng
tff(fact_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
% one_reorient
tff(fact_69_mod__and__dist,axiom,
! [P: assn,Q: assn,H2: 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)),H2))
<=> ( 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),H2))
& 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),H2)) ) ) ).
% mod_and_dist
tff(fact_70_ac__operator__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( syntax_ac_operator(A,F2)
<=> ( ! [A5: A,B4: A,C3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B4)),C3) = aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B4),C3))
& ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F2,A5),B4) = aa(A,A,aa(A,fun(A,A),F2,B4),A5) ) ) ).
% ac_operator_def
tff(fact_71_ac__operator_Oright__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),C2)),B2) ) ) ).
% ac_operator.right_commute
tff(fact_72_ac__operator_Oleft__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) = aa(A,A,aa(A,fun(A,A),F2,B2),aa(A,A,aa(A,fun(A,A),F2,A3),C2)) ) ) ).
% ac_operator.left_commute
tff(fact_73_ac__operator_Oright__assoc,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ).
% ac_operator.right_assoc
tff(fact_74_ac__operator_Oleft__assoc,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2) ) ) ).
% ac_operator.left_assoc
tff(fact_75_ac__operator_Ocommute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] :
( syntax_ac_operator(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = aa(A,A,aa(A,fun(A,A),F2,B2),A3) ) ) ).
% ac_operator.commute
tff(fact_76_ac__operator_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A6: A,B5: A,C4: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A6),B5)),C4) = aa(A,A,aa(A,fun(A,A),F2,A6),aa(A,A,aa(A,fun(A,A),F2,B5),C4))
=> ( ! [A6: A,B5: A] : aa(A,A,aa(A,fun(A,A),F2,A6),B5) = aa(A,A,aa(A,fun(A,A),F2,B5),A6)
=> syntax_ac_operator(A,F2) ) ) ).
% ac_operator.intro
tff(fact_77_mod__starE,axiom,
! [A3: assn,B2: assn,H2: 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)),H2))
=> ~ ( ? [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_78_mod__starD,axiom,
! [A4: assn,B3: assn,H2: 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)),H2))
=> ? [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_79_entails__def,axiom,
! [P: assn,Q: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),Q))
<=> ! [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,Q),H)) ) ) ).
% entails_def
tff(fact_80_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_81_entailsD,axiom,
! [P: assn,Q: assn,H2: 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),H2))
=> 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),H2)) ) ) ).
% entailsD
tff(fact_82_ent__fwd,axiom,
! [P: assn,H2: 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),H2))
=> ( 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),H2)) ) ) ).
% ent_fwd
tff(fact_83_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_84_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_85_ent__conjE2,axiom,
! [B3: assn,C5: assn,A4: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),C5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),B3)),C5)) ) ).
% ent_conjE2
tff(fact_86_ent__conjE1,axiom,
! [A4: assn,C5: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A4),B3)),C5)) ) ).
% ent_conjE1
tff(fact_87_ent__conjI,axiom,
! [A4: assn,B3: assn,C5: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),B3))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B3),C5))) ) ) ).
% ent_conjI
tff(fact_88_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_89_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_90_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_91_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn(one_one(assn)) ).
% is_pure_assn_basic_simps(2)
tff(fact_92_SLN__def,axiom,
sln = one_one(assn) ).
% SLN_def
tff(fact_93_mult_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> syntax_ac_operator(A,times_times(A)) ) ).
% mult.ac_operator_axioms
tff(fact_94_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_95_inf__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F2: 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)),F2),G),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ).
% inf_apply
tff(fact_96_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_97_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_98_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_99_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_100_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_101_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_102_precise__def,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
( precise(A,B,R)
<=> ! [A5: A,A7: A,H: product_prod(heap_ext(product_unit),set(nat)),P5: B,F3: assn,F4: 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),P5)),F3)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A7),P5)),F4))),H))
=> ( A5 = A7 ) ) ) ).
% precise_def
tff(fact_103_preciseI,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn))] :
( ! [A6: A,A8: A,H3: product_prod(heap_ext(product_unit),set(nat)),P6: B,F5: assn,F6: 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),P6)),F5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A8),P6)),F6))),H3))
=> ( A6 = A8 ) )
=> precise(A,B,R) ) ).
% preciseI
tff(fact_104_preciseD,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P7: B,F: assn,A9: A,F7: assn,H2: 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),P7)),F)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R,A9),P7)),F7))),H2))
=> ( A3 = A9 ) ) ) ).
% preciseD
tff(fact_105_preciseD_H,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,assn)),A3: A,P7: B,F: assn,H2: product_prod(heap_ext(product_unit),set(nat)),A9: 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),P7)),F)),H2))
=> ( 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,A9),P7)),F7)),H2))
=> ( A3 = A9 ) ) ) ) ).
% preciseD'
tff(fact_106_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F2: 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)),F2),G),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ).
% inf_fun_def
tff(fact_107_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_108_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_109_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_110_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_111_inf_Oassoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).
% inf.assoc
tff(fact_112_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_113_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_114_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_115_inf_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).
% inf.left_commute
tff(fact_116_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_117_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_118_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_119_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_120_ent__false__iff,axiom,
! [P: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),bot_bot(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)) ) ).
% ent_false_iff
tff(fact_121_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_122_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,option(A))] : map_filter(B,A,F2,nil(B)) = nil(A) ).
% map_filter_simps(2)
tff(fact_123_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_124_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_125_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_126_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] : enumerate(A,N,nil(A)) = nil(product_prod(nat,A)) ).
% enumerate_simps(1)
tff(fact_127_mod__star__trueI,axiom,
! [P: assn,H2: 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),H2))
=> 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))),H2)) ) ).
% mod_star_trueI
tff(fact_128_mod__star__trueE,axiom,
! [P: assn,H2: 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))),H2))
=> ~ ! [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)) ) ).
% mod_star_trueE
tff(fact_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_pure__false,axiom,
pure_assn(fFalse) = bot_bot(assn) ).
% pure_false
tff(fact_145_pure__assn__eq__false__iff,axiom,
! [P: bool] :
( ( pure_assn(P) = bot_bot(assn) )
<=> ~ pp(P) ) ).
% pure_assn_eq_false_iff
tff(fact_146_assn__basic__inequalities_I1_J,axiom,
top_top(assn) != one_one(assn) ).
% assn_basic_inequalities(1)
tff(fact_147_assn__basic__inequalities_I5_J,axiom,
top_top(assn) != bot_bot(assn) ).
% assn_basic_inequalities(5)
tff(fact_148_assn__basic__inequalities_I3_J,axiom,
bot_bot(assn) != one_one(assn) ).
% assn_basic_inequalities(3)
tff(fact_149_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,Z2),A3) = A3 ) ) ).
% monoid.left_neutral
tff(fact_150_monoid_Oright__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),Z2) = A3 ) ) ).
% monoid.right_neutral
tff(fact_151_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,A3: A] :
( comm_monoid(A,F2,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),Z2) = A3 ) ) ).
% comm_monoid.comm_neutral
tff(fact_152_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_153_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_154_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.monoid_axioms
tff(fact_155_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_156_mod__false,axiom,
! [H2: 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)),H2)) ).
% mod_false
tff(fact_157_ent__false,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,bot_bot(assn)),P)) ).
% ent_false
tff(fact_158_ent__true,axiom,
! [P: assn] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),top_top(assn))) ).
% ent_true
tff(fact_159_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_160_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_161_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_162_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_163_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn(bot_bot(assn)) ).
% is_pure_assn_basic_simps(1)
tff(fact_164_rotate1_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).
% rotate1.simps(1)
tff(fact_165_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_166_sngr__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P7: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P7,X)),sngr_assn(A,P7,Y)) = bot_bot(assn) ) ).
% sngr_same_false
tff(fact_167_snga__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P7: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P7,X)),snga_assn(A,P7,Y)) = bot_bot(assn) ) ).
% snga_same_false
tff(fact_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
=> monoid_axioms(A,F2,Z2) ) ).
% monoid.axioms(2)
tff(fact_176_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
=> comm_monoid_axioms(A,F2,Z2) ) ).
% comm_monoid.axioms(2)
tff(fact_177_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_178_entt__refl,axiom,
! [A4: assn] : entailst(A4,A4) ).
% entt_refl
tff(fact_179_entt__true,axiom,
! [A4: assn] : entailst(A4,top_top(assn)) ).
% entt_true
tff(fact_180_entt__emp,axiom,
! [A4: assn] : entailst(A4,one_one(assn)) ).
% entt_emp
tff(fact_181_entt__trans,axiom,
! [A4: assn,B3: assn,C5: assn] :
( entailst(A4,B3)
=> ( entailst(B3,C5)
=> entailst(A4,C5) ) ) ).
% entt_trans
tff(fact_182_monoid__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A6) = A6
=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,A6),Z2) = A6
=> monoid_axioms(A,F2,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_183_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,A6),Z2) = A6
=> comm_monoid_axioms(A,F2,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_184_monoid__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F2,Z2)
<=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A5) = A5
& ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),Z2) = A5 ) ) ).
% monoid_axioms_def
tff(fact_185_comm__monoid__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F2,Z2)
<=> ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),Z2) = A5 ) ).
% comm_monoid_axioms_def
tff(fact_186_entt__frame__fwd,axiom,
! [P: assn,Q: assn,A4: assn,F: assn,B3: assn] :
( entailst(P,Q)
=> ( entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F))
=> ( entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F),B3)
=> entailst(A4,B3) ) ) ) ).
% entt_frame_fwd
tff(fact_187_entt__star__mono,axiom,
! [A4: assn,B3: assn,C5: assn,D2: assn] :
( entailst(A4,B3)
=> ( entailst(C5,D2)
=> entailst(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),C5),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),D2)) ) ) ).
% entt_star_mono
tff(fact_188_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_189_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F2,Z2)
=> comm_monoid(A,F2,Z2) ) ).
% semilattice_neutr.axioms(2)
tff(fact_190_monoid__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
<=> ( semigroup(A,F2)
& monoid_axioms(A,F2,Z2) ) ) ).
% monoid_def
tff(fact_191_monoid_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semigroup(A,F2)
=> ( monoid_axioms(A,F2,Z2)
=> monoid(A,F2,Z2) ) ) ).
% monoid.intro
tff(fact_192_comm__monoid__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
<=> ( abel_semigroup(A,F2)
& comm_monoid_axioms(A,F2,Z2) ) ) ).
% comm_monoid_def
tff(fact_193_comm__monoid_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( abel_semigroup(A,F2)
=> ( comm_monoid_axioms(A,F2,Z2)
=> comm_monoid(A,F2,Z2) ) ) ).
% comm_monoid.intro
tff(fact_194_semilattice__neutr__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F2,Z2)
<=> ( semilattice(A,F2)
& comm_monoid(A,F2,Z2) ) ) ).
% semilattice_neutr_def
tff(fact_195_semilattice__neutr_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semilattice(A,F2)
=> ( comm_monoid(A,F2,Z2)
=> semilattice_neutr(A,F2,Z2) ) ) ).
% semilattice_neutr.intro
tff(fact_196_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [P7: ref(A),X: A,H2: 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,P7,X)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(3)
tff(fact_197_mod__true,axiom,
! [H2: 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)),H2))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H2)) ) ).
% mod_true
tff(fact_198_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( heap(B)
=> ! [Q3: array(B),Y: list(B),H2: 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,Q3,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)),H2),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(4)
tff(fact_199_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H2: 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)),H2),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)),H2),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)),H2),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(6)
tff(fact_200_mod__h__bot__iff_I1_J,axiom,
! [B2: bool,H2: 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)),H2),bot_bot(set(nat)))))
<=> pp(B2) ) ).
% mod_h_bot_iff(1)
tff(fact_201_in__range__empty,axiom,
! [H2: 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)),H2),bot_bot(set(nat))))) ).
% in_range_empty
tff(fact_202_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: 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)),H2),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)),H2),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)),H2),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(5)
tff(fact_203_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semilattice(A,F2)
=> abel_semigroup(A,F2) ) ).
% semilattice.axioms(1)
tff(fact_204_semilattice_Oidem,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A] :
( semilattice(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),A3) = A3 ) ) ).
% semilattice.idem
tff(fact_205_semilattice_Oleft__idem,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,A3),B2)) = aa(A,A,aa(A,fun(A,A),F2,A3),B2) ) ) ).
% semilattice.left_idem
tff(fact_206_semilattice_Oright__idem,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F2,A3),B2) ) ) ).
% semilattice.right_idem
tff(fact_207_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
=> semigroup(A,F2) ) ).
% abel_semigroup.axioms(1)
tff(fact_208_semigroup_Oassoc,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ).
% semigroup.assoc
tff(fact_209_semigroup_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A6: A,B5: A,C4: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A6),B5)),C4) = aa(A,A,aa(A,fun(A,A),F2,A6),aa(A,A,aa(A,fun(A,A),F2,B5),C4))
=> semigroup(A,F2) ) ).
% semigroup.intro
tff(fact_210_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_211_abel__semigroup_Ocommute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),A3: A,B2: A] :
( abel_semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = aa(A,A,aa(A,fun(A,A),F2,B2),A3) ) ) ).
% abel_semigroup.commute
tff(fact_212_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),B2: A,A3: A,C2: A] :
( abel_semigroup(A,F2)
=> ( aa(A,A,aa(A,fun(A,A),F2,B2),aa(A,A,aa(A,fun(A,A),F2,A3),C2)) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)) ) ) ).
% abel_semigroup.left_commute
tff(fact_213_semigroup__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semigroup(A,F2)
<=> ! [A5: A,B4: A,C3: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,A5),B4)),C3) = aa(A,A,aa(A,fun(A,A),F2,A5),aa(A,A,aa(A,fun(A,A),F2,B4),C3)) ) ).
% semigroup_def
tff(fact_214_mod__h__bot__indep,axiom,
! [P: assn,H2: 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)),H2),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_215_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_216_models__in__range,axiom,
! [P: assn,H2: 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),H2))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H2)) ) ).
% models_in_range
tff(fact_217_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> abel_semigroup(A,inf_inf(A)) ) ).
% inf.abel_semigroup_axioms
tff(fact_218_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_219_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice(A,inf_inf(A)) ) ).
% inf.semilattice_axioms
tff(fact_220_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semigroup(A,inf_inf(A)) ) ).
% inf.semigroup_axioms
tff(fact_221_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F2,Z2)
=> abel_semigroup(A,F2) ) ).
% comm_monoid.axioms(1)
tff(fact_222_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F2,Z2)
=> semigroup(A,F2) ) ).
% monoid.axioms(1)
tff(fact_223_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F2,Z2)
=> semilattice(A,F2) ) ).
% semilattice_neutr.axioms(1)
tff(fact_224_mod__h__bot__iff_I2_J,axiom,
! [H2: 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)),H2),bot_bot(set(nat))))) ).
% mod_h_bot_iff(2)
tff(fact_225_mod__emp__simp,axiom,
! [H2: 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)),H2),bot_bot(set(nat))))) ).
% mod_emp_simp
tff(fact_226_abel__semigroup_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semigroup(A,F2)
=> ( abel_s757365448890700780axioms(A,F2)
=> abel_semigroup(A,F2) ) ) ).
% abel_semigroup.intro
tff(fact_227_abel__semigroup__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
<=> ( semigroup(A,F2)
& abel_s757365448890700780axioms(A,F2) ) ) ).
% abel_semigroup_def
tff(fact_228_semilattice_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
=> ( semilattice_axioms(A,F2)
=> semilattice(A,F2) ) ) ).
% semilattice.intro
tff(fact_229_semilattice__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semilattice(A,F2)
<=> ( abel_semigroup(A,F2)
& semilattice_axioms(A,F2) ) ) ).
% semilattice_def
tff(fact_230_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_231_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_232_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_233_pure__assn__raw_Osimps,axiom,
! [B: $tType,A: $tType,B2: bool,H2: 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)),H2),As2)))
<=> ( ( As2 = bot_bot(set(B)) )
& pp(B2) ) ) ).
% pure_assn_raw.simps
tff(fact_234_mod__star__conv,axiom,
! [A4: assn,B3: assn,H2: 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)),H2))
<=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
( ( H2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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_235_star__assnI,axiom,
! [P: assn,H2: heap_ext(product_unit),As2: set(nat),Q: assn,As3: 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)),H2),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)),H2),As3)))
=> ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As3) = 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)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As3)))) ) ) ) ).
% star_assnI
tff(fact_236_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_237_sup__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F2: 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)),F2),G),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ).
% sup_apply
tff(fact_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_in__range__dist__union,axiom,
! [H2: heap_ext(product_unit),As2: set(nat),As3: 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)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As3))))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3))) ) ) ).
% in_range_dist_union
tff(fact_256_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F2: 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)),F2),G),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ).
% sup_fun_def
tff(fact_257_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_258_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).
% sup.left_commute
tff(fact_259_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_260_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_261_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_262_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).
% sup.assoc
tff(fact_263_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_264_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_265_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_266_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_267_semilattice__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,A6),A6) = A6
=> semilattice_axioms(A,F2) ) ).
% semilattice_axioms.intro
tff(fact_268_semilattice__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semilattice_axioms(A,F2)
<=> ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,A5),A5) = A5 ) ).
% semilattice_axioms_def
tff(fact_269_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( ! [A6: A,B5: A] : aa(A,A,aa(A,fun(A,A),F2,A6),B5) = aa(A,A,aa(A,fun(A,A),F2,B5),A6)
=> abel_s757365448890700780axioms(A,F2) ) ).
% abel_semigroup_axioms.intro
tff(fact_270_abel__semigroup__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F2)
<=> ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F2,A5),B4) = aa(A,A,aa(A,fun(A,A),F2,B4),A5) ) ).
% abel_semigroup_axioms_def
tff(fact_271_distrib__imp1,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z3: 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),Z3)) = 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),Z3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),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_272_distrib__imp2,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z3: 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),Z3)) = 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),Z3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> abel_semigroup(A,sup_sup(A)) ) ).
% sup.abel_semigroup_axioms
tff(fact_282_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice(A,sup_sup(A)) ) ).
% sup.semilattice_axioms
tff(fact_283_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semigroup(A,sup_sup(A)) ) ).
% sup.semigroup_axioms
tff(fact_284_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_285_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.monoid_axioms
tff(fact_286_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_287_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_288_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( abel_semigroup(A,F2)
=> abel_s757365448890700780axioms(A,F2) ) ).
% abel_semigroup.axioms(2)
tff(fact_289_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semilattice(A,F2)
=> semilattice_axioms(A,F2) ) ).
% semilattice.axioms(2)
tff(fact_290_one__assn__raw_Osimps,axiom,
! [H2: 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)),H2),As2)))
<=> ( As2 = bot_bot(set(nat)) ) ) ).
% one_assn_raw.simps
tff(fact_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_wand__assnI,axiom,
! [H2: 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)),H2),As2)))
=> ( ! [H4: heap_ext(product_unit),As4: set(nat)] :
( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As4) = bot_bot(set(nat)) )
=> ( relH(As2,H2,H4)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2)))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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)),H4),As4)))
=> 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)),H4),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,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)),H2),As2))) ) ) ).
% wand_assnI
tff(fact_302_IntI,axiom,
! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C2,A4))
=> ( pp(member2(A,C2,B3))
=> pp(member2(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).
% IntI
tff(fact_303_Int__iff,axiom,
! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
<=> ( pp(member2(A,C2,A4))
& pp(member2(A,C2,B3)) ) ) ).
% Int_iff
tff(fact_304_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_305_mod__or__dist,axiom,
! [P: assn,Q: assn,H2: 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)),H2))
<=> ( 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),H2))
| 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),H2)) ) ) ).
% mod_or_dist
tff(fact_306_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_307_relH__dist__union,axiom,
! [As2: set(nat),As3: set(nat),H2: 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),As3),H2,H5)
<=> ( relH(As2,H2,H5)
& relH(As3,H2,H5) ) ) ).
% relH_dist_union
tff(fact_308_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H2: 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)),H2),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)),H2),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)),H2),bot_bot(set(nat))))) ) ) ).
% mod_h_bot_iff(7)
tff(fact_309_relH__sym,axiom,
! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H2,H5)
=> relH(As2,H5,H2) ) ).
% relH_sym
tff(fact_310_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_311_star__or__dist2,axiom,
! [C5: assn,A4: assn,B3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C5),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),C5),A4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C5),B3)) ).
% star_or_dist2
tff(fact_312_star__or__dist1,axiom,
! [A4: assn,B3: assn,C5: 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)),C5) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),C5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B3),C5)) ).
% star_or_dist1
tff(fact_313_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_314_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_315_ent__disjI2_H,axiom,
! [A4: assn,C5: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C5))) ) ).
% ent_disjI2'
tff(fact_316_ent__disjI1_H,axiom,
! [A4: assn,B3: assn,C5: 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),C5))) ) ).
% ent_disjI1'
tff(fact_317_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_318_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_319_ent__disjE,axiom,
! [A4: assn,C5: assn,B3: assn] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,A4),C5))
=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,B3),C5))
=> pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3)),C5)) ) ) ).
% ent_disjE
tff(fact_320_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_321_entt__disjD1,axiom,
! [A4: assn,B3: assn,C5: assn] :
( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3),C5)
=> entailst(A4,C5) ) ).
% entt_disjD1
tff(fact_322_entt__disjD2,axiom,
! [A4: assn,B3: assn,C5: assn] :
( entailst(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A4),B3),C5)
=> entailst(B3,C5) ) ).
% entt_disjD2
tff(fact_323_entt__disjI1_H,axiom,
! [A4: assn,B3: assn,C5: assn] :
( entailst(A4,B3)
=> entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C5)) ) ).
% entt_disjI1'
tff(fact_324_entt__disjI2_H,axiom,
! [A4: assn,C5: assn,B3: assn] :
( entailst(A4,C5)
=> entailst(A4,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B3),C5)) ) ).
% entt_disjI2'
tff(fact_325_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_326_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_327_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_328_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_329_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_330_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_331_mod__relH,axiom,
! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),P: assn] :
( relH(As2,H2,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)),H2),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_332_relH__in__rangeI_I2_J,axiom,
! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H2,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_333_relH__in__rangeI_I1_J,axiom,
! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H2,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)),H2),As2))) ) ).
% relH_in_rangeI(1)
tff(fact_334_relH__refl,axiom,
! [H2: 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)),H2),As2)))
=> relH(As2,H2,H2) ) ).
% relH_refl
tff(fact_335_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_336_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_337_Int__left__commute,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5)) = 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),C5)) ).
% Int_left_commute
tff(fact_338_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_339_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_340_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_341_Int__assoc,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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)),C5) = 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),C5)) ).
% Int_assoc
tff(fact_342_IntD2,axiom,
! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> pp(member2(A,C2,B3)) ) ).
% IntD2
tff(fact_343_IntD1,axiom,
! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> pp(member2(A,C2,A4)) ) ).
% IntD1
tff(fact_344_IntE,axiom,
! [A: $tType,C2: A,A4: set(A),B3: set(A)] :
( pp(member2(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)))
=> ~ ( pp(member2(A,C2,A4))
=> ~ pp(member2(A,C2,B3)) ) ) ).
% IntE
tff(fact_345_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_346_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_347_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_348_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_349_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_350_Un__Int__distrib2,axiom,
! [A: $tType,B3: set(A),C5: 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),C5)),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)),C5),A4)) ).
% Un_Int_distrib2
tff(fact_351_Int__Un__distrib2,axiom,
! [A: $tType,B3: set(A),C5: 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),C5)),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)),C5),A4)) ).
% Int_Un_distrib2
tff(fact_352_Un__Int__distrib,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5)) = 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),C5)) ).
% Un_Int_distrib
tff(fact_353_Int__Un__distrib,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5)) = 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),C5)) ).
% Int_Un_distrib
tff(fact_354_Un__Int__crazy,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),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),C5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C5),A4)) ).
% Un_Int_crazy
tff(fact_355_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)))
& ! [H4: heap_ext(product_unit),As4: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As4) = bot_bot(set(nat)) )
& relH(As,H3,H4)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) )
=> 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)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As4)))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_356_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)))
& ! [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(2)
tff(fact_357_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)))
& ! [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(1)
tff(fact_358_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),H2: 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)),H2),As2)))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
& ! [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)),As2),As6) = bot_bot(set(nat)) )
& relH(As2,H2,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),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)),H7),As6))) )
=> 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)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6)))) ) ) ) ).
% wand_raw.simps
tff(fact_359_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_360_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_361_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_362_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),H2: 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)),H2),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)),H2),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)),H2),As22))) ) ) ).
% times_assn_raw.simps
tff(fact_363_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_364_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_365_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),H4: 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,H4)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,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),As))) ) ) )
=> pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,P)) ) ) ).
% properI
tff(fact_366_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_367_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_368_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_369_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_370_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_371_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_372_mod__not__dist,axiom,
! [P: assn,H2: 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)),H2))
<=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H2))
& ~ 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),H2)) ) ) ).
% mod_not_dist
tff(fact_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_properD1,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H2: 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)),H2),As2)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ) ).
% properD1
tff(fact_400_proper__iff,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),As2: set(nat),H2: 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,H2,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)),H2),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_401_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))
<=> ! [H: heap_ext(product_unit),H7: heap_ext(product_unit),As7: 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),As7)))
=> 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),As7))) )
& ( ( 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),As7)))
& relH(As7,H,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),As7))) )
=> 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)),H7),As7))) ) ) ) ).
% proper_def
tff(fact_402_properD2,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H2: 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)),H2),As2)))
=> ( relH(As2,H2,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_403_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_404_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_405_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_406_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_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_sngr__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : sngr_assn(A,R2,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,sngr_assn_raw(A,R2,X)) ) ).
% sngr_assn_def
tff(fact_427_snga__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),A3: list(A)] : snga_assn(A,R2,A3) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,snga_assn_raw(A,R2,A3)) ) ).
% snga_assn_def
tff(fact_428_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,set(A))] : list_collect_set(B,A,F2,nil(B)) = bot_bot(set(A)) ).
% list_collect_set_simps(1)
tff(fact_429_mod__pure,axiom,
! [B2: bool,H2: 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)),H2))
<=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) )
& pp(B2) ) ) ).
% mod_pure
tff(fact_430_snga__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A)] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,snga_assn_raw(A,R2,X))) ) ).
% snga_assn_proper
tff(fact_431_sngr__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,sngr_assn_raw(A,R2,X))) ) ).
% sngr_assn_proper
tff(fact_432_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_433_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_434_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_435_UNIV__coset,axiom,
! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).
% UNIV_coset
tff(fact_436_mod__emp,axiom,
! [H2: 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)),H2))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) ) ) ).
% mod_emp
tff(fact_437_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_438_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_finite(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_439_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F2,aa(list(C),list(B),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_440_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_441_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_442_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_443_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_444_sup__neg__inf,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [P7: A,Q3: A,R2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P7),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P7),aa(A,A,uminus_uminus(A),Q3))),R2)) ) ) ).
% sup_neg_inf
tff(fact_445_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_446_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_447_relH__def,axiom,
! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As2,H2,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)),H2),As2)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2)))
& ! [T2: typerep,X4: nat] :
( pp(member2(nat,X4,As2))
=> ( ( refs(product_unit,H2,T2,X4) = refs(product_unit,H5,T2,X4) )
& ( arrays(product_unit,H2,T2,X4) = arrays(product_unit,H5,T2,X4) ) ) ) ) ) ).
% relH_def
tff(fact_448_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_449_mult__cancel__right,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
tff(fact_450_mult__cancel__left,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
tff(fact_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_inf_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
<=> ( 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),C2)) ) ) ) ).
% inf.bounded_iff
tff(fact_461_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_462_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_463_sup_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% sup.bounded_iff
tff(fact_464_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = nil(A) )
<=> ( Xs = nil(B) ) ) ).
% map_is_Nil_conv
tff(fact_465_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs) )
<=> ( Xs = nil(B) ) ) ).
% Nil_is_map_conv
tff(fact_466_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A3: list(A)] :
( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A3) = nil(B) )
<=> ( A3 = nil(A) ) ) ).
% list.map_disc_iff
tff(fact_467_Int__subset__iff,axiom,
! [A: $tType,C5: 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)),C5),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)),C5),A4))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B3)) ) ) ).
% Int_subset_iff
tff(fact_468_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_469_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_470_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_471_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_472_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_473_mult__cancel__right2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_right2
tff(fact_474_mult__cancel__right1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B2: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
<=> ( ( C2 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_right1
tff(fact_475_mult__cancel__left2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_left2
tff(fact_476_mult__cancel__left1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B2: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
<=> ( ( C2 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_left1
tff(fact_477_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite(A),A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A4) = nil(A) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
tff(fact_478_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_479_nths__map,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),I: 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),F2),Xs)),I) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(nat),list(B),aa(list(B),fun(set(nat),list(B)),nths(B),Xs),I)) ).
% nths_map
tff(fact_480_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_481_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))))] :
~ ! [P4: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q4: 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)))),P4),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))),Q4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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_482_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ordere2520102378445227354miring(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_right_mono
tff(fact_491_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_right_mono_neg
tff(fact_492_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_left_mono
tff(fact_493_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_494_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_left_mono_neg
tff(fact_495_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_496_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_497_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_mono'
tff(fact_498_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_mono
tff(fact_499_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_500_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_501_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_502_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))))] :
~ ! [R3: 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)))),R3),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_503_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))))] :
~ ! [R3: 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)))),R3),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_504_mult__left__le,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),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),C2)),A3)) ) ) ) ).
% mult_left_le
tff(fact_505_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_506_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_507_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_508_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_509_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_510_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: fun(A,B)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),nil(A)) = nil(B) ).
% list.simps(8)
tff(fact_511_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_512_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(bool,product_prod(A,set(B)))] :
~ ! [B5: 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))),B5),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_513_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_514_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_515_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_516_inf_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
=> 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)),C2)) ) ) ).
% inf.coboundedI2
tff(fact_517_inf_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).
% inf.coboundedI1
tff(fact_518_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_519_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_520_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_521_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_522_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_523_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_524_inf_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% inf.boundedI
tff(fact_525_inf_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% inf.boundedE
tff(fact_526_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_527_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_528_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_529_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_530_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_531_inf__unique,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [F2: 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),F2,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),F2,X2),Y2)),Y2))
=> ( ! [X2: A,Y2: A,Z3: 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),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3))) ) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).
% inf_unique
tff(fact_532_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_533_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_534_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_535_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_536_inf__mono,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),C2),D3))) ) ) ) ).
% inf_mono
tff(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_sup_Omono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,D3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),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),C2),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).
% sup.mono
tff(fact_552_sup__mono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),C2),D3))) ) ) ) ).
% sup_mono
tff(fact_553_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_554_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_555_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_556_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_557_sup__unique,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [F2: 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),F2,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),F2,X2),Y2)))
=> ( ! [X2: A,Y2: A,Z3: 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),Z3),X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3)),X2)) ) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).
% sup_unique
tff(fact_558_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_559_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_560_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_561_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_562_sup_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% sup.boundedE
tff(fact_563_sup_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2)),A3)) ) ) ) ).
% sup.boundedI
tff(fact_564_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_565_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_566_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_567_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_568_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_569_sup_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI1
tff(fact_570_sup_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI2
tff(fact_571_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_572_Int__greatest,axiom,
! [A: $tType,C5: 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)),C5),A4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ).
% Int_greatest
tff(fact_573_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_574_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_575_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_576_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_577_Int__mono,axiom,
! [A: $tType,A4: set(A),C5: 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),C5))
=> ( 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)),C5),D2))) ) ) ).
% Int_mono
tff(fact_578_inter__eq__subsetI,axiom,
! [A: $tType,S: set(A),S2: 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),S2))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S2) )
=> ( 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_579_mult__right__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
tff(fact_580_mult__left__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
tff(fact_581_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_582_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_583_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_584_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_585_rotate1__map,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rotate1(B),Xs)) ).
% rotate1_map
tff(fact_586_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_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(A),B3))
=> ( A4 = B3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_587_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_588_relH__subset,axiom,
! [Bs: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),As2: set(nat)] :
( relH(Bs,H2,H5)
=> ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As2),Bs))
=> relH(As2,H2,H5) ) ) ).
% relH_subset
tff(fact_589_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_590_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_591_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_592_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_593_disjoint__mono,axiom,
! [A: $tType,A3: set(A),A9: set(A),B2: set(A),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A9))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B2),B6))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A9),B6) = 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_594_Un__Int__assoc__eq,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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)),C5) = 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),C5)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),A4)) ) ).
% Un_Int_assoc_eq
tff(fact_595_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_596_in__range__subset,axiom,
! [As2: set(nat),As3: set(nat),H2: heap_ext(product_unit)] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As2),As3))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3)))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ) ).
% in_range_subset
tff(fact_597_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_598_finite__Int,axiom,
! [A: $tType,F: set(A),G2: set(A)] :
( ( pp(aa(set(A),bool,finite_finite(A),F))
| pp(aa(set(A),bool,finite_finite(A),G2)) )
=> pp(aa(set(A),bool,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F),G2))) ) ).
% finite_Int
tff(fact_599_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)))
& ! [H4: heap_ext(product_unit),As4: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As4) = bot_bot(set(nat)) )
& relH(As,H3,H4)
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As4))) )
=> 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)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As4)))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_600_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)))
& ! [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(2)
tff(fact_601_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)))
& ! [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)))) ) ) )
=> ~ 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_602_mod__star__trueE_H,axiom,
! [P: assn,H2: 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))),H2))
=> ~ ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H4) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H2) )
=> ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H4)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2)))
=> ~ 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)) ) ) ) ).
% mod_star_trueE'
tff(fact_603_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] : map(A,B) = map_tailrec(A,B) ).
% map_eq_map_tailrec
tff(fact_604_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: fun(B,set(A)),X: fun(C,B),A3: C] : list_collect_set(B,A,F2,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),F2,aa(C,B,X,A3)) ).
% list_collect_set_map_simps(2)
tff(fact_605_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_606_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_607_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_608_mult__le__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),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_609_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_610_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_611_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_612_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_613_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F2: 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)),F2),Fs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
| ( ( aa(A,nat,F2,X) = aa(A,nat,F2,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_614_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_615_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_616_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_617_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_618_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_619_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_620_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_621_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,set(A)),A3: B] : list_collect_set(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F2,A3) ).
% list_collect_set_simps(2)
tff(fact_622_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_623_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_624_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_625_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,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
=> ! [Y2: B,Ys2: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys2) )
=> ( ( 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),Xs3),Ys2) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_626_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_627_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_628_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_629_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_630_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_631_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_632_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_633_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_634_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_635_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)))
=> ( ! [Ys3: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),Ys3))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys3: 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),Ys3)))
=> 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),Ys3))) ) )
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys3: 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)),Ys3))
=> 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),Ys3))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_636_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_637_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_638_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_639_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_640_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_641_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y3: A,Ys4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) ).
% neq_Nil_conv
tff(fact_642_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,Ys3: 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),Ys3)))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
=> 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),Ys3))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% list_induct2'
tff(fact_643_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_644_merge_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: product_prod(list(A),list(A))] :
( ! [L2: 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)),L2)
=> ( ! [V: A,Va: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)),nil(A))
=> ~ ! [X1: A,L1: list(A),X23: A,L2: 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),L2)) ) ) ) ).
% merge.cases
tff(fact_645_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod(list(A),list(A))] :
( ! [Ys3: 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)),Ys3)
=> ( ! [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,Ys3: 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),Ys3)) ) ) ).
% shuffles.cases
tff(fact_646_successively_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P4: 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)),P4),nil(A))
=> ( ! [P4: 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)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
=> ~ ! [P4: 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)),P4),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_647_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P4: 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)),P4),nil(A))
=> ~ ! [P4: fun(A,fun(A,bool)),X2: A,Ys3: 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)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3)) ) ).
% sorted_wrt.cases
tff(fact_648_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: product_prod(fun(A,B),list(A))] :
( ! [F8: 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)),F8),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
=> ( ! [F8: fun(A,B),X2: A,Y2: A,Zs2: 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)),F8),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),Zs2)))
=> ~ ! [A6: 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)),A6),nil(A)) ) ) ) ).
% arg_min_list.cases
tff(fact_649_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
( ! [F8: 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))),F8),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))
=> ~ ! [F8: fun(A,B),A6: 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))),F8),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),A6),As)),Bs2)) ) ).
% map_tailrec_rev.cases
tff(fact_650_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
( ! [Xs1: 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)),Xs1),Xs22)),nil(A))
=> ( ! [Xs1: 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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
=> ~ ! [Xs1: 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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) ) ) ).
% mergesort_by_rel_split.cases
tff(fact_651_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,Ys3: 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),Ys3)))
=> ( ! [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)),V: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),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),V),Va))) ) ) ).
% mergesort_by_rel_merge.cases
tff(fact_652_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_653_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))] :
( ! [P4: 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))),P4),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)))
=> ~ ! [P4: 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))),P4),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_654_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))] :
( ! [P4: 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))),P4),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)))
=> ( ! [P4: fun(A,fun(B,bool)),A6: A,As: list(A),B5: 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))),P4),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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
=> ( ! [P4: fun(A,fun(B,bool)),V: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),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),V),Va)),nil(B)))
=> ~ ! [P4: fun(A,fun(B,bool)),V: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),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),V),Va))) ) ) ) ).
% list_all_zip.cases
tff(fact_655_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
( ! [F8: 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))),F8),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)))
=> ( ! [F8: fun(A,fun(B,C)),A6: A,As: list(A),B5: 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))),F8),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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))
=> ( ! [A6: fun(A,fun(B,C)),V: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),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),V),Va)),nil(B)))
=> ~ ! [A6: fun(A,fun(B,C)),V: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A6),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),V),Va))) ) ) ) ).
% zipf.cases
tff(fact_656_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_657_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_658_measures__less,axiom,
! [A: $tType,F2: 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,F2,X)),aa(A,nat,F2,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)),F2),Fs)))) ) ).
% measures_less
tff(fact_659_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_660_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_661_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_662_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_663_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% inf.strict_boundedE
tff(fact_664_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_665_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
=> 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)),C2)) ) ) ).
% inf.strict_coboundedI1
tff(fact_666_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
=> 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)),C2)) ) ) ).
% inf.strict_coboundedI2
tff(fact_667_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI2
tff(fact_668_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI1
tff(fact_669_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_670_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% sup.strict_boundedE
tff(fact_671_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_672_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_673_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_674_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_675_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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,F2,Z4) = Y )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) = Ys ) ) ) ).
% map_eq_Cons_conv
tff(fact_676_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),F2: 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),F2),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,F2,Z4) )
& ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) ) ) ) ).
% Cons_eq_map_conv
tff(fact_677_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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,F2,Z3) = Y )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) = Ys ) ) ) ).
% map_eq_Cons_D
tff(fact_678_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),F2: 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),F2),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,F2,Z3) )
& ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) ) ) ) ).
% Cons_eq_map_D
tff(fact_679_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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,F2,X21)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),X22)) ).
% list.simps(9)
tff(fact_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% mult_less_cancel_left_neg
tff(fact_694_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_less_cancel_left_pos
tff(fact_695_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_strict_left_mono_neg
tff(fact_696_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_strict_left_mono
tff(fact_697_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
& 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),C2),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_698_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_strict_right_mono_neg
tff(fact_699_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_strict_right_mono
tff(fact_700_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
& 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),C2),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_701_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord2810124833399127020strict(A)
=> ! [A3: A,B2: A,C2: 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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_measures__lesseq,axiom,
! [A: $tType,F2: 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,F2,X)),aa(A,nat,F2,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)),F2),Fs)))) ) ) ).
% measures_lesseq
tff(fact_710_mult__less__le__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_711_mult__le__less__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_712_mult__right__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_right_le_imp_le
tff(fact_713_mult__left__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_left_le_imp_le
tff(fact_714_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_le_cancel_left_pos
tff(fact_715_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% mult_le_cancel_left_neg
tff(fact_716_mult__less__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_right
tff(fact_717_mult__strict__mono_H,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_strict_mono'
tff(fact_718_mult__right__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_right_less_imp_less
tff(fact_719_mult__less__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_left
tff(fact_720_mult__strict__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: 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),C2),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)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).
% mult_strict_mono
tff(fact_721_mult__left__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_left_less_imp_less
tff(fact_722_mult__le__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% mult_le_cancel_right
tff(fact_723_mult__le__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% mult_le_cancel_left
tff(fact_724_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_725_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_726_mult__less__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: 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),C2)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),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_727_mult__less__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),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_728_mult__less__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: 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),C2),A3)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),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_729_mult__less__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> 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),C2),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_730_mult__le__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: 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),C2)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),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_731_mult__le__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),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_732_mult__le__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: 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),C2),A3)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2),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_733_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% field_le_mult_one_interval
tff(fact_734_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_735_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_736_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_737_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_738_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod(list(A),list(A))] :
( ! [Ys3: 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)),Ys3)
=> ~ ! [X2: A,Xs2: list(A),Ys3: 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)),Ys3) ) ).
% subset_eq_mset_impl.cases
tff(fact_739_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_740_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% sgn_neg
tff(fact_741_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_742_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_743_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C2: 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),C2) )
<=> ( B2 = C2 ) ) ) ).
% add_left_cancel
tff(fact_744_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C2: 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),C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% add_right_cancel
tff(fact_745_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_746_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_left
tff(fact_747_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_right
tff(fact_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_left
tff(fact_758_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_right
tff(fact_759_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_760_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_761_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_762_sgn__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).
% sgn_1
tff(fact_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_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_772_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_773_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_774_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_775_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_776_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_777_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_778_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_779_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).
% sgn_pos
tff(fact_780_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_781_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F2: 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,F2,Z2,Less_eq,Less)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = Z2 )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
tff(fact_782_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F2: 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,F2,Z2,Less_eq,Less)
=> ( ( Z2 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
tff(fact_783_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( K = L ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_784_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_785_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_786_add_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% add.right_assoc
tff(fact_787_add_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% add.right_commute
tff(fact_788_add_Oassoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% add.assoc
tff(fact_789_add_Oleft__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: 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),C2) )
<=> ( B2 = C2 ) ) ) ).
% add.left_cancel
tff(fact_790_add_Oright__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A3: A,C2: 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),C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% add.right_cancel
tff(fact_791_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_792_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_793_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C2: 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),C2) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
tff(fact_794_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C2: 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),C2),A3) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
tff(fact_795_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_796_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = 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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_797_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_798_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_mono
tff(fact_799_add__left__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% add_left_mono
tff(fact_800_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))
=> ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).
% less_eqE
tff(fact_801_add__right__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).
% add_right_mono
tff(fact_802_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))
<=> ? [C3: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).
% le_iff_add
tff(fact_803_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_left
tff(fact_804_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_right
tff(fact_805_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_806_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_807_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_808_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_809_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = 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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_810_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_811_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_strict_mono
tff(fact_812_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% add_strict_left_mono
tff(fact_813_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).
% add_strict_right_mono
tff(fact_814_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_left
tff(fact_815_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_right
tff(fact_816_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,E2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E2)),C2) ) ).
% combine_common_factor
tff(fact_817_distrib__right,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% distrib_right
tff(fact_818_distrib__left,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).
% distrib_left
tff(fact_819_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% comm_semiring_class.distrib
tff(fact_820_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).
% ring_class.ring_distribs(1)
tff(fact_821_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% ring_class.ring_distribs(2)
tff(fact_822_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_823_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_824_sgn__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,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),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).
% sgn_mult
tff(fact_825_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_826_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_827_add_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> syntax_ac_operator(A,plus_plus(A)) ) ).
% add.ac_operator_axioms
tff(fact_828_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_829_add__decreasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,C2: 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),C2),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),C2)),B2)) ) ) ) ).
% add_decreasing
tff(fact_830_add__increasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% add_increasing
tff(fact_831_add__decreasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),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),C2)),B2)) ) ) ) ).
% add_decreasing2
tff(fact_832_add__increasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> ( 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),C2))) ) ) ) ).
% add_increasing2
tff(fact_833_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_834_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_835_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_836_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_837_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_838_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I2: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),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),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_839_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_le_less_mono
tff(fact_840_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: 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),C2),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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_less_le_mono
tff(fact_841_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_842_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_843_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))
=> ~ ! [C4: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) )
=> ( C4 = zero_zero(A) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_844_pos__add__strict,axiom,
! [A: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% pos_add_strict
tff(fact_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_sgn__minus__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% sgn_minus_1
tff(fact_853_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_854_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_855_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
=> semilattice_neutr(A,F2,Z2) ) ).
% semilattice_neutr_order.axioms(1)
tff(fact_856_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_857_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_858_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_859_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_860_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_861_add__strict__increasing,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% add_strict_increasing
tff(fact_862_add__strict__increasing2,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% add_strict_increasing2
tff(fact_863_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_864_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_865_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_866_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% sgn_1_pos
tff(fact_867_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_868_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( ( X = zero_zero(A) )
=> ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
& ( ( X != zero_zero(A) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).
% sgn_if
tff(fact_869_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% sgn_1_neg
tff(fact_870_less__one,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
<=> ( N = zero_zero(nat) ) ) ).
% less_one
tff(fact_871_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_872_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_873_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_874_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_875_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [R2: A,A3: A,B2: A,C2: A,D3: A] :
( ( R2 != zero_zero(A) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).
% add_scale_eq_noteq
tff(fact_876_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_877_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_878_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),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),C2)) ) ) ) ).
% crossproduct_noteq
tff(fact_879_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_880_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
<=> ( ( M2 = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_881_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = one_one(nat) )
<=> ( ( M2 = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_882_nat__mult__1,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).
% nat_mult_1
tff(fact_883_nat__mult__1__right,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).
% nat_mult_1_right
tff(fact_884_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
=> ( ( N = one_one(nat) )
| ( M2 = zero_zero(nat) ) ) ) ).
% mult_eq_self_implies_10
tff(fact_885_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_886_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_887_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_888_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_889_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),Pivot: A] : linorder_part(B,A,F2,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_890_le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: 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,C2))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_891_minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: 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,C2))),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_892_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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,C2))))
<=> 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),C2))) ) ) ) ).
% neg_le_minus_divide_eq
tff(fact_893_times__divide__eq__right,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).
% times_divide_eq_right
tff(fact_894_divide__divide__eq__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A,C2: A] : divide_divide(A,A3,divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2) ) ).
% divide_divide_eq_right
tff(fact_895_divide__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A3,B2),C2) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% divide_divide_eq_left
tff(fact_896_times__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C2) ) ).
% times_divide_eq_left
tff(fact_897_div__by__1,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : divide_divide(A,A3,one_one(A)) = A3 ) ).
% div_by_1
tff(fact_898_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_899_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).
% power_one_right
tff(fact_900_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_901_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_902_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_903_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_904_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2 = zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
& ( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_divide__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] : divide_divide(A,A3,aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B2)) ) ).
% divide_sgn
tff(fact_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A3,B2),C2) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).
% divide_divide_eq_left'
tff(fact_944_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_945_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_946_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_947_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_948_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( A3 = divide_divide(A,B2,C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_949_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( ( divide_divide(A,B2,C2) = A3 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_950_eq__divide__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 )
=> ( A3 = divide_divide(A,B2,C2) ) ) ) ) ).
% eq_divide_imp
tff(fact_951_divide__eq__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
=> ( divide_divide(A,B2,C2) = A3 ) ) ) ) ).
% divide_eq_imp
tff(fact_952_eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = divide_divide(A,B2,C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq
tff(fact_953_divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( divide_divide(A,B2,C2) = A3 )
<=> ( ( ( C2 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq
tff(fact_954_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_955_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_956_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_957_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_958_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_959_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_960_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A3,B2),C2) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_961_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: 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),C2),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,C2,A3)),divide_divide(A,C2,B2))) ) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_962_divide__strict__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C2: 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)),C2))
=> ( 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,C2,A3)),divide_divide(A,C2,B2))) ) ) ) ) ).
% divide_strict_left_mono
tff(fact_963_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_964_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_965_pos__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ).
% pos_less_divide_eq
tff(fact_966_pos__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),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),C2))) ) ) ) ).
% pos_divide_less_eq
tff(fact_967_neg__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C2)))
<=> 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),C2))) ) ) ) ).
% neg_less_divide_eq
tff(fact_968_neg__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),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),C2)),B2)) ) ) ) ).
% neg_divide_less_eq
tff(fact_969_less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq
tff(fact_970_divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% divide_less_eq
tff(fact_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N2: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),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),N2))) ) ) ) ).
% power_strict_increasing
tff(fact_982_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_983_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_984_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N2: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),N2))) ) ) ) ).
% power_increasing
tff(fact_985_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_986_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_987_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( C2 = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_988_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) = C2 )
<=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_989_minus__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) = A3 )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_990_eq__minus__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_991_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_992_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_993_divide__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: 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),C2),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,C2,A3)),divide_divide(A,C2,B2))) ) ) ) ) ).
% divide_left_mono_neg
tff(fact_994_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_995_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_996_pos__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C2)))
<=> 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),C2)),B2)) ) ) ) ).
% pos_le_divide_eq
tff(fact_997_pos__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),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),C2))) ) ) ) ).
% pos_divide_le_eq
tff(fact_998_neg__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C2)))
<=> 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),C2))) ) ) ) ).
% neg_le_divide_eq
tff(fact_999_neg__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),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),C2)),B2)) ) ) ) ).
% neg_divide_le_eq
tff(fact_1000_divide__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C2: 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)),C2))
=> ( 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,C2,A3)),divide_divide(A,C2,B2))) ) ) ) ) ).
% divide_left_mono
tff(fact_1001_le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1002_divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1003_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_1004_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_1005_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N2: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),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),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_strict_decreasing
tff(fact_1006_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N2: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_decreasing
tff(fact_1007_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_1008_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_1009_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),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),C2))) ) ) ) ).
% pos_minus_divide_less_eq
tff(fact_1010_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_less_minus_divide_eq
tff(fact_1011_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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,C2))),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),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_less_eq
tff(fact_1012_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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,C2))))
<=> 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),C2))) ) ) ) ).
% neg_less_minus_divide_eq
tff(fact_1013_minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: 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,C2))),A3))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1014_less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1015_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_1016_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_1017_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_1018_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),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),C2))) ) ) ) ).
% pos_minus_divide_le_eq
tff(fact_1019_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
<=> 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),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_le_minus_divide_eq
tff(fact_1020_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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,C2))),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),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_le_eq
tff(fact_1021_div__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: 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),C2)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self4
tff(fact_1022_div__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: 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),C2),B2)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self3
tff(fact_1023_div__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C2: 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),C2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self2
tff(fact_1024_div__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C2: 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),C2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A3,B2)) ) ) ) ).
% div_mult_self1
tff(fact_1025_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_1026_div__mult__mult1__if,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2 = zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
& ( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).
% div_mult_mult1_if
tff(fact_1027_div__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A3,B2) ) ) ) ).
% div_mult_mult2
tff(fact_1028_div__mult__mult1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A3,B2) ) ) ) ).
% div_mult_mult1
tff(fact_1029_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_1030_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M2,N)),M2)) ) ) ).
% div_less_dividend
tff(fact_1031_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
=> ( ( divide_divide(nat,M2,N) = M2 )
<=> ( N = one_one(nat) ) ) ) ).
% div_eq_dividend_iff
tff(fact_1032_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_1033_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_1034_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_1035_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),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)),C2))
=> 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))),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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))),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1036_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C2: 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,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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))),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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))),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1037_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_1038_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),C2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% distrib_right_NO_MATCH
tff(fact_1039_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ) ) ).
% distrib_left_NO_MATCH
tff(fact_1040_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_1041_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_1042_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_1043_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_1044_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : minus_minus(A,A3,A3) = zero_zero(A) ) ).
% diff_self
tff(fact_1045_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : minus_minus(A,A3,zero_zero(A)) = A3 ) ).
% diff_0_right
tff(fact_1046_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_1047_diff__zero,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : minus_minus(A,A3,zero_zero(A)) = A3 ) ).
% diff_zero
tff(fact_1048_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_1049_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [V2: num,W: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),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),V2),W))),Z2) ) ).
% mult_numeral_left_semiring_numeral
tff(fact_1050_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_1051_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_1052_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_1053_add__diff__cancel__left,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [C2: A,A3: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = minus_minus(A,A3,B2) ) ).
% add_diff_cancel_left
tff(fact_1054_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_1055_add__diff__cancel__right,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = minus_minus(A,A3,B2) ) ).
% add_diff_cancel_right
tff(fact_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_distrib__right__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [A3: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).
% distrib_right_numeral
tff(fact_1065_distrib__left__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).
% distrib_left_numeral
tff(fact_1066_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),minus_minus(A,B2,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).
% right_diff_distrib_numeral
tff(fact_1067_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [A3: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),aa(num,A,numeral_numeral(A),V2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).
% left_diff_distrib_numeral
tff(fact_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C2,D3) )
=> ( ( A3 = B2 )
<=> ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_1089_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : minus_minus(A,minus_minus(A,A3,C2),B2) = minus_minus(A,minus_minus(A,A3,B2),C2) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1090_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: 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),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,C2)),minus_minus(A,B2,D3))) ) ) ) ).
% diff_mono
tff(fact_1091_diff__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: 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,C2,A3)),minus_minus(A,C2,B2))) ) ) ).
% diff_left_mono
tff(fact_1092_diff__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: 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,C2)),minus_minus(A,B2,C2))) ) ) ).
% diff_right_mono
tff(fact_1093_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C2,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),C2),D3)) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_1094_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_1095_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: 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),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,C2)),minus_minus(A,B2,D3))) ) ) ) ).
% diff_strict_mono
tff(fact_1096_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C2,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),C2),D3)) ) ) ) ).
% diff_eq_diff_less
tff(fact_1097_diff__strict__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: 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,C2,A3)),minus_minus(A,C2,B2))) ) ) ).
% diff_strict_left_mono
tff(fact_1098_diff__strict__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: 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,C2)),minus_minus(A,B2,C2))) ) ) ).
% diff_strict_right_mono
tff(fact_1099_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_1100_diff__eq__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus(A,A3,B2) = C2 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).
% diff_eq_eq
tff(fact_1101_eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( ( A3 = minus_minus(A,C2,B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).
% eq_diff_eq
tff(fact_1102_add__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),minus_minus(A,B2,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).
% add_diff_eq
tff(fact_1103_diff__diff__eq2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : minus_minus(A,A3,minus_minus(A,B2,C2)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2),B2) ) ).
% diff_diff_eq2
tff(fact_1104_diff__add__eq,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,A3,B2)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2),B2) ) ).
% diff_add_eq
tff(fact_1105_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = minus_minus(A,minus_minus(A,A3,C2),B2) ) ).
% diff_add_eq_diff_diff_swap
tff(fact_1106_add__implies__diff,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [C2: A,B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A3 )
=> ( C2 = minus_minus(A,A3,B2) ) ) ) ).
% add_implies_diff
tff(fact_1107_diff__diff__eq,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A,C2: A] : minus_minus(A,minus_minus(A,A3,B2),C2) = minus_minus(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% diff_diff_eq
tff(fact_1108_right__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),minus_minus(A,B2,C2)) = 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),C2)) ) ).
% right_diff_distrib'
tff(fact_1109_left__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,C2)),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),C2),A3)) ) ).
% left_diff_distrib'
tff(fact_1110_right__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),minus_minus(A,B2,C2)) = 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),C2)) ) ).
% right_diff_distrib
tff(fact_1111_left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% left_diff_distrib
tff(fact_1112_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_1113_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( nO_MATCH(B,A,divide_divide(B,X,Y),C2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,A3,B2)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_1114_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C2: 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,C2)) = 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),C2)) ) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_1115_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_1116_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_1117_diff__le__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),minus_minus(A,A3,B2)),C2))
<=> 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),C2),B2))) ) ) ).
% diff_le_eq
tff(fact_1118_le__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),minus_minus(A,C2,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)),C2)) ) ) ).
% le_diff_eq
tff(fact_1119_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_1120_le__add__diff,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2),A3))) ) ) ).
% le_add_diff
tff(fact_1121_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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),C2),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),C2),A3)),B2)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1122_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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),C2),minus_minus(A,B2,A3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1123_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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),C2),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),minus_minus(A,B2,A3)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1124_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1125_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),minus_minus(A,B2,A3)),C2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1126_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( minus_minus(A,C2,minus_minus(A,B2,A3)) = minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3),B2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1127_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_1128_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: 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) = C2 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1129_less__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),minus_minus(A,C2,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)),C2)) ) ) ).
% less_diff_eq
tff(fact_1130_diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,A3,B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% diff_less_eq
tff(fact_1131_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2) = D3 ) ) ) ).
% eq_add_iff1
tff(fact_1132_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
<=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E2)),D3) ) ) ) ).
% eq_add_iff2
tff(fact_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2)),D3)) ) ) ).
% le_add_iff1
tff(fact_1144_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E2)),D3))) ) ) ).
% le_add_iff2
tff(fact_1145_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),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)),E2)),C2)),D3)) ) ) ).
% less_add_iff1
tff(fact_1146_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,B2,A3)),E2)),D3))) ) ) ).
% less_add_iff2
tff(fact_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W: num] :
( ( divide_divide(A,B2,C2) = aa(num,A,numeral_numeral(A),W) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_1158_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B2: A,C2: A] :
( ( aa(num,A,numeral_numeral(A),W) = divide_divide(A,B2,C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2 ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_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_1170_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_1171_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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)),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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)),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1172_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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)),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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)),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1173_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W: num] :
( ( divide_divide(A,B2,C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
<=> ( ( ( C2 != 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))),C2) ) )
& ( ( C2 = 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_1174_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num,B2: A,C2: A] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = divide_divide(A,B2,C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B2 ) )
& ( ( C2 = 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_1175_mult__eq__if,axiom,
! [M2: nat,N: nat] :
( ( ( M2 = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = zero_zero(nat) ) )
& ( ( M2 != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,M2,one_one(nat))),N)) ) ) ) ).
% mult_eq_if
tff(fact_1176_scaling__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [U: A,V2: A,R2: A,S3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),S3))
=> 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),R2),minus_minus(A,V2,U)),S3))),V2)) ) ) ) ) ).
% scaling_mono
tff(fact_1177_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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)),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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)),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1178_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C2: 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,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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)),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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)),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1179_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W: num,B2: A,C2: 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,C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> 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))),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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))),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1180_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),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)),C2))
=> 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))),C2))) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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))),C2)),B2)) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),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_1181_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_1182_power__eq__if,axiom,
! [A: $tType] :
( power(A)
=> ! [M2: nat,P7: A] :
( ( ( M2 = zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P7),M2) = one_one(A) ) )
& ( ( M2 != zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P7),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),P7),aa(nat,A,aa(A,fun(nat,A),power_power(A),P7),minus_minus(nat,M2,one_one(nat)))) ) ) ) ) ).
% power_eq_if
tff(fact_1183_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V2: 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),V2))),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),V2),W)))),Y) ) ).
% semiring_norm(170)
tff(fact_1184_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),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),V2),W)))),Y) ) ).
% semiring_norm(171)
tff(fact_1185_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V2: 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),V2))),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),V2),W))),Y) ) ).
% semiring_norm(172)
tff(fact_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).
% slice_Nil
tff(fact_1196_slice__eq__bounds__empty,axiom,
! [A: $tType,I2: nat,Xs: list(A)] : slice(A,I2,I2,Xs) = nil(A) ).
% slice_eq_bounds_empty
tff(fact_1197_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_1198_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_1199_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_1200_Int__Diff,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),minus_minus(set(A),B3,C5)) ).
% Int_Diff
tff(fact_1201_Diff__Int2,axiom,
! [A: $tType,A4: set(A),C5: 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),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5),B3) ).
% Diff_Int2
tff(fact_1202_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_1203_Diff__Int__distrib,axiom,
! [A: $tType,C5: 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)),C5),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)),C5),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),B3)) ).
% Diff_Int_distrib
tff(fact_1204_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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)),C5) = minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ).
% Diff_Int_distrib2
tff(fact_1205_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_1206_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_1207_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_1208_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_1209_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_1210_Diff__Un,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5)) = 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,C5)) ).
% Diff_Un
tff(fact_1211_Diff__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A),C5: 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),C5)) = 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,C5)) ).
% Diff_Int
tff(fact_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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_1218_power_Opower__eq__if,axiom,
! [A: $tType,M2: nat,One: A,Times: fun(A,fun(A,A)),P7: A] :
( ( ( M2 = zero_zero(nat) )
=> ( power2(A,One,Times,P7,M2) = One ) )
& ( ( M2 != zero_zero(nat) )
=> ( power2(A,One,Times,P7,M2) = aa(A,A,aa(A,fun(A,A),Times,P7),power2(A,One,Times,P7,minus_minus(nat,M2,one_one(nat)))) ) ) ) ).
% power.power_eq_if
tff(fact_1219_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_1220_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_1221_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_1222_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_1223_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_1224_Suc__diff__1,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,nat,suc,minus_minus(nat,N,one_one(nat))) = N ) ) ).
% Suc_diff_1
tff(fact_1225_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_1226_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_1227_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_1228_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_1229_diff__Suc__1,axiom,
! [N: nat] : minus_minus(nat,aa(nat,nat,suc,N),one_one(nat)) = N ).
% diff_Suc_1
tff(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list(A),V2: 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),V2)) = aa(nat,A,nth(A,Xs),minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2),one_one(nat))) ).
% nth_Cons_numeral
tff(fact_1237_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_1238_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_1239_One__nat__def,axiom,
one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).
% One_nat_def
tff(fact_1240_Suc__eq__plus1,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).
% Suc_eq_plus1
tff(fact_1241_plus__1__eq__Suc,axiom,
aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).
% plus_1_eq_Suc
tff(fact_1242_Suc__eq__plus1__left,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).
% Suc_eq_plus1_left
tff(fact_1243_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] : minus_minus(nat,M2,aa(nat,nat,suc,N)) = minus_minus(nat,minus_minus(nat,M2,one_one(nat)),N) ).
% diff_Suc_eq_diff_pred
tff(fact_1244_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_1245_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_1246_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_1247_numerals_I1_J,axiom,
aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).
% numerals(1)
tff(fact_1248_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_1249_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_1250_nat__induct__non__zero,axiom,
! [N: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,P,one_one(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_induct_non_zero
tff(fact_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( minus_minus(nat,aa(nat,nat,suc,M2),N) = minus_minus(nat,M2,minus_minus(nat,N,one_one(nat))) ) ) ).
% Suc_diff_eq_diff_pred
tff(fact_1258_Suc__pred_H,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( N = aa(nat,nat,suc,minus_minus(nat,N,one_one(nat))) ) ) ).
% Suc_pred'
tff(fact_1259_add__eq__if,axiom,
! [M2: nat,N: nat] :
( ( ( M2 = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = N ) )
& ( ( M2 != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),minus_minus(nat,M2,one_one(nat))),N)) ) ) ) ).
% add_eq_if
tff(fact_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_Suc__1,axiom,
aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).
% Suc_1
tff(fact_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,B2,A3) ) ) ).
% pos_zdiv_mult_2
tff(fact_1284_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
=> ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A3) ) ) ).
% neg_zdiv_mult_2
tff(fact_1285_minus__1__div__exp__eq__int,axiom,
! [N: nat] : divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).
% minus_1_div_exp_eq_int
tff(fact_1286_int__bit__induct,axiom,
! [P: fun(int,bool),K: int] :
( pp(aa(int,bool,P,zero_zero(int)))
=> ( pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),one_one(int))))
=> ( ! [K2: int] :
( pp(aa(int,bool,P,K2))
=> ( ( K2 != zero_zero(int) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
=> ( ! [K2: int] :
( pp(aa(int,bool,P,K2))
=> ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
=> pp(aa(int,bool,P,K)) ) ) ) ) ).
% int_bit_induct
tff(fact_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_nat__1__add__1,axiom,
aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).
% nat_1_add_1
tff(fact_1294_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_1295_nat__induct2,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( pp(aa(nat,bool,P,one_one(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_induct2
tff(fact_1296_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_1297_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_1298_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_1299_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_1300_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).
% ex_power_ivl2
tff(fact_1301_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).
% ex_power_ivl1
tff(fact_1302_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,R2: A,Q3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),one_one(A))),minus_minus(A,R2,aa(num,A,numeral_numeral(A),L))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),R2) ) ) ) ) ).
% divmod_step_eq
tff(fact_1303_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_1304_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_1305_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_1306_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_1307_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_1308_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_1309_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_1310_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( pp(dvd_dvd(nat,M2,one_one(nat)))
<=> ( M2 = one_one(nat) ) ) ).
% nat_dvd_1_iff_1
tff(fact_1311_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3)))
<=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_1312_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
<=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_1313_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( C2 = zero_zero(A) )
| pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% dvd_mult_cancel_right
tff(fact_1314_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( C2 = zero_zero(A) )
| pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% dvd_mult_cancel_left
tff(fact_1315_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3))))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_1316_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: 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),C2),A3)),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_1317_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_1318_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_1319_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_1320_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_1321_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_1322_mod__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: 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),C2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self4
tff(fact_1323_mod__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: 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),C2),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self3
tff(fact_1324_mod__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C2: 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),C2)),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self2
tff(fact_1325_mod__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: 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),C2),B2)),B2) = modulo_modulo(A,A3,B2) ) ).
% mod_mult_self1
tff(fact_1326_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_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
=> ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_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_1344_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_1345_mod2__gr__0,axiom,
! [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
<=> ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).
% mod2_gr_0
tff(fact_1346_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_1347_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_1348_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_1349_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_1350_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = minus_minus(nat,N,one_one(nat)) ) ) ).
% odd_two_times_div_two_nat
tff(fact_1351_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_1352_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_1353_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_1354_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
=> ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% div_pos_neg_trivial
tff(fact_1355_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
=> ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = minus_minus(int,minus_minus(int,L,one_one(int)),modulo_modulo(int,minus_minus(int,K,one_one(int)),L)) ) ) ).
% minus_mod_int_eq
tff(fact_1356_zmod__minus1,axiom,
! [B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = minus_minus(int,B2,one_one(int)) ) ) ).
% zmod_minus1
tff(fact_1357_div__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
=> ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,minus_minus(int,K,L),L)),one_one(int)) ) ) ) ).
% div_pos_geq
tff(fact_1358_int__div__less__self,axiom,
! [X: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,X,K)),X)) ) ) ).
% int_div_less_self
tff(fact_1359_div__eq__minus1,axiom,
! [B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).
% div_eq_minus1
tff(fact_1360_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2 != zero_zero(int) )
=> ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
=> ( divide_divide(int,aa(int,int,uminus_uminus(int),A3),B2) = aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)) ) )
& ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
=> ( divide_divide(int,aa(int,int,uminus_uminus(int),A3),B2) = minus_minus(int,aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)),one_one(int)) ) ) ) ) ).
% zdiv_zminus1_eq_if
tff(fact_1361_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2 != zero_zero(int) )
=> ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
=> ( divide_divide(int,A3,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)) ) )
& ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
=> ( divide_divide(int,A3,aa(int,int,uminus_uminus(int),B2)) = minus_minus(int,aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)),one_one(int)) ) ) ) ) ).
% zdiv_zminus2_eq_if
tff(fact_1362_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_1363_mod__mult__right__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).
% mod_mult_right_eq
tff(fact_1364_mod__mult__left__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).
% mod_mult_left_eq
tff(fact_1365_mult__mod__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).
% mult_mod_right
tff(fact_1366_mod__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C2) ) ).
% mod_mult_mult2
tff(fact_1367_mod__mult__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,A9: A,B2: A,B6: A] :
( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A9,C2) )
=> ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
=> ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A9),B6),C2) ) ) ) ) ).
% mod_mult_cong
tff(fact_1368_mod__mult__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ).
% mod_mult_eq
tff(fact_1369_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_1370_dvd__mult__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2))
=> pp(dvd_dvd(A,B2,C2)) ) ) ).
% dvd_mult_right
tff(fact_1371_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,C2,D3))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ).
% mult_dvd_mono
tff(fact_1372_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_1373_dvd__mult__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2))
=> pp(dvd_dvd(A,A3,C2)) ) ) ).
% dvd_mult_left
tff(fact_1374_dvd__mult2,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,B2))
=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).
% dvd_mult2
tff(fact_1375_dvd__mult,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,A3,C2))
=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).
% dvd_mult
tff(fact_1376_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_1377_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_1378_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_1379_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_1380_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_1381_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : pp(dvd_dvd(A,one_one(A),A3)) ) ).
% one_dvd
tff(fact_1382_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_1383_mod__eqE,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
=> ~ ! [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),C2),D4)) ) ) ).
% mod_eqE
tff(fact_1384_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,A3))) ) ) ).
% pos_zmod_mult_2
tff(fact_1385_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
=> ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A3)),one_one(int)) ) ) ).
% neg_zmod_mult_2
tff(fact_1386_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_1387_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
tff(fact_1388_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: 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),C2) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
tff(fact_1389_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: 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),C2))
<=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_1390_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: 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),C2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_1391_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: 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),C2))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_1392_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: 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),C2),B2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_1393_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_1394_dvd__div__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C2) ) ) ) ).
% dvd_div_mult
tff(fact_1395_div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ) ) ).
% div_mult_swap
tff(fact_1396_div__div__eq__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,B2))
=> ( pp(dvd_dvd(A,B2,A3))
=> ( divide_divide(A,A3,divide_divide(A,B2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),C2) ) ) ) ) ).
% div_div_eq_right
tff(fact_1397_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2),A3))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A3,B2),C2) ) ) ) ).
% dvd_div_mult2_eq
tff(fact_1398_dvd__mult__imp__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2))
=> pp(dvd_dvd(A,A3,divide_divide(A,B2,C2))) ) ) ).
% dvd_mult_imp_div
tff(fact_1399_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,D3: A,C2: A] :
( pp(dvd_dvd(A,B2,A3))
=> ( pp(dvd_dvd(A,D3,C2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),divide_divide(A,C2,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_1400_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,divide_divide(A,C2,B2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_div_unit_iff
tff(fact_1401_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,divide_divide(A,A3,B2),C2))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% div_unit_dvd_iff
tff(fact_1402_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( divide_divide(A,B2,A3) = divide_divide(A,C2,A3) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
tff(fact_1403_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_1404_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_1405_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).
% cancel_div_mod_rules(2)
tff(fact_1406_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).
% cancel_div_mod_rules(1)
tff(fact_1407_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_1408_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_1409_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_1410_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_1411_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_1412_div__mult1__eq,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = 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,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C2)),C2)) ) ).
% div_mult1_eq
tff(fact_1413_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_1414_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_1415_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_1416_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_1417_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_1418_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C4) ) ) ) ).
% unit_dvdE
tff(fact_1419_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D2: A,T3: 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),T3)))
<=> 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))),T3))) ) ) ) ).
% inf_period(3)
tff(fact_1420_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D2: A,T3: 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),T3)))
<=> ~ 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))),T3))) ) ) ) ).
% inf_period(4)
tff(fact_1421_dvd__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,A3,B2))
=> ( ( divide_divide(A,B2,A3) = C2 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_1422_div__dvd__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,A3))
=> ( pp(dvd_dvd(A,divide_divide(A,A3,B2),C2))
<=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).
% div_dvd_iff_mult
tff(fact_1423_dvd__div__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( pp(dvd_dvd(A,C2,B2))
=> ( pp(dvd_dvd(A,A3,divide_divide(A,B2,C2)))
<=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2)) ) ) ) ) ).
% dvd_div_iff_mult
tff(fact_1424_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( A3 != zero_zero(A) )
=> ( ( C2 != zero_zero(A) )
=> ( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,C2,D3))
=> ( ( divide_divide(A,B2,A3) = divide_divide(A,D3,C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_1425_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_1426_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,C2,one_one(A)))
=> ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A3,B2),C2) ) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_1427_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ) ) ).
% unit_div_mult_swap
tff(fact_1428_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: 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)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2) ) ) ) ).
% unit_div_commute
tff(fact_1429_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,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),C2)) = divide_divide(A,divide_divide(A,A3,B2),C2) ) ) ) ) ).
% div_mult_unit2
tff(fact_1430_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( A3 = divide_divide(A,C2,B2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C2 ) ) ) ) ).
% unit_eq_div2
tff(fact_1431_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( divide_divide(A,A3,B2) = C2 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).
% unit_eq_div1
tff(fact_1432_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_1433_power__dvd__imp__le,axiom,
! [I2: nat,M2: nat,N: nat] :
( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).
% power_dvd_imp_le
tff(fact_1434_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [B5: A] :
( ( B5 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B5,one_one(A)))
=> ( ( divide_divide(A,one_one(A),A3) = B5 )
=> ( ( divide_divide(A,one_one(A),B5) = A3 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B5) = one_one(A) )
=> ( divide_divide(A,C2,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B5) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_1435_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_1436_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_1437_evenE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ~ ! [B5: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B5) ) ) ).
% evenE
tff(fact_1438_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_1439_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_1440_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_1441_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
=> ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2),M2))
<=> ( N = one_one(nat) ) ) ) ).
% dvd_mult_cancel2
tff(fact_1442_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
=> ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N),M2))
<=> ( N = one_one(nat) ) ) ) ).
% dvd_mult_cancel1
tff(fact_1443_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_1444_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_1445_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,divide_divide(A,A3,B2),C2))),modulo_modulo(A,A3,B2)) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1446_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_1447_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
=> ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2) = one_one(nat) ) ) ).
% Suc_times_mod_eq
tff(fact_1448_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_1449_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A3))
=> ~ ! [B5: 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))),B5)),one_one(A)) ) ) ).
% oddE
tff(fact_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z2)) ) ).
% zle_add1_eq_le
tff(fact_1458_verit__le__mono__div,axiom,
! [A4: nat,B3: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A4),B3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A4,N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B3,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),divide_divide(nat,B3,N))) ) ) ).
% verit_le_mono_div
tff(fact_1459_verit__le__mono__div__int,axiom,
! [A4: int,B3: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A4),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A4,N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B3,N)),zero_zero(int)),one_one(int),zero_zero(int)))),divide_divide(int,B3,N))) ) ) ).
% verit_le_mono_div_int
tff(fact_1460_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),minus_minus(int,Z2,one_one(int))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2)) ) ).
% zle_diff1_eq
tff(fact_1461_one__div__minus__numeral,axiom,
! [N: num] : divide_divide(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).
% one_div_minus_numeral
tff(fact_1462_minus__one__div__numeral,axiom,
! [N: num] : divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).
% minus_one_div_numeral
tff(fact_1463_one__mod__minus__numeral,axiom,
! [N: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).
% one_mod_minus_numeral
tff(fact_1464_minus__one__mod__numeral,axiom,
! [N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).
% minus_one_mod_numeral
tff(fact_1465_int__less__induct,axiom,
! [I2: int,K: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K))
=> ( pp(aa(int,bool,P,minus_minus(int,K,one_one(int))))
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I3),K))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,minus_minus(int,I3,one_one(int)))) ) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% int_less_induct
tff(fact_1466_int__le__induct,axiom,
! [I2: int,K: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
=> ( pp(aa(int,bool,P,K))
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,minus_minus(int,I3,one_one(int)))) ) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% int_le_induct
tff(fact_1467_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2))
| ( W = Z2 ) ) ) ).
% zless_add1_eq
tff(fact_1468_int__gr__induct,axiom,
! [K: int,I2: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I3))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% int_gr_induct
tff(fact_1469_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).
% int_one_le_iff_zero_less
tff(fact_1470_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M2))
=> ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
<=> ( ( M2 = one_one(int) )
& ( N = one_one(int) ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_1471_zsgn__def,axiom,
! [I2: int] :
( ( ( I2 = zero_zero(int) )
=> ( aa(int,int,sgn_sgn(int),I2) = zero_zero(int) ) )
& ( ( I2 != zero_zero(int) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
=> ( aa(int,int,sgn_sgn(int),I2) = one_one(int) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
=> ( aa(int,int,sgn_sgn(int),I2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).
% zsgn_def
tff(fact_1472_odd__nonzero,axiom,
! [Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2) != zero_zero(int) ).
% odd_nonzero
tff(fact_1473_odd__less__0__iff,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ).
% odd_less_0_iff
tff(fact_1474_int__ge__induct,axiom,
! [K: int,I2: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
=> ( pp(aa(int,bool,P,K))
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% int_ge_induct
tff(fact_1475_add1__zle__eq,axiom,
! [W: int,Z2: 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),W),one_one(int))),Z2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2)) ) ).
% add1_zle_eq
tff(fact_1476_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z2)) ) ).
% zless_imp_add1_zle
tff(fact_1477_int__induct,axiom,
! [P: fun(int,bool),K: int,I2: int] :
( pp(aa(int,bool,P,K))
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I3))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
=> ( ! [I3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K))
=> ( pp(aa(int,bool,P,I3))
=> pp(aa(int,bool,P,minus_minus(int,I3,one_one(int)))) ) )
=> pp(aa(int,bool,P,I2)) ) ) ) ).
% int_induct
tff(fact_1478_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
=> ( ( M2 = one_one(int) )
| ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
tff(fact_1479_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
<=> ( ( ( M2 = one_one(int) )
& ( N = one_one(int) ) )
| ( ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) )
& ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
% zmult_eq_1_iff
tff(fact_1480_le__imp__0__less,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2))) ) ).
% le_imp_0_less
tff(fact_1481_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_1482_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_1483_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
=> ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2),one_one(int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_1484_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_1485_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_1486_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_1487_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_1488_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_1489_zminus1__lemma,axiom,
! [A3: int,B2: int,Q3: int,R2: int] :
( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> ( ( B2 != zero_zero(int) )
=> eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),aa(int,int,uminus_uminus(int),Q3),minus_minus(int,aa(int,int,uminus_uminus(int),Q3),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),zero_zero(int),minus_minus(int,B2,R2)))) ) ) ).
% zminus1_lemma
tff(fact_1490_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_1491_divmod__BitM__2__eq,axiom,
! [M2: num] : unique8689654367752047608divmod(int,bitM(M2),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),minus_minus(int,aa(num,int,numeral_numeral(int),M2),one_one(int))),one_one(int)) ).
% divmod_BitM_2_eq
tff(fact_1492_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_1493_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_1494_and__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = zero_zero(int) )
| ( L = zero_zero(int) ) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = zero_zero(int) ) )
& ( ~ ( ( K = zero_zero(int) )
| ( L = zero_zero(int) ) )
=> ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = L ) )
& ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = K ) )
& ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).
% and_int_unfold
tff(fact_1495_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_1496_upto__aux__rec,axiom,
! [J: int,I2: int,Js: list(int)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
=> ( upto_aux(I2,J,Js) = Js ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
=> ( upto_aux(I2,J,Js) = upto_aux(I2,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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_sgn__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,sgn_sgn(A),A3)),aa(A,A,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_1511_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_1512_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_1513_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_1514_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_1515_and__minus__numerals_I6_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N)))),one_one(int)) = one_one(int) ).
% and_minus_numerals(6)
tff(fact_1516_and__minus__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N)))) = one_one(int) ).
% and_minus_numerals(2)
tff(fact_1517_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_1518_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_1519_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_1520_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_1521_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_1522_and__minus__numerals_I5_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = zero_zero(int) ).
% and_minus_numerals(5)
tff(fact_1523_and__minus__numerals_I1_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = zero_zero(int) ).
% and_minus_numerals(1)
tff(fact_1524_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_1525_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_1526_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_1527_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_1528_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_1529_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_1530_zmod__numeral__Bit1,axiom,
! [V2: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit1(V2)),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V2),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).
% zmod_numeral_Bit1
tff(fact_1531_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_bit1
tff(fact_1532_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_1533_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_1534_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_1535_of__bool__def,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P7: bool] :
( ( pp(P7)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P7) = one_one(A) ) )
& ( ~ pp(P7)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P7) = zero_zero(A) ) ) ) ) ).
% of_bool_def
tff(fact_1536_split__of__bool,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P7: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P7)))
<=> ( ( pp(P7)
=> pp(aa(A,bool,P,one_one(A))) )
& ( ~ pp(P7)
=> pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool
tff(fact_1537_split__of__bool__asm,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P7: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P7)))
<=> ~ ( ( pp(P7)
& ~ pp(aa(A,bool,P,one_one(A))) )
| ( ~ pp(P7)
& ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool_asm
tff(fact_1538_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_1539_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit(int,N,K) != zero_zero(int) )
=> ( bit_se2584673776208193580ke_bit(int,N,minus_minus(int,K,one_one(int))) = minus_minus(int,bit_se2584673776208193580ke_bit(int,N,K),one_one(int)) ) ) ).
% take_bit_decr_eq
tff(fact_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,N,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% take_bit_Suc_minus_bit1
tff(fact_1547_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_1548_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_1549_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_1550_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_1551_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_1552_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [A6: A] :
( ( divide_divide(A,A6,aa(num,A,numeral_numeral(A),bit0(one2))) = A6 )
=> pp(aa(A,bool,P,A6)) )
=> ( ! [A6: A,B5: bool] :
( pp(aa(A,bool,P,A6))
=> ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A6)),aa(num,A,numeral_numeral(A),bit0(one2))) = A6 )
=> 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),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A6)))) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% bits_induct
tff(fact_1553_push__bit__minus__one,axiom,
! [N: nat] : bit_se4730199178511100633sh_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).
% push_bit_minus_one
tff(fact_1554_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_1555_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit(int,N,K) != minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N),one_one(int)) )
=> ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),bit_se2584673776208193580ke_bit(int,N,K)) ) ) ).
% take_bit_incr_eq
tff(fact_1556_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_1557_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_1558_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_1559_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_1560_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_1561_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_minus_bit1
tff(fact_1562_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_bit1
tff(fact_1563_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% take_bit_numeral_minus_bit1
tff(fact_1564_and__int_Osimps,axiom,
! [K: int,L: int] :
( ( ( pp(member2(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L))))) ) )
& ( ~ ( pp(member2(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).
% and_int.simps
tff(fact_1565_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( ( ( pp(member2(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa))))) ) )
& ( ~ ( pp(member2(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).
% and_int.elims
tff(fact_1566_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_1567_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_1568_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_1569_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)),insert3(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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_1570_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)),insert3(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_1571_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)),insert3(A),A3),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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_1572_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C5: set(A),B3: set(A)] :
( pp(member2(A,A3,C5))
=> ( 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)),insert3(A),A3),B3)),C5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ) ) ).
% Int_insert_left_if1
tff(fact_1573_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C5: set(A),B3: set(A)] :
( ~ pp(member2(A,A3,C5))
=> ( 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)),insert3(A),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5) ) ) ).
% Int_insert_left_if0
tff(fact_1574_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)),insert3(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_1575_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)),insert3(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_1576_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)),insert3(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_1577_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)),insert3(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_1578_upto__Nil,axiom,
! [I2: int,J: int] :
( ( upto(I2,J) = nil(int) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).
% upto_Nil
tff(fact_1579_upto__Nil2,axiom,
! [I2: int,J: int] :
( ( nil(int) = upto(I2,J) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).
% upto_Nil2
tff(fact_1580_upto__empty,axiom,
! [J: int,I2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
=> ( upto(I2,J) = nil(int) ) ) ).
% upto_empty
tff(fact_1581_upto__single,axiom,
! [I2: int] : upto(I2,I2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),nil(int)) ).
% upto_single
tff(fact_1582_and__nat__numerals_I2_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = one_one(nat) ).
% and_nat_numerals(2)
tff(fact_1583_and__nat__numerals_I4_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit1(X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).
% and_nat_numerals(4)
tff(fact_1584_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_1585_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_1586_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)),insert3(A),A3),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(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_1587_Int__insert__left,axiom,
! [A: $tType,A3: A,C5: set(A),B3: set(A)] :
( ( pp(member2(A,A3,C5))
=> ( 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)),insert3(A),A3),B3)),C5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ) )
& ( ~ pp(member2(A,A3,C5))
=> ( 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)),insert3(A),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5) ) ) ) ).
% Int_insert_left
tff(fact_1588_upto__code,axiom,
! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil(int)) ).
% upto_code
tff(fact_1589_pred__numeral__def,axiom,
! [K: num] : pred_numeral(K) = minus_minus(nat,aa(num,nat,numeral_numeral(nat),K),one_one(nat)) ).
% pred_numeral_def
tff(fact_1590_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_1591_upto_Osimps,axiom,
! [I2: int,J: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = nil(int) ) ) ) ).
% upto.simps
tff(fact_1592_upto__rec1,axiom,
! [I2: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ).
% upto_rec1
tff(fact_1593_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_1594_and__int_Opsimps,axiom,
! [K: int,L: int] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
=> ( ( ( pp(member2(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L))))) ) )
& ( ~ ( pp(member2(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).
% and_int.psimps
tff(fact_1595_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
=> ~ ( ( ( ( pp(member2(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa))))) ) )
& ( ~ ( pp(member2(int,X,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,Xa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) )
=> ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).
% and_int.pelims
tff(fact_1596_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_1597_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
<=> pp(dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))) ) ).
% take_bit_eq_mask_iff_exp_dvd
tff(fact_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_flip__bit__nat__def,axiom,
! [M2: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),bit_se4730199178511100633sh_bit(nat,M2,one_one(nat))) ).
% flip_bit_nat_def
tff(fact_1611_flip__bit__int__def,axiom,
! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).
% flip_bit_int_def
tff(fact_1612_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_1613_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit(int,N,K) = bit_se2239418461657761734s_mask(int,N) )
<=> ( bit_se2584673776208193580ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).
% take_bit_eq_mask_iff
tff(fact_1614_mask__half__int,axiom,
! [N: nat] : divide_divide(int,bit_se2239418461657761734s_mask(int,N),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,minus_minus(nat,N,one_one(nat))) ).
% mask_half_int
tff(fact_1615_mask__nat__def,axiom,
! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),one_one(nat)) ).
% mask_nat_def
tff(fact_1616_mask__int__def,axiom,
! [N: nat] : bit_se2239418461657761734s_mask(int,N) = minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N),one_one(int)) ).
% mask_int_def
tff(fact_1617_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_1618_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
=> ( ! [K2: int,L3: int] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L3))
=> ( ( ~ ( pp(member2(int,K2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member2(int,L3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),bit0(one2))))) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L3)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% and_int.pinduct
tff(fact_1619_upto_Opsimps,axiom,
! [I2: 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),I2),J))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = nil(int) ) ) ) ) ).
% upto.psimps
tff(fact_1620_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_1621_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,J2: int] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J2))
=> 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))),J2)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J2)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% upto.pinduct
tff(fact_1622_nat0__intermed__int__val,axiom,
! [N: nat,F2: fun(nat,int),K: int] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(nat,int,F2,I3)))),one_one(int))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
& ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_1623_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_1624_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_1625_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_1626_and__not__numerals_I8_J,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% and_not_numerals(8)
tff(fact_1627_abs__idempotent,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : abs_abs(A,abs_abs(A,A3)) = abs_abs(A,A3) ) ).
% abs_idempotent
tff(fact_1628_abs__0__eq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( zero_zero(A) = abs_abs(A,A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_0_eq
tff(fact_1629_abs__eq__0,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( abs_abs(A,A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_eq_0
tff(fact_1630_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_1631_abs__add__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B2)) ) ).
% abs_add_abs
tff(fact_1632_abs__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% abs_mult_self_eq
tff(fact_1633_abs__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
% abs_1
tff(fact_1634_abs__minus__cancel,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : abs_abs(A,aa(A,A,uminus_uminus(A),A3)) = abs_abs(A,A3) ) ).
% abs_minus_cancel
tff(fact_1635_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))
=> ( abs_abs(A,A3) = A3 ) ) ) ).
% abs_of_nonneg
tff(fact_1636_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),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_1637_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),abs_abs(A,A3)),zero_zero(A)))
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_le_zero_iff
tff(fact_1638_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)),abs_abs(A,A3)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% zero_less_abs_iff
tff(fact_1639_abs__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).
% abs_neg_one
tff(fact_1640_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_1641_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_1642_zdvd1__eq,axiom,
! [X: int] :
( pp(dvd_dvd(int,X,one_one(int)))
<=> ( abs_abs(int,X) = one_one(int) ) ) ).
% zdvd1_eq
tff(fact_1643_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)))
=> ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_nonpos
tff(fact_1644_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_1645_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_1646_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_1647_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_1648_abs__sgn__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).
% abs_sgn_eq_1
tff(fact_1649_or__minus__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N))) ).
% or_minus_numerals(2)
tff(fact_1650_or__minus__numerals_I6_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(N))) ).
% or_minus_numerals(6)
tff(fact_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_zabs__less__one__iff,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),abs_abs(int,Z2)),one_one(int)))
<=> ( Z2 = zero_zero(int) ) ) ).
% zabs_less_one_iff
tff(fact_1658_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_1659_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_1660_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_1661_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_1662_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_1663_or__minus__minus__numerals,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),minus_minus(int,aa(num,int,numeral_numeral(int),M2),one_one(int))),minus_minus(int,aa(num,int,numeral_numeral(int),N),one_one(int)))) ).
% or_minus_minus_numerals
tff(fact_1664_and__minus__minus__numerals,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),minus_minus(int,aa(num,int,numeral_numeral(int),M2),one_one(int))),minus_minus(int,aa(num,int,numeral_numeral(int),N),one_one(int)))) ).
% and_minus_minus_numerals
tff(fact_1665_or__not__numerals_I1_J,axiom,
aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).
% or_not_numerals(1)
tff(fact_1666_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),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_1667_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),abs_abs(A,A3))) ) ).
% abs_ge_self
tff(fact_1668_abs__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: 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),abs_abs(A,A3)),abs_abs(A,B2)) ) ).
% abs_mult
tff(fact_1669_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_1670_abs__minus__commute,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : abs_abs(A,minus_minus(A,A3,B2)) = abs_abs(A,minus_minus(A,B2,A3)) ) ).
% abs_minus_commute
tff(fact_1671_set__bit__nat__def,axiom,
! [M2: nat,N: nat] : bit_se5668285175392031749et_bit(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),bit_se4730199178511100633sh_bit(nat,M2,one_one(nat))) ).
% set_bit_nat_def
tff(fact_1672_or__not__numerals_I4_J,axiom,
! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).
% or_not_numerals(4)
tff(fact_1673_or__not__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).
% or_not_numerals(2)
tff(fact_1674_or__not__numerals_I3_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).
% or_not_numerals(3)
tff(fact_1675_or__not__numerals_I7_J,axiom,
! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).
% or_not_numerals(7)
tff(fact_1676_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_1677_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)),abs_abs(A,A3))) ) ).
% abs_ge_zero
tff(fact_1678_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),abs_abs(A,A3)),zero_zero(A))) ) ).
% abs_not_less_zero
tff(fact_1679_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))
=> ( abs_abs(A,A3) = A3 ) ) ) ).
% abs_of_pos
tff(fact_1680_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),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),abs_abs(A,A3)),abs_abs(A,B2)))) ) ).
% abs_triangle_ineq
tff(fact_1681_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,abs_abs(A,A3),abs_abs(A,B2))),abs_abs(A,minus_minus(A,B2,A3)))) ) ).
% abs_triangle_ineq2_sym
tff(fact_1682_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),abs_abs(A,minus_minus(A,abs_abs(A,A3),abs_abs(A,B2)))),abs_abs(A,minus_minus(A,A3,B2)))) ) ).
% abs_triangle_ineq3
tff(fact_1683_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,abs_abs(A,A3),abs_abs(A,B2))),abs_abs(A,minus_minus(A,A3,B2)))) ) ).
% abs_triangle_ineq2
tff(fact_1684_abs__mult__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),abs_abs(A,A3)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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),abs_abs(A,A3)),abs_abs(A,B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3))) ) ) ) ).
% abs_mult_less
tff(fact_1685_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),abs_abs(A,A3)),B2)) ) ) ) ).
% abs_leI
tff(fact_1686_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),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_1687_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),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_1688_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)),abs_abs(A,A3))) ) ).
% abs_ge_minus_self
tff(fact_1689_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: A] : abs_abs(A,K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).
% linordered_idom_class.abs_sgn
tff(fact_1690_abs__mult__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).
% abs_mult_sgn
tff(fact_1691_sgn__mult__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),abs_abs(A,A3)) = A3 ) ).
% sgn_mult_abs
tff(fact_1692_mult__sgn__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),abs_abs(A,X)) = X ) ).
% mult_sgn_abs
tff(fact_1693_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( abs_abs(int,aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)) = one_one(int) )
=> ( abs_abs(int,M2) = one_one(int) ) ) ).
% abs_zmult_eq_1
tff(fact_1694_set__bit__int__def,axiom,
! [N: nat,K: int] : bit_se5668285175392031749et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,N,one_one(int))) ).
% set_bit_int_def
tff(fact_1695_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_1696_or__not__numerals_I5_J,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(5)
tff(fact_1697_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),abs_abs(A,X)),E)) )
=> ( X = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_1698_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),abs_abs(A,Y)),X) = abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).
% abs_mult_pos
tff(fact_1699_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))) ) )
=> ( 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),abs_abs(A,A3)),abs_abs(A,B2)) ) ) ) ).
% abs_eq_mult
tff(fact_1700_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),abs_abs(A,A3))),zero_zero(A))) ) ).
% abs_minus_le_zero
tff(fact_1701_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),abs_abs(A,minus_minus(A,A3,B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B2)))) ) ).
% abs_triangle_ineq4
tff(fact_1702_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(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),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,minus_minus(A,A3,C2))),abs_abs(A,minus_minus(A,B2,D3))))) ) ).
% abs_diff_triangle_ineq
tff(fact_1703_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)))
=> ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_neg
tff(fact_1704_abs__sgn__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( ( A3 = zero_zero(A) )
=> ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = zero_zero(A) ) )
& ( ( A3 != zero_zero(A) )
=> ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ) ).
% abs_sgn_eq
tff(fact_1705_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_1706_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_1707_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_1708_not__int__def,axiom,
! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = minus_minus(int,aa(int,int,uminus_uminus(int),K),one_one(int)) ).
% not_int_def
tff(fact_1709_and__not__numerals_I1_J,axiom,
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).
% and_not_numerals(1)
tff(fact_1710_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_1711_or__not__numerals_I8_J,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(8)
tff(fact_1712_or__not__numerals_I9_J,axiom,
! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(9)
tff(fact_1713_unset__bit__int__def,axiom,
! [N: nat,K: int] : bit_se2638667681897837118et_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),bit_se4730199178511100633sh_bit(int,N,one_one(int)))) ).
% unset_bit_int_def
tff(fact_1714_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)),abs_abs(A,X)))) ) ).
% abs_add_one_gt_zero
tff(fact_1715_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_1716_and__not__numerals_I4_J,axiom,
! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M2)) ).
% and_not_numerals(4)
tff(fact_1717_and__not__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = one_one(int) ).
% and_not_numerals(2)
tff(fact_1718_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2 != zero_zero(int) )
=> ( pp(dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M2),N),M2))
<=> ( abs_abs(int,N) = one_one(int) ) ) ) ).
% zdvd_mult_cancel1
tff(fact_1719_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_1720_abs__square__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
<=> ( abs_abs(A,X) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_1721_and__not__numerals_I7_J,axiom,
! [M2: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M2)) ).
% and_not_numerals(7)
tff(fact_1722_and__not__numerals_I3_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(N)))) = zero_zero(int) ).
% and_not_numerals(3)
tff(fact_1723_xor__int__unfold,axiom,
! [K: int,L: int] :
( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
& ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K) ) )
& ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( K = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = L ) )
& ( ( K != zero_zero(int) )
=> ( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = K ) )
& ( ( L != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
tff(fact_1724_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),abs_abs(A,X)),one_one(A))) ) ) ).
% abs_square_le_1
tff(fact_1725_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),abs_abs(A,X)),one_one(A))) ) ) ).
% abs_square_less_1
tff(fact_1726_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F2: fun(nat,int),K: int] :
( ! [I3: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3)),aa(nat,int,F2,I3)))),one_one(int))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,M2)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
& ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).
% nat_intermed_int_val
tff(fact_1727_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),minus_minus(int,X,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,minus_minus(int,X,Z2))),one_one(int))),D3))),Z2)) ) ).
% decr_lemma
tff(fact_1728_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,minus_minus(int,X,Z2))),one_one(int))),D3)))) ) ).
% incr_lemma
tff(fact_1729_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_1730_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_1731_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_1732_nat__ivt__aux,axiom,
! [N: nat,F2: fun(nat,int),K: int] :
( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3)),aa(nat,int,F2,I3)))),one_one(int))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
& ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).
% nat_ivt_aux
tff(fact_1733_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_1734_or__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
| ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
& ( ~ ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
| ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( ( ( K = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = L ) )
& ( ( K != zero_zero(int) )
=> ( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = K ) )
& ( ( L != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).
% or_int_unfold
tff(fact_1735_or__minus__numerals_I5_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).
% or_minus_numerals(5)
tff(fact_1736_or__minus__numerals_I1_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).
% or_minus_numerals(1)
tff(fact_1737_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_1738_drop__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( bit_se4197421643247451524op_bit(A,N,A3) = A3 ) )
& ( ( N != zero_zero(nat) )
=> ( bit_se4197421643247451524op_bit(A,N,A3) = bit_se4197421643247451524op_bit(A,minus_minus(nat,N,one_one(nat)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).
% drop_bit_rec
tff(fact_1739_bit__rec,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,A3,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,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))),minus_minus(nat,N,one_one(nat)))) ) ) ) ) ).
% bit_rec
tff(fact_1740_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [J2: nat] : ~ pp(bit_se5641148757651400278ts_bit(A,A3,aa(nat,nat,suc,J2)))
=> ( 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_1741_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_1742_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_1743_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_1744_max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% max.bounded_iff
tff(fact_1745_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_1746_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_1747_length__map,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).
% length_map
tff(fact_1748_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_1749_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_1750_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_1751_drop__bit__minus__one,axiom,
! [N: nat] : bit_se4197421643247451524op_bit(int,N,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).
% drop_bit_minus_one
tff(fact_1752_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A9: list(A),B6: 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)),A9) = aa(list(B),nat,size_size(list(B)),B6) )
=> ( ( 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),A9),B6) )
<=> ( ( A3 = A9 )
& ( B2 = B6 ) ) ) ) ) ).
% zip_eq_zip_same_len
tff(fact_1753_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_1754_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_1755_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_1756_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_1757_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_1758_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_1759_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_1760_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_1761_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_1762_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_1763_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list(A),F2: 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),F2),Xs)),N) = aa(A,B,F2,aa(nat,A,nth(A,Xs),N)) ) ) ).
% nth_map
tff(fact_1764_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_1765_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_1766_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_1767_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_1768_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_1769_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),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)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).
% nth_zip
tff(fact_1770_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_1771_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_1772_max_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).
% max.left_commute
tff(fact_1773_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_1774_max_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).
% max.assoc
tff(fact_1775_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
~ ! [L1: list(A),L2: 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),L2) )
=> ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L2) )
=> ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L2) ) ) ) ).
% pair_list_split
tff(fact_1776_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_1777_max_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI2
tff(fact_1778_max_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI1
tff(fact_1779_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_1780_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_1781_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_1782_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_1783_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_1784_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_1785_max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: 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),C2),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),C2)),A3)) ) ) ) ).
% max.boundedI
tff(fact_1786_max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% max.boundedE
tff(fact_1787_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_1788_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_1789_max_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,D3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),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),C2),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).
% max.mono
tff(fact_1790_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI2
tff(fact_1791_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI1
tff(fact_1792_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_1793_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ) ) ).
% max.strict_boundedE
tff(fact_1794_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_1795_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_1796_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_1797_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_1798_sup__max,axiom,
! [A: $tType] :
( ( semilattice_sup(A)
& linorder(A) )
=> ( sup_sup(A) = ord_max(A) ) ) ).
% sup_max
tff(fact_1799_length__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( ! [Xs2: list(A)] :
( ! [Ys5: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys5)),aa(list(A),nat,size_size(list(A)),Xs2)))
=> pp(aa(list(A),bool,P,Ys5)) )
=> pp(aa(list(A),bool,P,Xs2)) )
=> pp(aa(list(A),bool,P,Xs)) ) ).
% length_induct
tff(fact_1800_finite__maxlen,axiom,
! [A: $tType,M: set(list(A))] :
( pp(aa(set(list(A)),bool,finite_finite(list(A)),M))
=> ? [N3: 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)),N3)) ) ) ).
% finite_maxlen
tff(fact_1801_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: 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),F2),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_1802_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_max(A)) ) ).
% max.abel_semigroup_axioms
tff(fact_1803_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A9: list(A),B6: 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)),A9) = aa(list(B),nat,size_size(list(B)),B6) )
=> ( ( 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),A9),B6) )
=> ( ( A3 = A9 )
& ( B2 = B6 ) ) ) ) ) ).
% zip_inj
tff(fact_1804_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_max(A)) ) ).
% max.semilattice_axioms
tff(fact_1805_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_max(A)) ) ).
% max.semigroup_axioms
tff(fact_1806_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_1807_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_1808_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_1809_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: 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)),Ws) )
=> ( 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,Ys3: list(B),Z3: C,Zs2: list(C),W3: D,Ws2: list(D)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
=> ( ( aa(list(C),nat,size_size(list(C)),Zs2) = 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,Xs2),Ys3),Zs2),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,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),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W3),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,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).
% list_induct4
tff(fact_1810_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,Ys3: list(B),Z3: C,Zs2: list(C)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
=> ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
=> ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs2),Ys3),Zs2))
=> 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),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2))) ) ) )
=> 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_1811_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,Ys3: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
=> 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),Ys3))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ).
% list_induct2
tff(fact_1812_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,Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).
% length_Suc_conv
tff(fact_1813_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,Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).
% Suc_length_conv
tff(fact_1814_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_1815_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_1816_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_1817_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) )
& ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1818_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
=> ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),X_12)) )
<=> ? [Xs4: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs4) = K )
& ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
=> pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),aa(nat,A,nth(A,Xs4),I4))) ) ) ) ).
% Skolem_list_nth
tff(fact_1819_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_1820_nths__all,axiom,
! [A: $tType,Xs: list(A),I: 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,I)) )
=> ( aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),I) = Xs ) ) ).
% nths_all
tff(fact_1821_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_1822_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_1823_length__code,axiom,
! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).
% length_code
tff(fact_1824_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_1825_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( pp(bit_se5641148757651400278ts_bit(int,minus_minus(int,aa(int,int,uminus_uminus(int),K),one_one(int)),N))
<=> ~ pp(bit_se5641148757651400278ts_bit(int,K,N)) ) ).
% bit_not_int_iff'
tff(fact_1826_length__compl__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [E: A,L3: list(A)] :
( ! [Ll: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
=> pp(aa(list(A),bool,P,Ll)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),L3))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_induct
tff(fact_1827_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
<=> ? [X4: A,Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).
% Suc_le_length_iff
tff(fact_1828_list__decomp__1,axiom,
! [A: $tType,L: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
=> ? [A6: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),nil(A)) ) ).
% list_decomp_1
tff(fact_1829_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))
<=> ? [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N4))) ) ) ).
% list_ex_length
tff(fact_1830_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_1831_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_1832_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_1833_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( pp(bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K),N))
<=> pp(bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),minus_minus(int,K,one_one(int))),N)) ) ).
% bit_minus_int_iff
tff(fact_1834_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_1835_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),M3))
=> ( pp(bit_se5641148757651400278ts_bit(int,K,M3))
<=> pp(bit_se5641148757651400278ts_bit(int,K,N3)) ) )
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( pp(bit_se5641148757651400278ts_bit(int,K,minus_minus(nat,N3,one_one(nat))))
<=> ~ pp(bit_se5641148757651400278ts_bit(int,K,N3)) ) ) ) ).
% int_bit_bound
tff(fact_1836_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)) )
=> ? [A6: A,B5: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),nil(A))) ) ).
% list_decomp_2
tff(fact_1837_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_1838_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_1839_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_1840_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_1841_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_1842_in__set__enumerate__eq,axiom,
! [A: $tType,P7: product_prod(nat,A),N: nat,Xs: list(A)] :
( pp(member2(product_prod(nat,A),P7,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),P7)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P7)),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),P7),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P7) ) ) ) ).
% in_set_enumerate_eq
tff(fact_1843_sorted__in__between,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I2: nat,J: nat,L: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),I2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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),I2)),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),I2),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_1844_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R2: 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,R2)))
<=> ( 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),R2)) )
| ( ( 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,R2))) ) ) ) ).
% Cons_lenlex_iff
tff(fact_1845_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: 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,R2)))
<=> ( ( 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),R2))
& ( 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,R2))) ) ) ) ).
% Cons_in_lex
tff(fact_1846_nat__numeral__diff__1,axiom,
! [V2: num] : minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2),one_one(nat)) = nat2(minus_minus(int,aa(num,int,numeral_numeral(int),V2),one_one(int))) ).
% nat_numeral_diff_1
tff(fact_1847_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_1848_List_Ofinite__set,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(set(A),bool,finite_finite(A),aa(list(A),set(A),set2(A),Xs))) ).
% List.finite_set
tff(fact_1849_map__eq__conv,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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,F2,X4) = aa(B,A,G,X4) ) ) ) ).
% map_eq_conv
tff(fact_1850_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_1851_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_1852_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_1853_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_1854_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)),insert3(A),X21),aa(list(A),set(A),set2(A),X22)) ).
% list.simps(15)
tff(fact_1855_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_1856_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list(A),R2: 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,R2)))
<=> ( Ns != nil(A) ) ) ).
% Nil_lenlex_iff1
tff(fact_1857_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_1858_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)),insert3(A),X),aa(list(A),set(A),set2(A),Xs)) ).
% List.set_insert
tff(fact_1859_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_1860_nat__1,axiom,
nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).
% nat_1
tff(fact_1861_length__upto,axiom,
! [I2: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I2,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),minus_minus(int,J,I2)),one_one(int))) ).
% length_upto
tff(fact_1862_one__less__nat__eq,axiom,
! [Z2: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),nat2(Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ).
% one_less_nat_eq
tff(fact_1863_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_1864_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_1865_infinite__UNIV__listI,axiom,
! [A: $tType] : ~ pp(aa(set(list(A)),bool,finite_finite(list(A)),top_top(set(list(A))))) ).
% infinite_UNIV_listI
tff(fact_1866_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_1867_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_1868_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_1869_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_1870_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_1871_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_1872_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_1873_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_1874_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_1875_list_Oset__cases,axiom,
! [A: $tType,E2: A,A3: list(A)] :
( pp(member2(A,E2,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),E2),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,E2,aa(list(A),set(A),set2(A),Z22))) ) ) ) ).
% list.set_cases
tff(fact_1876_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_1877_finite__list,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A4 ) ).
% finite_list
tff(fact_1878_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_1879_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
( ( X = Ya )
=> ( ! [Z3: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),Ya)))
=> ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),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_1880_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list(A),F2: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),X)))
=> ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),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_1881_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list(A),Xa: list(A),F2: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),X)))
=> ( pp(member2(A,Za,aa(list(A),set(A),set2(A),Xa)))
=> ( ( aa(A,B,F2,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),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_1882_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs) ) ) ).
% map_ext
tff(fact_1883_map__idI,axiom,
! [A: $tType,Xs: list(A),F2: fun(A,A)] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,A,F2,X2) = X2 ) )
=> ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F2),Xs) = Xs ) ) ).
% map_idI
tff(fact_1884_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: 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,F2,X2) = aa(A,B,G,X2) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys) ) ) ) ).
% map_cong
tff(fact_1885_ex__map__conv,axiom,
! [B: $tType,A: $tType,Ys: list(B),F2: fun(A,B)] :
( ? [Xs4: list(A)] : Ys = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs4)
<=> ! [X4: B] :
( pp(member2(B,X4,aa(list(B),set(B),set2(B),Ys)))
=> ? [Xa2: A] : X4 = aa(A,B,F2,Xa2) ) ) ).
% ex_map_conv
tff(fact_1886_in__set__nthsD,axiom,
! [A: $tType,X: A,Xs: list(A),I: 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),I))))
=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_nthsD
tff(fact_1887_notin__set__nthsI,axiom,
! [A: $tType,X: A,Xs: list(A),I: 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),I)))) ) ).
% notin_set_nthsI
tff(fact_1888_list__ex__cong,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),F2: 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,F2,X2))
<=> pp(aa(A,bool,G,X2)) ) )
=> ( pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),F2),Xs))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_ex(A),G),Ys)) ) ) ) ).
% list_ex_cong
tff(fact_1889_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_1890_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_1891_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_1892_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: 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),F2,X2) = aa(A,list(B),G,X2) ) )
=> ( bind(A,B,Xs,F2) = bind(A,B,Ys,G) ) ) ) ).
% list_bind_cong
tff(fact_1893_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_1894_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_1895_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_1896_sorted0,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).
% sorted0
tff(fact_1897_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less(A),nil(A)) ) ).
% strict_sorted_simps(1)
tff(fact_1898_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_1899_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_1900_sorted__wrt__less__idx,axiom,
! [Ns: list(nat),I2: nat] :
( sorted_wrt(nat,ord_less(nat),Ns)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2))) ) ) ).
% sorted_wrt_less_idx
tff(fact_1901_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_1902_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_1903_sorted__nths,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I: 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),I)) ) ) ).
% sorted_nths
tff(fact_1904_sorted__upto,axiom,
! [M2: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M2,N)) ).
% sorted_upto
tff(fact_1905_sorted__wrt__upto,axiom,
! [I2: int,J: int] : sorted_wrt(int,ord_less(int),upto(I2,J)) ).
% sorted_wrt_upto
tff(fact_1906_nat__one__as__int,axiom,
one_one(nat) = nat2(one_one(int)) ).
% nat_one_as_int
tff(fact_1907_empty__set,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).
% empty_set
tff(fact_1908_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_1909_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_1910_set__nths__subset,axiom,
! [A: $tType,Xs: list(A),I: 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),I))),aa(list(A),set(A),set2(A),Xs))) ).
% set_nths_subset
tff(fact_1911_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_1912_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_1913_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_1914_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_1915_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_1916_sorted__wrt__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I2: nat,J: nat] :
( sorted_wrt(A,P,Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).
% sorted_wrt_nth_less
tff(fact_1917_sorted__wrt__iff__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( sorted_wrt(A,P,Xs)
<=> ! [I4: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% sorted_wrt_iff_nth_less
tff(fact_1918_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_1919_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs12: 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)),Xs12),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)),Xs12),Xs23) ).
% mergesort_by_rel_split.simps(1)
tff(fact_1920_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_1921_Nil__notin__lex,axiom,
! [A: $tType,Ys: list(A),R2: 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,R2))) ).
% Nil_notin_lex
tff(fact_1922_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list(A),R2: 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,R2))) ).
% Nil2_notin_lex
tff(fact_1923_lenlex__irreflexive,axiom,
! [A: $tType,R2: 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),R2))
=> ~ 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,R2))) ) ).
% lenlex_irreflexive
tff(fact_1924_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_1925_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list(A),R2: 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,R2))) ).
% Nil_lenlex_iff2
tff(fact_1926_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_1927_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_1928_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1929_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_1930_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)) )
<=> ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) ) ) ).
% all_set_conv_all_nth
tff(fact_1931_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_1932_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I4: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).
% sorted_iff_nth_mono_less
tff(fact_1933_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_1934_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs12: 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)),Xs12),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),Xs12)),Xs23) ).
% mergesort_by_rel_split.simps(2)
tff(fact_1935_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 )
=> ( ! [Xs1: 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)),Xs1),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)),Xs1),Xs22) ) ) )
=> ( ! [Xs1: 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)),Xs1),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),Xs1)),Xs22) ) ) )
=> ~ ! [Xs1: 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)),Xs1),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),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs2) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
tff(fact_1936_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),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),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4)))) ) ) ) ).
% sorted_iff_nth_Suc
tff(fact_1937_lenlex__length,axiom,
! [A: $tType,Ms: list(A),Ns: list(A),R2: 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,R2)))
=> 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_1938_sorted__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I2: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),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),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).
% sorted_nth_mono
tff(fact_1939_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I4: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).
% sorted_iff_nth_mono
tff(fact_1940_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,nat,suc,nat2(Z2)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).
% Suc_nat_eq_nat_zadd1
tff(fact_1941_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_1942_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_1943_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_1944_in__set__zip,axiom,
! [A: $tType,B: $tType,P7: product_prod(A,B),Xs: list(A),Ys: list(B)] :
( pp(member2(product_prod(A,B),P7,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))))
<=> ? [N4: nat] :
( ( aa(nat,A,nth(A,Xs),N4) = aa(product_prod(A,B),A,product_fst(A,B),P7) )
& ( aa(nat,B,nth(B,Ys),N4) = aa(product_prod(A,B),B,product_snd(A,B),P7) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).
% in_set_zip
tff(fact_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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)),insert3(A),X),bot_bot(set(A)))) ).
% set_removeAll
tff(fact_1952_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))))
=> ( 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))))
=> ( mergesort_by_rel(A,R,Xs) = merges9089515139780605204_merge(A,R,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))),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_1953_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] : mergesort_by_rel(A,R,nil(A)) = nil(A) ).
% mergesort_by_rel_simps(1)
tff(fact_1954_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_1955_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_1956_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X: 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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),V2: A,Va2: list(A)] : merges9089515139780605204_merge(A,R,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ).
% mergesort_by_rel_merge.simps(3)
tff(fact_1965_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,Ys3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ~ ( ( 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),Ys3))) ) )
& ( ~ 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),Ys3)) ) ) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
tff(fact_1966_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_1967_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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_1968_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_1969_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_1970_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))))
=> ( 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))))
=> ( mergesort_by_rel(A,R,Xs) = merges9089515139780605204_merge(A,R,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))),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_1971_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: 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,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))),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_1972_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_1973_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: 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,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))),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_1974_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_1975_upto__rec2,axiom,
! [I2: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( upto(I2,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,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_1976_division__decomp,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
=> ? [B7: A,C6: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),C6) )
& pp(dvd_dvd(A,B7,B2))
& pp(dvd_dvd(A,C6,C2)) ) ) ) ).
% division_decomp
tff(fact_1977_dvd__productE,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [P7: A,A3: A,B2: A] :
( pp(dvd_dvd(A,P7,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> ~ ! [X2: A,Y2: A] :
( ( P7 = 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_1978_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V2: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).
% neg_numeral_le_ceiling
tff(fact_1979_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V2: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).
% ceiling_less_neg_numeral
tff(fact_1980_normalize__denom__zero,axiom,
! [P7: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P7),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
% normalize_denom_zero
tff(fact_1981_append_Oassoc,axiom,
! [A: $tType,A3: list(A),B2: list(A),C2: 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)),C2) = 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),C2)) ).
% append.assoc
tff(fact_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_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_1993_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_1994_append__eq__append__conv,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
| ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
<=> ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
tff(fact_1995_map__append,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),Xs)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys)) ).
% map_append
tff(fact_1996_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
=> ( upt(I2,J) = nil(nat) ) ) ).
% upt_conv_Nil
tff(fact_1997_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_1998_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_1999_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_2000_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list(A),E12: A,E23: A,L22: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L22))) )
<=> ( ( L12 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L22 = nil(A) ) ) ) ).
% list_ee_eq_leel(1)
tff(fact_2001_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list(A),E12: A,E23: A,L22: list(A),E1: A,E22: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L22))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
<=> ( ( L12 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L22 = nil(A) ) ) ) ).
% list_ee_eq_leel(2)
tff(fact_2002_list__se__match_I1_J,axiom,
! [A: $tType,L12: list(A),L22: list(A),A3: A] :
( ( L12 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L22) = 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)) )
& ( L22 = nil(A) ) ) ) ) ).
% list_se_match(1)
tff(fact_2003_list__se__match_I2_J,axiom,
! [A: $tType,L22: list(A),L12: list(A),A3: A] :
( ( L22 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L22) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( L22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(2)
tff(fact_2004_list__se__match_I3_J,axiom,
! [A: $tType,L12: list(A),A3: A,L22: 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),L22) )
<=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
& ( L22 = nil(A) ) ) ) ) ).
% list_se_match(3)
tff(fact_2005_list__se__match_I4_J,axiom,
! [A: $tType,L22: list(A),A3: A,L12: list(A)] :
( ( L22 != 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),L22) )
<=> ( ( L12 = nil(A) )
& ( L22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(4)
tff(fact_2006_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E2: A,L12: list(A),E3: A,L22: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),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),E3),L22)) )
<=> ( ( L12 = nil(A) )
& ( E3 = E2 )
& ( L22 = nil(A) ) ) ) ).
% list_e_eq_lel(1)
tff(fact_2007_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list(A),E3: A,L22: list(A),E2: 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),E3),L22)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) )
<=> ( ( L12 = nil(A) )
& ( E3 = E2 )
& ( L22 = nil(A) ) ) ) ).
% list_e_eq_lel(2)
tff(fact_2008_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_2009_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( upt(I2,J) = nil(nat) )
<=> ( ( J = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2)) ) ) ).
% upt_eq_Nil_conv
tff(fact_2010_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
=> ( 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),Us),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),Us)),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_2011_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_2012_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list(B),F2: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),F2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,X)),bind(B,A,Xs,F2)) ).
% bind_simps(2)
tff(fact_2013_length__upt,axiom,
! [I2: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I2,J)) = minus_minus(nat,J,I2) ).
% length_upt
tff(fact_2014_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_2015_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_2016_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_2017_nth__upt,axiom,
! [I2: 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),I2),K)),J))
=> ( aa(nat,nat,nth(nat,upt(I2,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) ) ) ).
% nth_upt
tff(fact_2018_ceiling__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).
% ceiling_less_one
tff(fact_2019_one__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).
% one_le_ceiling
tff(fact_2020_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_2021_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_2022_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_2023_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_2024_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_2025_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_2026_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V2: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),minus_minus(A,aa(num,A,numeral_numeral(A),V2),one_one(A)))) ) ) ).
% ceiling_less_numeral
tff(fact_2027_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V2: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(num,A,numeral_numeral(A),V2),one_one(A))),X)) ) ) ).
% numeral_le_ceiling
tff(fact_2028_append__eq__appendI,axiom,
! [A: $tType,Xs: list(A),Xs12: list(A),Zs: list(A),Ys: list(A),Us: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs12) = Zs )
=> ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs12),Us) )
=> ( 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),Us) ) ) ) ).
% append_eq_appendI
tff(fact_2029_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) )
<=> ? [Us2: list(A)] :
( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us2) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts ) )
| ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = Zs )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts) ) ) ) ) ).
% append_eq_append_conv2
tff(fact_2030_append_Osemigroup__axioms,axiom,
! [A: $tType] : semigroup(list(A),append(A)) ).
% append.semigroup_axioms
tff(fact_2031_upt__0,axiom,
! [I2: nat] : upt(I2,zero_zero(nat)) = nil(nat) ).
% upt_0
tff(fact_2032_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_2033_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list(nat),Q3: 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,Q3) )
<=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M2),Q3) ) ) ).
% upt_conv_Cons_Cons
tff(fact_2034_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_2035_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs12: 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),Xs12) = Ys )
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs12),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_2036_append_Omonoid__axioms,axiom,
! [A: $tType] : monoid(list(A),append(A),nil(A)) ).
% append.monoid_axioms
tff(fact_2037_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_2038_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_2039_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_2040_List_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F2: 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),F2),Xs) )
<=> ? [Us2: list(B),Vs2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
& ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
& ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs2) ) ) ) ).
% List.append_eq_map_conv
tff(fact_2041_List_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
<=> ? [Us2: list(B),Vs2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
& ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
& ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs2) ) ) ) ).
% List.map_eq_append_conv
tff(fact_2042_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_2043_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
=> ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) ) ).
% upt_conv_Cons
tff(fact_2044_upt__rec,axiom,
! [I2: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
=> ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
=> ( upt(I2,J) = nil(nat) ) ) ) ).
% upt_rec
tff(fact_2045_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_2046_rev__exhaust,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ~ ! [Ys3: list(A),Y2: 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),Y2),nil(A))) ) ).
% rev_exhaust
tff(fact_2047_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 ) )
| ? [Ys6: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys6) = Ys )
& ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) ) ) ) ) ).
% Cons_eq_append_conv
tff(fact_2048_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) ) )
| ? [Ys6: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys6) )
& ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) = Xs ) ) ) ) ).
% append_eq_Cons_conv
tff(fact_2049_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_2050_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) ) )
=> ~ ! [Ys2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) != Xs ) ) ) ) ).
% list_append_eq_Cons_cases
tff(fact_2051_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) ) )
=> ~ ! [Ys2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) != Xs ) ) ) ) ).
% list_Cons_eq_append_cases
tff(fact_2052_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,Ys3: list(B)] :
( ( Ys3 != 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),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B))))) )
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
=> ( ( Xs2 != nil(A) )
=> ( ( Ys3 != 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),Ys3),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_2053_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
<=> ? [Xs4: list(A),X4: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))) ) ).
% neq_Nil_rev_conv
tff(fact_2054_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,Ys3: 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),Ys3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),nil(B)))))
=> ( ! [X2: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
=> 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),Ys3),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_2055_neq__Nil__revE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [Ll2: list(A),E: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),nil(A))) ) ).
% neq_Nil_revE
tff(fact_2056_split__list,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)) ) ).
% split_list
tff(fact_2057_split__list__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))) ) ) ).
% split_list_last
tff(fact_2058_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)) )
=> ? [Ys3: list(A),X2: 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),X2),Zs2))
& pp(aa(A,bool,P,X2)) ) ) ).
% split_list_prop
tff(fact_2059_split__list__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))) ) ) ).
% split_list_first
tff(fact_2060_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)) )
=> ~ ! [Ys3: list(A),X2: 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),X2),Zs2))
=> ~ pp(aa(A,bool,P,X2)) ) ) ).
% split_list_propE
tff(fact_2061_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),Xs5: 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),Xs5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys7)) )
<=> ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
tff(fact_2062_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
<=> ? [Ys4: list(A),Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) ) ).
% in_set_conv_decomp
tff(fact_2063_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)) )
=> ? [Ys3: list(A),X2: 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),X2),Zs2)) )
& pp(aa(A,bool,P,X2))
& ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Zs2)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).
% split_list_last_prop
tff(fact_2064_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)) )
=> ? [Ys3: list(A),X2: 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),X2),Zs2))
& pp(aa(A,bool,P,X2))
& ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys3)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).
% split_list_first_prop
tff(fact_2065_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)) )
=> ~ ! [Ys3: list(A),X2: 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),X2),Zs2)) )
=> ( pp(aa(A,bool,P,X2))
=> ~ ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Zs2)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ) ).
% split_list_last_propE
tff(fact_2066_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)) )
=> ~ ! [Ys3: list(A),X2: 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),X2),Zs2))
=> ( pp(aa(A,bool,P,X2))
=> ~ ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys3)))
=> ~ pp(aa(A,bool,P,Xa3)) ) ) ) ) ).
% split_list_first_propE
tff(fact_2067_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)))
<=> ? [Ys4: list(A),Zs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Zs3))) ) ) ).
% in_set_conv_decomp_last
tff(fact_2068_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)))
<=> ? [Ys4: list(A),Zs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
& ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Ys4))) ) ) ).
% in_set_conv_decomp_first
tff(fact_2069_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)) )
<=> ? [Ys4: list(A),X4: A,Zs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3)) )
& pp(aa(A,bool,P,X4))
& ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Zs3)))
=> ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).
% split_list_last_prop_iff
tff(fact_2070_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)) )
<=> ? [Ys4: list(A),X4: A] :
( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3))
& pp(aa(A,bool,P,X4))
& ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys4)))
=> ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).
% split_list_first_prop_iff
tff(fact_2071_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_2072_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M2,N)) ).
% sorted_wrt_upt
tff(fact_2073_sorted__upt,axiom,
! [M2: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M2,N)) ).
% sorted_upt
tff(fact_2074_lex__append__leftI,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),R2: 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,R2)))
=> 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,R2))) ) ).
% lex_append_leftI
tff(fact_2075_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,list(A)),X: B,Xs: list(B)] : maps(B,A,F2,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),F2,X)),maps(B,A,F2,Xs)) ).
% maps_simps(1)
tff(fact_2076_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs: list(nat)] :
( ( upt(I2,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),I2),J))
& ( I2 = X )
& ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_2077_upto__aux__def,axiom,
! [I2: int,J: int,Js: list(int)] : upto_aux(I2,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),Js) ).
% upto_aux_def
tff(fact_2078_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,Xs3: list(A),Y2: A,Ys2: 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))),Xs3)) )
& ( 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))),Ys2)) ) ) ) ) ).
% same_length_different
tff(fact_2079_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_2080_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_2081_lex__append__leftD,axiom,
! [A: $tType,R2: 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),R2))
=> ( 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,R2)))
=> 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,R2))) ) ) ).
% lex_append_leftD
tff(fact_2082_lex__append__left__iff,axiom,
! [A: $tType,R2: 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),R2))
=> ( 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,R2)))
<=> 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,R2))) ) ) ).
% lex_append_left_iff
tff(fact_2083_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_2084_lex__append__rightI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: 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,R2)))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
=> 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),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),lex(A,R2))) ) ) ).
% lex_append_rightI
tff(fact_2085_lenlex__append1,axiom,
! [A: $tType,Us: 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)),Us),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),Us),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_2086_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_2087_nth__map__upt,axiom,
! [A: $tType,I2: nat,N: nat,M2: nat,F2: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),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),F2),upt(M2,N))),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I2)) ) ) ).
% nth_map_upt
tff(fact_2088_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Ys4: list(A),Y3: A] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).
% length_Suc_rev_conv
tff(fact_2089_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,Ys4: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).
% length_Suc_conv_rev
tff(fact_2090_length__compl__rev__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [L3: list(A),E: A] :
( ! [Ll: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
=> pp(aa(list(A),bool,P,Ll)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),nil(A))))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_rev_induct
tff(fact_2091_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_2092_upto__split2,axiom,
! [I2: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).
% upto_split2
tff(fact_2093_upto__split1,axiom,
! [I2: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,minus_minus(int,J,one_one(int)))),upto(J,K)) ) ) ) ).
% upto_split1
tff(fact_2094_map__upt__eqI,axiom,
! [A: $tType,Xs: list(A),N: nat,M2: nat,F2: 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,F2,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),F2),upt(M2,N)) = Xs ) ) ) ).
% map_upt_eqI
tff(fact_2095_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_2096_upto__split3,axiom,
! [I2: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,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_2097_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_2098_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V2: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).
% floor_le_neg_numeral
tff(fact_2099_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V2: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).
% neg_numeral_less_floor
tff(fact_2100_bezw__0,axiom,
! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).
% bezw_0
tff(fact_2101_floor__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).
% floor_le_one
tff(fact_2102_one__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) ) ) ).
% one_less_floor
tff(fact_2103_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V2: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).
% floor_le_numeral
tff(fact_2104_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_2105_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_2106_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_2107_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_2108_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_2109_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_2110_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V2: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).
% numeral_less_floor
tff(fact_2111_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
=> ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).
% upt_Suc_append
tff(fact_2112_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
=> ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,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),I2),J))
=> ( upt(I2,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).
% upt_Suc
tff(fact_2113_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
=> ( upt(I2,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(I2,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).
% upt_add_eq_append
tff(fact_2114_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_2115_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_2116_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),minus_minus(int,archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).
% ceiling_diff_floor_le_1
tff(fact_2117_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_2118_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( ( ( Xa = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Xa != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa))))) ) ) ) ) ).
% bezw.elims
tff(fact_2119_bezw_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero(nat) )
=> ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Y != zero_zero(nat) )
=> ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ) ).
% bezw.simps
tff(fact_2120_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_2121_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_2122_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_2123_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_2124_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P7: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> 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,P7,Q3))),one_one(A))),Q3)),P7)) ) ) ).
% ceiling_divide_lower
tff(fact_2125_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_2126_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_2127_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_2128_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_2129_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_2130_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_2131_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_2132_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_2133_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_2134_nth__upto,axiom,
! [I2: 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),I2),aa(nat,int,semiring_1_of_nat(int),K))),J))
=> ( aa(nat,int,nth(int,upto(I2,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).
% nth_upto
tff(fact_2135_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_2136_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_2137_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_2138_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_2139_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_2140_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_2141_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_2142_int__ops_I2_J,axiom,
aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
% int_ops(2)
tff(fact_2143_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_2144_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_2145_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))
=> ? [N3: 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),N3)),X))) ) ) ).
% ex_less_of_nat_mult
tff(fact_2146_int__ops_I4_J,axiom,
! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).
% int_ops(4)
tff(fact_2147_int__Suc,axiom,
! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).
% int_Suc
tff(fact_2148_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_2149_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_2150_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_2151_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),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_2152_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),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_2153_floor__exists,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [Z3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int))))) ) ) ).
% floor_exists
tff(fact_2154_floor__exists1,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [X2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X2)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int)))))
& ! [Y4: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int))))) )
=> ( Y4 = X2 ) ) ) ) ).
% floor_exists1
tff(fact_2155_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_2156_ceiling__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
=> ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
& ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
=> ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).
% ceiling_altdef
tff(fact_2157_Suc__as__int,axiom,
! [X3: nat] : aa(nat,nat,suc,X3) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).
% Suc_as_int
tff(fact_2158_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),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_2159_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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_2160_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T3: A] :
( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T3)))
<=> ! [I4: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A)))) )
=> pp(aa(int,bool,P,I4)) ) ) ) ).
% floor_split
tff(fact_2161_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_2162_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_2163_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
=> ~ ! [N3: 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,N3)))),E2)) ) ) ).
% nat_approx_posE
tff(fact_2164_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T3: A] :
( pp(aa(int,bool,P,archimedean_ceiling(A,T3)))
<=> ! [I4: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),minus_minus(A,aa(int,A,ring_1_of_int(A),I4),one_one(A))),T3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T3),aa(int,A,ring_1_of_int(A),I4))) )
=> pp(aa(int,bool,P,I4)) ) ) ) ).
% ceiling_split
tff(fact_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_floor__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).
% floor_correct
tff(fact_2174_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_2175_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_2176_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_2177_floor__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P7: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> 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,P7,Q3)))),Q3)),P7)) ) ) ).
% floor_divide_lower
tff(fact_2178_ceiling__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P7: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P7),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,P7,Q3)))),Q3))) ) ) ).
% ceiling_divide_upper
tff(fact_2179_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P7: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P7),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,P7,Q3)))),one_one(A))),Q3))) ) ) ).
% floor_divide_upper
tff(fact_2180_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_2181_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
=> ~ ( ( ( ( Xa = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Xa != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa))))) ) ) )
=> ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).
% bezw.pelims
tff(fact_2182_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_2183_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_2184_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_2185_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_2186_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_2187_pochhammer__1,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).
% pochhammer_1
tff(fact_2188_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_2189_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_2190_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(member2(A,one_one(A),ring_1_Ints(A))) ) ).
% Ints_1
tff(fact_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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)),abs_abs(A,X))) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_2199_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),abs_abs(A,X)),one_one(A)))
=> ( X = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_2200_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),abs_abs(A,minus_minus(A,X,Y))),one_one(A))) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_2201_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_2202_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_2203_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),minus_minus(A,R2,aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).
% pochhammer_absorb_comp
tff(fact_2204_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_2205_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_2206_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_2207_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_2208_nth__step__trancl,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),N: nat,M2: nat] :
( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),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,N3))),aa(nat,A,nth(A,Xs),N3)),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_2209_take__Suc__conv__app__nth,axiom,
! [A: $tType,I2: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,I2)),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),I2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).
% take_Suc_conv_app_nth
tff(fact_2210_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_2211_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(A),lattic643756798350308766er_Min(A,A4)),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set_nonempty
tff(fact_2212_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_2213_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_2214_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_2215_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_2216_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_2217_min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
<=> ( 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),C2)) ) ) ) ).
% min.bounded_iff
tff(fact_2218_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_2219_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_2220_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_2221_gbinomial__1,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).
% gbinomial_1
tff(fact_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_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_2234_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_2235_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_2236_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_2237_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_2238_take__upt,axiom,
! [I2: 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),I2),M2)),N))
=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),take(nat),M2),upt(I2,N)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M2)) ) ) ).
% take_upt
tff(fact_2239_nth__take,axiom,
! [A: $tType,I2: nat,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),I2) = aa(nat,A,nth(A,Xs),I2) ) ) ).
% nth_take
tff(fact_2240_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_2241_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_2242_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_2243_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_2244_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_2245_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_2246_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_2247_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_2248_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_2249_take__Cons__numeral,axiom,
! [A: $tType,V2: 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),V2)),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),V2),one_one(nat))),Xs)) ).
% take_Cons_numeral
tff(fact_2250_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_2251_inf__nat__def,axiom,
inf_inf(nat) = ord_min(nat) ).
% inf_nat_def
tff(fact_2252_min_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).
% min.assoc
tff(fact_2253_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_2254_min_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: 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),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).
% min.left_commute
tff(fact_2255_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_2256_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),bool)] :
( ! [Zs2: list(A),Ws2: list(B),N3: nat] :
( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws2) )
=> ( ( N3 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
=> ( ( Zs2 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N3),Xs) )
=> ( ( Ws2 = aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N3),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),Zs2),Ws2))) ) ) ) )
=> 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_2257_min_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),C2),D3))) ) ) ) ).
% min.mono
tff(fact_2258_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_2259_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_2260_min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% min.boundedE
tff(fact_2261_min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2))
=> 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),C2))) ) ) ) ).
% min.boundedI
tff(fact_2262_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_2263_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_2264_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_2265_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_2266_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_2267_min_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).
% min.coboundedI1
tff(fact_2268_min_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
=> 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)),C2)) ) ) ).
% min.coboundedI2
tff(fact_2269_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_2270_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_2271_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% min.strict_boundedE
tff(fact_2272_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_2273_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
=> 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)),C2)) ) ) ).
% min.strict_coboundedI1
tff(fact_2274_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
=> 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)),C2)) ) ) ).
% min.strict_coboundedI2
tff(fact_2275_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_2276_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_2277_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_2278_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_2279_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_2280_inf__min,axiom,
! [A: $tType] :
( ( semilattice_inf(A)
& linorder(A) )
=> ( inf_inf(A) = ord_min(A) ) ) ).
% inf_min
tff(fact_2281_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Xs)) ).
% take_map
tff(fact_2282_sorted__wrt__take,axiom,
! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
( sorted_wrt(A,F2,Xs)
=> sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ) ).
% sorted_wrt_take
tff(fact_2283_max__min__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ).
% max_min_distrib1
tff(fact_2284_max__min__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).
% max_min_distrib2
tff(fact_2285_min__max__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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),C2)),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),C2),A3)) ) ).
% min_max_distrib1
tff(fact_2286_min__max__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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),C2)) = 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),C2)) ) ).
% min_max_distrib2
tff(fact_2287_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_min(A)) ) ).
% min.abel_semigroup_axioms
tff(fact_2288_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_min(A)) ) ).
% min.semilattice_axioms
tff(fact_2289_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_min(A)) ) ).
% min.semigroup_axioms
tff(fact_2290_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_2291_rotate__map,axiom,
! [A: $tType,B: $tType,N: nat,F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),rotate(B),N),Xs)) ).
% rotate_map
tff(fact_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_rotate__append,axiom,
! [A: $tType,L: list(A),Q3: 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),Q3)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q3),L) ).
% rotate_append
tff(fact_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_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_2314_dvd__fact,axiom,
! [M2: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> pp(dvd_dvd(nat,M2,semiring_char_0_fact(nat,N))) ) ) ).
% dvd_fact
tff(fact_2315_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_2316_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_2317_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_2318_min__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P7: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P7) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P7)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P7)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P7) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P7)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P7)) ) ) ) ) ).
% min_mult_distrib_right
tff(fact_2319_max__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P7: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P7) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P7)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P7)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P7) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P7)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P7)) ) ) ) ) ).
% max_mult_distrib_right
tff(fact_2320_min__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P7: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P7),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),P7),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P7),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P7),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),P7),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P7),Y)) ) ) ) ) ).
% min_mult_distrib_left
tff(fact_2321_max__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P7: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P7),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),P7),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P7),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P7))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P7),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),P7),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P7),Y)) ) ) ) ) ).
% max_mult_distrib_left
tff(fact_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> ~ ! [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)),R2)) ) ) ) ) ).
% lex_take_index
tff(fact_2332_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_2333_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_2334_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_2335_fact__code,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N,one_one(nat))) ) ).
% fact_code
tff(fact_2336_lexord__take__index__conv,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2)))
<=> ( ( 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 ) )
| ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
& ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I4),X) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I4),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),I4)),aa(nat,A,nth(A,Y),I4)),R2)) ) ) ) ).
% lexord_take_index_conv
tff(fact_2337_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_2338_complete__linorder__inf__min,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ( inf_inf(A) = ord_min(A) ) ) ).
% complete_linorder_inf_min
tff(fact_2339_choose__two,axiom,
! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),minus_minus(nat,N,one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% choose_two
tff(fact_2340_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_2341_binomial__n__n,axiom,
! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).
% binomial_n_n
tff(fact_2342_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_2343_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_2344_binomial__n__0,axiom,
! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).
% binomial_n_0
tff(fact_2345_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_2346_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_2347_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_2348_last__upt,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
=> ( last(nat,upt(I2,J)) = minus_minus(nat,J,one_one(nat)) ) ) ).
% last_upt
tff(fact_2349_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R2: 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,R2)))
<=> ( 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),R2))
| ( ( 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,R2))) ) ) ) ).
% lexord_cons_cons
tff(fact_2350_lexord__Nil__left,axiom,
! [A: $tType,Y: list(A),R2: 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,R2)))
<=> ? [A5: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),X4) ) ).
% lexord_Nil_left
tff(fact_2351_choose__one,axiom,
! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).
% choose_one
tff(fact_2352_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_2353_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_2354_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_2355_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_2356_last__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: 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),F2),Xs)) = aa(A,B,F2,last(A,Xs)) ) ) ).
% last_map
tff(fact_2357_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_2358_longest__common__suffix,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
? [Ss: list(A),Xs3: list(A),Ys2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),Ss) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Ss) )
& ( ( Xs3 = nil(A) )
| ( Ys2 = nil(A) )
| ( last(A,Xs3) != last(A,Ys2) ) ) ) ).
% longest_common_suffix
tff(fact_2359_lexord__irreflexive,axiom,
! [A: $tType,R2: 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),R2))
=> ~ 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,R2))) ) ).
% lexord_irreflexive
tff(fact_2360_lexord__linear,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: list(A),Y: list(A)] :
( ! [A6: A,B5: 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),A6),B5),R2))
| ( A6 = B5 )
| pp(member2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),A6),R2)) )
=> ( 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,R2)))
| ( 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,R2))) ) ) ).
% lexord_linear
tff(fact_2361_lexord__Nil__right,axiom,
! [A: $tType,X: list(A),R2: 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,R2))) ).
% lexord_Nil_right
tff(fact_2362_lexord__append__leftI,axiom,
! [A: $tType,U: list(A),V2: list(A),R2: 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),V2),lexord(A,R2)))
=> 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),V2)),lexord(A,R2))) ) ).
% lexord_append_leftI
tff(fact_2363_binomial__absorb__comp,axiom,
! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),minus_minus(nat,N,K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),K)) ).
% binomial_absorb_comp
tff(fact_2364_lexord__partial__trans,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
( ! [X2: A,Y2: A,Z3: 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),R2))
=> ( 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),Z3),R2))
=> 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),Z3),R2)) ) ) )
=> ( 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,R2)))
=> ( 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,R2)))
=> 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,R2))) ) ) ) ).
% lexord_partial_trans
tff(fact_2365_lexord__append__leftD,axiom,
! [A: $tType,X: list(A),U: list(A),V2: list(A),R2: 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),V2)),lexord(A,R2)))
=> ( ! [A6: 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),A6),A6),R2))
=> 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),V2),lexord(A,R2))) ) ) ).
% lexord_append_leftD
tff(fact_2366_lexord__append__rightI,axiom,
! [A: $tType,Y: list(A),X: list(A),R2: set(product_prod(A,A))] :
( ? [B8: A,Z5: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B8),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,R2))) ) ).
% lexord_append_rightI
tff(fact_2367_lexord__sufE,axiom,
! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: 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,R2)))
=> ( ( 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,R2))) ) ) ) ) ).
% lexord_sufE
tff(fact_2368_binomial__absorption,axiom,
! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),K)) ).
% binomial_absorption
tff(fact_2369_lexord__lex,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2)))
<=> ( 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,R2)))
& ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).
% lexord_lex
tff(fact_2370_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R2: 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),R2))
=> 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,R2))) ) ).
% lexord_append_left_rightI
tff(fact_2371_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: 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,R2)))
<=> ( ? [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),R2)) )
| 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,R2))) ) ) ).
% lexord_same_pref_iff
tff(fact_2372_lexord__sufI,axiom,
! [A: $tType,U: list(A),W: list(A),R2: set(product_prod(A,A)),V2: 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,R2)))
=> ( 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),V2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z2)),lexord(A,R2))) ) ) ).
% lexord_sufI
tff(fact_2373_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),minus_minus(nat,K,one_one(nat)))),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),K)) ) ) ) ).
% choose_reduce_nat
tff(fact_2374_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),minus_minus(nat,K,one_one(nat)))) ) ) ).
% times_binomial_minus1_eq
tff(fact_2375_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType,B2: nat,A3: nat,F2: fun(nat,fun(A,A)),Acc: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = Acc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc)) ) ) ) ).
% fold_atLeastAtMost_nat.simps
tff(fact_2376_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
tff(fact_2377_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_2378_binomial__code,axiom,
! [N: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
=> ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),minus_minus(nat,N,K)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
=> ( aa(nat,nat,binomial(N),K) = divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),minus_minus(nat,N,K)),one_one(nat)),N,one_one(nat)),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).
% binomial_code
tff(fact_2379_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(minus_minus(nat,N,one_one(nat))),K)) ) ) ).
% binomial_addition_formula
tff(fact_2380_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),aa(nat,nat,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_2381_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),aa(nat,nat,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_2382_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_2383_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_2384_id__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I2)),Xs))) ) ) ).
% id_take_nth_drop
tff(fact_2385_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),minus_minus(nat,J,aa(nat,nat,suc,I2))))
=> ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_2386_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_finite(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)),insert3(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_2387_lenlex__append2,axiom,
! [A: $tType,R: set(product_prod(A,A)),Us: 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),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),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_2388_list__update__overwrite,axiom,
! [A: $tType,Xs: list(A),I2: 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),I2),X)),I2),Y) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),Y) ).
% list_update_overwrite
tff(fact_2389_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_2390_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_2391_drop__upt,axiom,
! [M2: nat,I2: nat,J: nat] : aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),drop(nat),M2),upt(I2,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M2),J) ).
% drop_upt
tff(fact_2392_length__list__update,axiom,
! [A: $tType,Xs: list(A),I2: 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),I2),X)) = aa(list(A),nat,size_size(list(A)),Xs) ).
% length_list_update
tff(fact_2393_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_2394_list__update__id,axiom,
! [A: $tType,Xs: list(A),I2: nat] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),aa(nat,A,nth(A,Xs),I2)) = Xs ).
% list_update_id
tff(fact_2395_nth__list__update__neq,axiom,
! [A: $tType,I2: nat,J: nat,Xs: list(A),X: A] :
( ( I2 != 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),I2),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).
% nth_list_update_neq
tff(fact_2396_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_2397_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_2398_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_2399_list__update__beyond,axiom,
! [A: $tType,Xs: list(A),I2: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),X) = Xs ) ) ).
% list_update_beyond
tff(fact_2400_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_2401_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_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_nth__list__update__eq,axiom,
! [A: $tType,I2: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),X)),I2) = X ) ) ).
% nth_list_update_eq
tff(fact_2410_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_2411_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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,C2,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_greaterThanLessThan
tff(fact_2412_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( irrefl(A,R2)
=> ( 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,R2)))
<=> 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,R2))) ) ) ).
% lexord_same_pref_if_irrefl
tff(fact_2413_set__swap,axiom,
! [A: $tType,I2: nat,Xs: list(A),J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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),I2),aa(nat,A,nth(A,Xs),J))),J),aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).
% set_swap
tff(fact_2414_drop__Cons__numeral,axiom,
! [A: $tType,V2: 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),V2)),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),V2),one_one(nat))),Xs) ).
% drop_Cons_numeral
tff(fact_2415_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I2)) ) ) ).
% nth_drop
tff(fact_2416_lexord__irrefl,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
=> irrefl(list(A),lexord(A,R)) ) ).
% lexord_irrefl
tff(fact_2417_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_2418_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I2: 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),I2),X)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I2),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)),I2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_update
tff(fact_2419_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_2420_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_2421_list__update__swap,axiom,
! [A: $tType,I2: nat,I5: nat,Xs: list(A),X: A,X5: A] :
( ( I2 != I5 )
=> ( 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),I2),X)),I5),X5) = 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),I5),X5)),I2),X) ) ) ).
% list_update_swap
tff(fact_2422_irrefl__lex,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
=> irrefl(list(A),lex(A,R2)) ) ).
% irrefl_lex
tff(fact_2423_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_2424_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_2425_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_2426_list__update_Osimps_I1_J,axiom,
! [A: $tType,I2: nat,V2: 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)),I2),V2) = nil(A) ).
% list_update.simps(1)
tff(fact_2427_list__update__code_I1_J,axiom,
! [A: $tType,I2: 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)),I2),Y) = nil(A) ).
% list_update_code(1)
tff(fact_2428_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Xs)) ).
% drop_map
tff(fact_2429_map__update,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),Xs)),K),aa(B,A,F2,Y)) ).
% map_update
tff(fact_2430_sorted__wrt__drop,axiom,
! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
( sorted_wrt(A,F2,Xs)
=> sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ) ).
% sorted_wrt_drop
tff(fact_2431_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_2432_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_2433_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_2434_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_2435_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I2: 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)),I2),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),I2),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),I2),aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).
% update_zip
tff(fact_2436_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_2437_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_2438_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),I2: 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,I2)),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),I2),Y)) ).
% list_update_code(3)
tff(fact_2439_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_2440_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_2441_set__update__subsetI,axiom,
! [A: $tType,Xs: list(A),A4: set(A),X: A,I2: nat] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),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),I2),X))),A4)) ) ) ).
% set_update_subsetI
tff(fact_2442_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_2443_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_2444_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_2445_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_2446_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_2447_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_2448_upd__conv__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list(A),A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),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),I2),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,I2)),Xs))) ) ) ).
% upd_conv_take_nth_drop
tff(fact_2449_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_2450_set__update__subset__insert,axiom,
! [A: $tType,Xs: list(A),I2: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),aa(list(A),set(A),set2(A),Xs)))) ).
% set_update_subset_insert
tff(fact_2451_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_2452_list__update__append1,axiom,
! [A: $tType,I2: nat,Xs: list(A),Ys: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(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)),I2),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),I2),X)),Ys) ) ) ).
% list_update_append1
tff(fact_2453_nth__list__update,axiom,
! [A: $tType,I2: nat,Xs: list(A),J: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( ( I2 = 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),I2),X)),J) = X ) )
& ( ( I2 != 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),I2),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).
% nth_list_update
tff(fact_2454_list__update__same__conv,axiom,
! [A: $tType,I2: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I2),X) = Xs )
<=> ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).
% list_update_same_conv
tff(fact_2455_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_2456_take__add,axiom,
! [A: $tType,I2: 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),I2),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),I2),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),I2),Xs))) ).
% take_add
tff(fact_2457_lexl__not__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: list(A)] :
( irrefl(A,R2)
=> ~ 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,R2))) ) ).
% lexl_not_refl
tff(fact_2458_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),J))
=> ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J))) ) ) ).
% sorted_list_of_set_greaterThanLessThan
tff(fact_2459_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 )
<=> ? [Ls: list(A),Rs: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),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),Ls)))
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),Rs) ) ) ) ) ).
% remove1_split
tff(fact_2460_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),X),Xs))) ) ) ).
% sorted_map_remove1
tff(fact_2461_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_2462_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_2463_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_2464_Cons__nth__drop__Suc,axiom,
! [A: $tType,I2: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I2)),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I2),Xs) ) ) ).
% Cons_nth_drop_Suc
tff(fact_2465_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_2466_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_2467_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_2468_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_2469_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_2470_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_2471_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_2472_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)),insert3(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)),insert3(A),aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).
% set_update_distinct
tff(fact_2473_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),minus_minus(nat,J,I2)))
=> ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_2474_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_2475_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_2476_hd__upt,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
=> ( aa(list(nat),nat,hd(nat),upt(I2,J)) = I2 ) ) ).
% hd_upt
tff(fact_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_greaterThanAtMost
tff(fact_2488_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_2489_distinct__swap,axiom,
! [A: $tType,I2: nat,Xs: list(A),J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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),I2),aa(nat,A,nth(A,Xs),J))),J),aa(nat,A,nth(A,Xs),I2))))
<=> pp(aa(list(A),bool,distinct(A),Xs)) ) ) ) ).
% distinct_swap
tff(fact_2490_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_2491_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)),insert3(A),X),bot_bot(set(A)))) ) ) ).
% set_remove1_eq
tff(fact_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_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_2499_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_2500_distinct_Osimps_I1_J,axiom,
! [A: $tType] : pp(aa(list(A),bool,distinct(A),nil(A))) ).
% distinct.simps(1)
tff(fact_2501_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_2502_distinct__take,axiom,
! [A: $tType,Xs: list(A),I2: 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),I2),Xs))) ) ).
% distinct_take
tff(fact_2503_distinct__upt,axiom,
! [I2: nat,J: nat] : pp(aa(list(nat),bool,distinct(nat),upt(I2,J))) ).
% distinct_upt
tff(fact_2504_distinct__drop,axiom,
! [A: $tType,Xs: list(A),I2: 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),I2),Xs))) ) ).
% distinct_drop
tff(fact_2505_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_2506_distinct__upto,axiom,
! [I2: int,J: int] : pp(aa(list(int),bool,distinct(int),upto(I2,J))) ).
% distinct_upto
tff(fact_2507_butlast_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).
% butlast.simps(1)
tff(fact_2508_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_2509_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_2510_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_2511_distinct__nthsI,axiom,
! [A: $tType,Xs: list(A),I: 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),I))) ) ).
% distinct_nthsI
tff(fact_2512_map__butlast,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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),F2),Xs)) ).
% map_butlast
tff(fact_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_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_2521_finite__distinct__list,axiom,
! [A: $tType,A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_2522_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_2523_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_2524_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_2525_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list(A),F2: 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),F2),A3)) = aa(A,B,F2,aa(list(A),A,hd(A),A3)) ) ) ).
% list.map_sel(1)
tff(fact_2526_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: 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),F2),Xs)) = aa(A,B,F2,aa(list(A),A,hd(A),Xs)) ) ) ).
% hd_map
tff(fact_2527_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_2528_longest__common__prefix,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
? [Ps: list(A),Xs3: list(A),Ys2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),Xs3) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),Ys2) )
& ( ( Xs3 = nil(A) )
| ( Ys2 = nil(A) )
| ( aa(list(A),A,hd(A),Xs3) != aa(list(A),A,hd(A),Ys2) ) ) ) ).
% longest_common_prefix
tff(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_not__distinct__decomp,axiom,
! [A: $tType,Ws: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),Ws))
=> ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y2: A] : Ws = 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),Ys3),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))),Zs2)))) ) ).
% not_distinct_decomp
tff(fact_2540_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),As2))
<=> ? [Xs4: list(A),Y3: A,Ys4: list(A)] :
( pp(member2(A,Y3,aa(list(A),set(A),set2(A),Xs4)))
& pp(aa(list(A),bool,distinct(A),Xs4))
& ( As2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4)) ) ) ) ).
% not_distinct_conv_prefix
tff(fact_2541_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_2542_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_2543_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list(A),I2: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J) )
<=> ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
tff(fact_2544_distinct__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(list(A),bool,distinct(A),Xs))
<=> ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( I4 != J3 )
=> ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).
% distinct_conv_nth
tff(fact_2545_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_2546_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,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),C2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).
% Ioc_disjoint
tff(fact_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_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_2554_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_2555_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_2556_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ~ pp(aa(list(A),bool,distinct(A),Xs))
=> ~ ! [Y2: A,Ys3: list(A)] :
( pp(aa(list(A),bool,distinct(A),Ys3))
=> ( pp(member2(A,Y2,aa(list(A),set(A),set2(A),Ys3)))
=> ! [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(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))),Zs2)) ) ) ) ).
% not_distinct_split_distinct
tff(fact_2557_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_2558_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_2559_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_2560_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_2561_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_2562_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_2563_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)),insert3(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(A),X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).
% Iio_Int_singleton
tff(fact_2564_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_2565_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_2566_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_2567_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_2568_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I2)),J))
=> ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J))) ) ) ).
% sorted_list_of_set_greaterThanAtMost
tff(fact_2569_greaterThanLessThan__upto,axiom,
! [I2: int,J: int] : set_or5935395276787703475ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),minus_minus(int,J,one_one(int)))) ).
% greaterThanLessThan_upto
tff(fact_2570_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_2571_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_2572_distinct__list__update,axiom,
! [A: $tType,Xs: list(A),A3: A,I2: 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)),insert3(A),aa(nat,A,nth(A,Xs),I2)),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),I2),A3))) ) ) ).
% distinct_list_update
tff(fact_2573_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list(A),I2: nat,J: nat] :
( pp(aa(list(A),bool,distinct(A),Vs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),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),I2),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_2574_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
<=> ? [Y3: A,N4: 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),N4)),Y3),R2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),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),N4),Y3) ) ) ) ).
% listrel1_iff_update
tff(fact_2575_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)))
& ! [Ys4: list(A)] :
( pp(member2(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> pp(aa(list(A),bool,distinct(A),Ys4)) )
& ! [Ys4: list(A),Zs3: list(A)] :
( ( pp(member2(list(A),Ys4,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)))
& ( Ys4 != 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),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).
% distinct_concat_iff
tff(fact_2576_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_finite(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_2577_zipf__zip,axiom,
! [A: $tType,B: $tType,L12: list(A),L22: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L22) )
=> ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L12,L22) = aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),L12),L22) ) ) ).
% zipf_zip
tff(fact_2578_quotient__of__number_I5_J,axiom,
! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).
% quotient_of_number(5)
tff(fact_2579_cppi,axiom,
! [D2: int,P: fun(int,bool),P2: fun(int,bool),A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ? [Z5: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),X2))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,A4))
=> ( X2 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) ) )
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P2,X2))
<=> pp(aa(int,bool,P2,minus_minus(int,X2,aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
=> ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
<=> ( ? [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
& pp(aa(int,bool,P2,X4)) )
| ? [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
& ? [Xa2: int] :
( pp(member2(int,Xa2,A4))
& pp(aa(int,bool,P,minus_minus(int,Xa2,X4))) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_2580_card__UNIV__unit,axiom,
finite_card(product_unit,top_top(set(product_unit))) = one_one(nat) ).
% card_UNIV_unit
tff(fact_2581_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_2582_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_2583_rat__one__code,axiom,
quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).
% rat_one_code
tff(fact_2584_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_2585_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_2586_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: 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,R2)))
<=> ( ( 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),R2))
& ( 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,R2))) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_2587_Int__atLeastAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastAtMost
tff(fact_2588_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_2589_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: 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,C2,D3)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastAtMostR1
tff(fact_2590_rat__zero__code,axiom,
quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
% rat_zero_code
tff(fact_2591_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set(A),B3: set(A)] :
( pp(member2(A,A3,A4))
=> ( ~ pp(member2(A,A3,B3))
=> ( finite_card(A,minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),B3))) = minus_minus(nat,finite_card(A,minus_minus(set(A),A4,B3)),one_one(nat)) ) ) ) ).
% card_Diff_insert
tff(fact_2592_card__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_card(int,set_or1337092689740270186AtMost(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),minus_minus(int,U,L)),one_one(int))) ).
% card_atLeastAtMost_int
tff(fact_2593_quotient__of__number_I4_J,axiom,
quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).
% quotient_of_number(4)
tff(fact_2594_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).
% quotient_of_number(3)
tff(fact_2595_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_card(int,set_or5935395276787703475ssThan(int,L,U)) = nat2(minus_minus(int,U,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).
% card_greaterThanLessThan_int
tff(fact_2596_listrel1__mono,axiom,
! [A: $tType,R2: 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))),R2),S3))
=> 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,R2)),listrel1(A,S3))) ) ).
% listrel1_mono
tff(fact_2597_sgn__rat__def,axiom,
! [A3: rat] :
( ( ( A3 = zero_zero(rat) )
=> ( aa(rat,rat,sgn_sgn(rat),A3) = zero_zero(rat) ) )
& ( ( A3 != zero_zero(rat) )
=> ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
=> ( aa(rat,rat,sgn_sgn(rat),A3) = one_one(rat) ) )
& ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
=> ( aa(rat,rat,sgn_sgn(rat),A3) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).
% sgn_rat_def
tff(fact_2598_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_2599_atLeastAtMost__upto,axiom,
! [I2: int,J: int] : set_or1337092689740270186AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,J)) ).
% atLeastAtMost_upto
tff(fact_2600_map__concat,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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),F2)),Xs)) ).
% map_concat
tff(fact_2601_List_Obind__def,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,list(B))] : bind(A,B,Xs,F2) = 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)),F2),Xs)) ).
% List.bind_def
tff(fact_2602_maps__def,axiom,
! [B: $tType,A: $tType,F2: fun(A,list(B)),Xs: list(A)] : maps(A,B,F2,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)),F2),Xs)) ).
% maps_def
tff(fact_2603_concat__map__maps,axiom,
! [A: $tType,B: $tType,F2: 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)),F2),Xs)) = maps(B,A,F2,Xs) ).
% concat_map_maps
tff(fact_2604_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_2605_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_2606_listrel1I2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> 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,R2))) ) ).
% listrel1I2
tff(fact_2607_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(A,fun(B,C))] : zipf(A,B,C,F2,nil(A),nil(B)) = nil(C) ).
% zipf.simps(1)
tff(fact_2608_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list(A),R2: 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,R2))) ).
% not_Nil_listrel1
tff(fact_2609_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list(A),R2: 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,R2))) ).
% not_listrel1_Nil
tff(fact_2610_listrel1__eq__len,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).
% listrel1_eq_len
tff(fact_2611_append__listrel1I,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: 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,R2)))
& ( Us = 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)),Us),Vs),listrel1(A,R2))) ) )
=> 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),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),listrel1(A,R2))) ) ).
% append_listrel1I
tff(fact_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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)),insert3(A),X2),bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_2618_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_2619_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_2620_bset_I1_J,axiom,
! [D2: int,B3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,B3))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,minus_minus(int,X2,D2))) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,B3))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,minus_minus(int,X2,D2))) ) )
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
& pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,minus_minus(int,X3,D2)))
& pp(aa(int,bool,Q,minus_minus(int,X3,D2))) ) ) ) ) ) ).
% bset(1)
tff(fact_2621_bset_I2_J,axiom,
! [D2: int,B3: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,B3))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,minus_minus(int,X2,D2))) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,B3))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,minus_minus(int,X2,D2))) ) )
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
| pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,minus_minus(int,X3,D2)))
| pp(aa(int,bool,Q,minus_minus(int,X3,D2))) ) ) ) ) ) ).
% bset(2)
tff(fact_2622_aset_I1_J,axiom,
! [D2: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,A4))
=> ( X2 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,A4))
=> ( X2 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) ) )
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
& pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)))
& pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2))) ) ) ) ) ) ).
% aset(1)
tff(fact_2623_aset_I2_J,axiom,
! [D2: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,A4))
=> ( X2 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,A4))
=> ( X2 != minus_minus(int,Xb2,Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) ) )
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
| pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)))
| pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2))) ) ) ) ) ) ).
% aset(2)
tff(fact_2624_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: 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),R2))
=> 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,R2))) ) ).
% listrel1I1
tff(fact_2625_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> ( ! [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),R2)) )
=> ~ ! [Zs2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
=> ~ 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),Zs2),listrel1(A,R2))) ) ) ) ).
% Cons_listrel1E1
tff(fact_2626_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: 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,R2)))
=> ( ! [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),R2)) )
=> ~ ! [Zs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
=> ~ 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)),Zs2),Ys),listrel1(A,R2))) ) ) ) ).
% Cons_listrel1E2
tff(fact_2627_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)) )
=> ? [Xss1: list(list(A)),Xs2: list(A),Xs3: 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),Xs2),Xs3)),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)),Xs2) )
& ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(list(A)),list(A),concat(A),Xss22)) ) ) ) ) ).
% concat_eq_appendD
tff(fact_2628_greaterThanAtMost__upto,axiom,
! [I2: int,J: int] : set_or3652927894154168847AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ).
% greaterThanAtMost_upto
tff(fact_2629_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_2630_card__1__singletonI,axiom,
! [A: $tType,S: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(A),X),bot_bot(set(A))) ) ) ) ) ).
% card_1_singletonI
tff(fact_2631_card__Un__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(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_2632_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_2633_bset_I9_J,axiom,
! [D3: int,D2: int,B3: set(int),T3: int] :
( pp(dvd_dvd(int,D3,D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3)))
=> pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),minus_minus(int,X3,D2)),T3))) ) ) ) ).
% bset(9)
tff(fact_2634_bset_I10_J,axiom,
! [D3: int,D2: int,B3: set(int),T3: int] :
( pp(dvd_dvd(int,D3,D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3)))
=> ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),minus_minus(int,X3,D2)),T3))) ) ) ) ).
% bset(10)
tff(fact_2635_aset_I9_J,axiom,
! [D3: int,D2: int,A4: set(int),T3: int] :
( pp(dvd_dvd(int,D3,D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3)))
=> pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)),T3))) ) ) ) ).
% aset(9)
tff(fact_2636_aset_I10_J,axiom,
! [D3: int,D2: int,A4: set(int),T3: int] :
( pp(dvd_dvd(int,D3,D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T3)))
=> ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)),T3))) ) ) ) ).
% aset(10)
tff(fact_2637_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
=> ( set_or1337092689740270186AtMost(int,M2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M2,N)) ) ) ).
% atLeastAtMostPlus1_int_conv
tff(fact_2638_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)) )
=> ? [Xss12: list(list(A)),Xs4: list(A),Xs6: 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),Xs4),Xs6)),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)),Xs4) )
& ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),aa(list(list(A)),list(A),concat(A),Xss23)) ) ) ) ) ) ).
% concat_eq_append_conv
tff(fact_2639_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us: 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),R2))
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
=> ( ( 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),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,R2))) ) ) ) ).
% listrel1I
tff(fact_2640_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> ~ ! [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),R2))
=> ! [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_2641_card__Un__disjoint,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(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_2642_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ~ ! [L3: list(A)] :
( sorted_wrt(A,ord_less(A),L3)
=> ( ( aa(list(A),set(A),set2(A),L3) = A4 )
=> ( aa(list(A),nat,size_size(list(A)),L3) != finite_card(A,A4) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
tff(fact_2643_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)),insert3(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)),insert3(A),X),bot_bot(set(A))))) = finite_card(A,A4) ) ) ) ).
% card_Diff_singleton_if
tff(fact_2644_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)),insert3(A),X),bot_bot(set(A))))) = minus_minus(nat,finite_card(A,A4),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_2645_periodic__finite__ex,axiom,
! [D3: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P,minus_minus(int,X2,aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) )
=> ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
<=> ? [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D3)))
& pp(aa(int,bool,P,X4)) ) ) ) ) ).
% periodic_finite_ex
tff(fact_2646_aset_I7_J,axiom,
! [D2: int,A4: set(int),T3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T3),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2))) ) ) ) ).
% aset(7)
tff(fact_2647_aset_I5_J,axiom,
! [D2: int,T3: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,T3,A4))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),T3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)),T3)) ) ) ) ) ).
% aset(5)
tff(fact_2648_aset_I4_J,axiom,
! [D2: int,T3: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,T3,A4))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( ( X3 != T3 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2) != T3 ) ) ) ) ) ).
% aset(4)
tff(fact_2649_aset_I3_J,axiom,
! [D2: int,T3: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T3),one_one(int)),A4))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( ( X3 = T3 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2) = T3 ) ) ) ) ) ).
% aset(3)
tff(fact_2650_bset_I7_J,axiom,
! [D2: int,T3: int,B3: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,T3,B3))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T3),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T3),minus_minus(int,X3,D2))) ) ) ) ) ).
% bset(7)
tff(fact_2651_bset_I5_J,axiom,
! [D2: int,B3: set(int),T3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),T3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),minus_minus(int,X3,D2)),T3)) ) ) ) ).
% bset(5)
tff(fact_2652_bset_I4_J,axiom,
! [D2: int,T3: int,B3: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,T3,B3))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( X3 != T3 )
=> ( minus_minus(int,X3,D2) != T3 ) ) ) ) ) ).
% bset(4)
tff(fact_2653_bset_I3_J,axiom,
! [D2: int,T3: int,B3: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,minus_minus(int,T3,one_one(int)),B3))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( X3 = T3 )
=> ( minus_minus(int,X3,D2) = T3 ) ) ) ) ) ).
% bset(3)
tff(fact_2654_rat__sgn__code,axiom,
! [P7: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P7)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P7)))),one_one(int)) ).
% rat_sgn_code
tff(fact_2655_simp__from__to,axiom,
! [J: int,I2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
=> ( set_or1337092689740270186AtMost(int,I2,J) = bot_bot(set(int)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
=> ( set_or1337092689740270186AtMost(int,I2,J) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert3(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ) ).
% simp_from_to
tff(fact_2656_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R2: 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,R2)))
<=> ( ( 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,R2)))
& ( 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),R2)) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_2657_distinct__concat,axiom,
! [A: $tType,Xs: list(list(A))] :
( pp(aa(list(list(A)),bool,distinct(list(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)) ) ) ) )
=> pp(aa(list(A),bool,distinct(A),aa(list(list(A)),list(A),concat(A),Xs))) ) ) ) ).
% distinct_concat
tff(fact_2658_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,Y)),minus_minus(nat,N,one_one(nat))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),Y))),N)) ) ) ).
% card_insert_le_m1
tff(fact_2659_aset_I8_J,axiom,
! [D2: int,A4: set(int),T3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T3),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2))) ) ) ) ).
% aset(8)
tff(fact_2660_aset_I6_J,axiom,
! [D2: int,T3: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T3),one_one(int)),A4))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,A4))
=> ( X3 != minus_minus(int,Xb3,Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),T3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)),T3)) ) ) ) ) ).
% aset(6)
tff(fact_2661_bset_I8_J,axiom,
! [D2: int,T3: int,B3: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( pp(member2(int,minus_minus(int,T3,one_one(int)),B3))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T3),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T3),minus_minus(int,X3,D2))) ) ) ) ) ).
% bset(8)
tff(fact_2662_bset_I6_J,axiom,
! [D2: int,B3: set(int),T3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ! [X3: int] :
( ! [Xa4: int] :
( pp(member2(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb3: int] :
( pp(member2(int,Xb3,B3))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),T3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),minus_minus(int,X3,D2)),T3)) ) ) ) ).
% bset(6)
tff(fact_2663_cpmi,axiom,
! [D2: int,P: fun(int,bool),P2: fun(int,bool),B3: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ? [Z5: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Z5))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa3: int] :
( pp(member2(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D2)))
=> ! [Xb2: int] :
( pp(member2(int,Xb2,B3))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,minus_minus(int,X2,D2))) ) )
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P2,X2))
<=> pp(aa(int,bool,P2,minus_minus(int,X2,aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
=> ( ? [X_12: int] : pp(aa(int,bool,P,X_12))
<=> ( ? [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
& pp(aa(int,bool,P2,X4)) )
| ? [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D2)))
& ? [Xa2: int] :
( pp(member2(int,Xa2,B3))
& pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X4))) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_2664_quotient__of__int,axiom,
! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),one_one(int)) ).
% quotient_of_int
tff(fact_2665_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_2666_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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_finite(B),A4))
=> ~ ! [L3: list(B)] :
( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3))
=> ( ( aa(list(B),set(B),set2(B),L3) = A4 )
=> ( aa(list(B),nat,size_size(list(B)),L3) != finite_card(B,A4) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
tff(fact_2667_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).
% Frct_code_post(4)
tff(fact_2668_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = divide_divide(rat,one_one(rat),aa(num,rat,numeral_numeral(rat),K)) ).
% Frct_code_post(5)
tff(fact_2669_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)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).
% card_disjoint_shuffles
tff(fact_2670_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_2671_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_2672_finite__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,finite_finite(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_2673_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_2674_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_2675_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_2676_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))),insert3(list(A)),Xs),bot_bot(set(list(A)))) ).
% shuffles.simps(2)
tff(fact_2677_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))),insert3(list(A)),Ys),bot_bot(set(list(A)))) ).
% shuffles.simps(1)
tff(fact_2678_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_2679_all__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N))
=> pp(aa(nat,bool,P,M4)) )
<=> ! [X4: nat] :
( pp(member2(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,P,X4)) ) ) ).
% all_nat_less
tff(fact_2680_ex__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N))
& pp(aa(nat,bool,P,M4)) )
<=> ? [X4: nat] :
( pp(member2(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
& pp(aa(nat,bool,P,X4)) ) ) ).
% ex_nat_less
tff(fact_2681_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,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
=> ! [Z3: A,Zs4: list(A)] :
( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
=> ( ( X2 = Z3 )
=> ~ pp(member2(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs3),Ys))) ) ) )
=> ~ ! [Y2: A,Ys2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ! [Z3: A,Zs4: list(A)] :
( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
=> ( ( Y2 = Z3 )
=> ~ pp(member2(list(A),Zs4,aa(list(A),set(list(A)),aa(list(A),fun(list(A),set(list(A))),shuffles(A),Xs),Ys2))) ) ) ) ) ) ) ) ).
% shufflesE
tff(fact_2682_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_2683_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_2684_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),F2: fun(B,A),Xs: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( pp(aa(list(A),bool,distinct(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)))
=> pp(aa(list(B),bool,distinct(B),Xs)) ) ) ).
% folding_insort_key.distinct_if_distinct_map
tff(fact_2685_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_2686_Frct__code__post_I3_J,axiom,
frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).
% Frct_code_post(3)
tff(fact_2687_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_2688_finite__trancl__ntranl,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
=> ( transitive_trancl(A,R) = transitive_ntrancl(A,minus_minus(nat,finite_card(product_prod(A,A),R),one_one(nat)),R) ) ) ).
% finite_trancl_ntranl
tff(fact_2689_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),F2: fun(B,A),A4: set(B),L: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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_finite(B),A4))
=> ( ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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),F2),A4) = L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_2690_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),F2: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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)),insert3(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(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),F2),A4)) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_2691_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),F2: fun(B,A),Xs: list(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),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),F2),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
tff(fact_2692_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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_finite(B),A4))
=> ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A4) = nil(B) )
<=> ( A4 = bot_bot(set(B)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_2693_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A4))) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_2694_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_2695_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),F2: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),bot_bot(set(B))) = nil(B) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_2696_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),F2: fun(B,A),A4: set(B),B3: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),A4) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),B3) )
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(B),B3))
=> ( A4 = B3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_2697_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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_finite(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),F2),A4)) = A4 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
tff(fact_2698_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),A4)) = finite_card(B,A4) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
tff(fact_2699_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A4)))) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_2700_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),F2: fun(B,A),A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A4))) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_2701_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),F2: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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)),insert3(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(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),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),minus_minus(set(B),A4,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_2702_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),F2: fun(B,A),X: B,A4: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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)),insert3(B),X),A4)),S))
=> ( pp(aa(set(B),bool,finite_finite(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),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(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),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A4)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_2703_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))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% listset.simps(1)
tff(fact_2704_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))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% lists_empty
tff(fact_2705_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: 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,R2)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(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),N4)),aa(nat,B,nth(B,Ys),N4)),R2)) ) ) ) ).
% listrel_iff_nth
tff(fact_2706_zero__rat__def,axiom,
zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).
% zero_rat_def
tff(fact_2707_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_2708_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_2709_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_2710_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_2711_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_2712_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_2713_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_2714_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_2715_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_2716_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_2717_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_2718_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_2719_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_2720_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) )
| ? [A5: A,L4: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),L4) )
& pp(member2(A,A5,A4))
& pp(member2(list(A),L4,aa(set(A),set(list(A)),lists(A),A4))) ) ) ) ).
% lists.simps
tff(fact_2721_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) )
=> ~ ! [A6: A,L3: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),L3) )
=> ( pp(member2(A,A6,A4))
=> ~ pp(member2(list(A),L3,aa(set(A),set(list(A)),lists(A),A4))) ) ) ) ) ).
% lists.cases
tff(fact_2722_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list(A),R2: 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,R2)))
=> ( Xs = nil(A) ) ) ).
% listrel_Nil2
tff(fact_2723_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list(B),R2: 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,R2)))
=> ( Xs = nil(B) ) ) ).
% listrel_Nil1
tff(fact_2724_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: 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,R2))) ).
% listrel.Nil
tff(fact_2725_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: 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,R2)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).
% listrel_eq_len
tff(fact_2726_listrel__mono,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: 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))),R2),S3))
=> 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,R2)),listrel(A,B,S3))) ) ).
% listrel_mono
tff(fact_2727_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: 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,R2)))
=> ~ ! [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),R2))
=> ~ 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,R2))) ) ) ) ).
% listrel_Cons2
tff(fact_2728_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: 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,R2)))
=> ~ ! [Y2: B,Ys3: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),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),Y),Y2),R2))
=> ~ 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),Ys3),listrel(A,B,R2))) ) ) ) ).
% listrel_Cons1
tff(fact_2729_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: 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),R2))
=> ( 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,R2)))
=> 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,R2))) ) ) ).
% listrel.Cons
tff(fact_2730_one__rat__def,axiom,
one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).
% one_rat_def
tff(fact_2731_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: 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,R2)))
=> ( ( ( 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) )
=> ! [Ys3: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),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),X2),Y2),R2))
=> ~ 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),Ys3),listrel(A,B,R2))) ) ) ) ) ) ).
% listrel.cases
tff(fact_2732_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: 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,R2)))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs4: list(A),Ys4: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys4) )
& 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),R2))
& 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)),Xs4),Ys4),listrel(A,B,R2))) ) ) ) ).
% listrel.simps
tff(fact_2733_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),R5))
=> ( refl_on(A,top_top(set(A)),R5)
=> 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,R2)),listrel(A,A,R5))) ) ) ).
% listrel1_subset_listrel
tff(fact_2734_zero__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat))) ).
% zero_rat.transfer
tff(fact_2735_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( image2(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( image2(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( image2(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_2736_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),Xs: list(B),Ys: list(B)] :
( inj_on(B,A,F2,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),F2),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),F2),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),F2),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),F2),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_2737_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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),F2),linord329482645794927042rt_key(B,A,F2,X,Xs))) ) ) ).
% sorted_insort_insert_key
tff(fact_2738_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I2: 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),I2),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),I2))) ) ) ) ) ).
% sorted_rev_nth_mono
tff(fact_2739_inj__on__rev,axiom,
! [A: $tType,A4: set(list(A))] : inj_on(list(A),list(A),rev(A),A4) ).
% inj_on_rev
tff(fact_2740_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_2741_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_2742_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set(list(A))] : image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),L5) = L5 ).
% op_conc_empty_img_id
tff(fact_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_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_2750_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_2751_list_Oset__map,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),V2: 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),F2),V2)) = image2(A,B,F2,aa(list(A),set(A),set2(A),V2)) ).
% list.set_map
tff(fact_2752_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
<=> ( Xs = Ys ) ) ) ).
% inj_map_eq_map
tff(fact_2753_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_2754_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_2755_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_2756_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_2757_inj__mapI,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] :
( inj_on(A,B,F2,top_top(set(A)))
=> inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A)))) ) ).
% inj_mapI
tff(fact_2758_inj__map,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] :
( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
<=> inj_on(A,B,F2,top_top(set(A))) ) ).
% inj_map
tff(fact_2759_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))
=> ( 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_2760_lists__image,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] : aa(set(A),set(list(A)),lists(A),image2(B,A,F2,A4)) = image2(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),set(list(B)),lists(B),A4)) ).
% lists_image
tff(fact_2761_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_2762_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_2763_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_2764_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),X: B,Xs: list(B)] :
( pp(member2(A,aa(B,A,F2,X),image2(B,A,F2,aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F2,X,Xs) = Xs ) ) ) ).
% insort_insert_key_triv
tff(fact_2765_rev_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).
% rev.simps(1)
tff(fact_2766_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_2767_rev__map,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rev(B),Xs)) ).
% rev_map
tff(fact_2768_listrel__refl__on,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,A4,R2)
=> refl_on(list(A),aa(set(A),set(list(A)),lists(A),A4),listrel(A,A,R2)) ) ).
% listrel_refl_on
tff(fact_2769_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),F2: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> inj_on(B,A,F2,S) ) ).
% folding_insort_key.inj_on
tff(fact_2770_inj__mapD,axiom,
! [B: $tType,A: $tType,F2: fun(A,B)] :
( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
=> inj_on(A,B,F2,top_top(set(A))) ) ).
% inj_mapD
tff(fact_2771_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_2772_map__injective,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
=> ( inj_on(B,A,F2,top_top(set(B)))
=> ( Xs = Ys ) ) ) ).
% map_injective
tff(fact_2773_image__Int__subset,axiom,
! [A: $tType,B: $tType,F2: 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)),image2(B,A,F2,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)),image2(B,A,F2,A4)),image2(B,A,F2,B3)))) ).
% image_Int_subset
tff(fact_2774_image__set,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : image2(B,A,F2,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),F2),Xs)) ).
% image_set
tff(fact_2775_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))),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)))),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_2776_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))),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_2777_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))),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_2778_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_2779_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_2780_distinct__insort__insert,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),F2: fun(B,A),X: B] :
( pp(aa(list(B),bool,distinct(B),Xs))
=> pp(aa(list(B),bool,distinct(B),linord329482645794927042rt_key(B,A,F2,X,Xs))) ) ) ).
% distinct_insort_insert
tff(fact_2781_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_2782_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
( inj_on(A,B,F2,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),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
<=> ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
tff(fact_2783_map__inj__on,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
=> ( inj_on(B,A,F2,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_2784_distinct__map,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs)))
<=> ( pp(aa(list(B),bool,distinct(B),Xs))
& inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs)) ) ) ).
% distinct_map
tff(fact_2785_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),X: A,Xs: list(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),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,F2,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) ) ) ).
% map_removeAll_inj
tff(fact_2786_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I: 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)),I) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ).
% nths_drop
tff(fact_2787_inj__on__nth,axiom,
! [A: $tType,Xs: list(A),I: set(nat)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ! [X2: nat] :
( pp(member2(nat,X2,I))
=> 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),I) ) ) ).
% inj_on_nth
tff(fact_2788_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_2789_rev__take,axiom,
! [A: $tType,I2: 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),I2),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),I2)),aa(list(A),list(A),rev(A),Xs)) ).
% rev_take
tff(fact_2790_rev__drop,axiom,
! [A: $tType,I2: 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),I2),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),I2)),aa(list(A),list(A),rev(A),Xs)) ).
% rev_drop
tff(fact_2791_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_2792_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),X: A,Xs: list(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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),F2),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,F2,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) ) ) ).
% map_removeAll_inj_on
tff(fact_2793_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_2794_image__Suc__atMost,axiom,
! [N: nat] : image2(nat,nat,suc,set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).
% image_Suc_atMost
tff(fact_2795_image__Suc__lessThan,axiom,
! [N: nat] : image2(nat,nat,suc,set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).
% image_Suc_lessThan
tff(fact_2796_one__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat))) ).
% one_rat.transfer
tff(fact_2797_subset__subseqs,axiom,
! [A: $tType,X6: set(A),Xs: list(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs)))
=> pp(member2(set(A),X6,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_2798_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_2799_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_2800_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))
<=> ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),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,I4))),aa(nat,A,nth(A,Xs),I4))) ) ) ) ).
% sorted_rev_iff_nth_Suc
tff(fact_2801_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))),insert3(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))),insert3(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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(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),Ys3)))),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)),Ys3))) ) ) ) ) ) ) ).
% shuffles.elims
tff(fact_2802_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))
<=> ! [I4: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),J3)),aa(nat,A,nth(A,Xs),I4))) ) ) ) ) ).
% sorted_rev_iff_nth_mono
tff(fact_2803_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B3: set(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3))
<=> ( inj_on(A,B,F2,A4)
& inj_on(A,B,F2,B3)
& ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),image2(A,B,F2,minus_minus(set(A),A4,B3))),image2(A,B,F2,minus_minus(set(A),B3,A4))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_2804_image__Int,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B3: set(A)] :
( inj_on(A,B,F2,top_top(set(A)))
=> ( image2(A,B,F2,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)),image2(A,B,F2,A4)),image2(A,B,F2,B3)) ) ) ).
% image_Int
tff(fact_2805_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A4: set(A),B3: set(A)] :
( inj_on(A,B,F2,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
=> ( image2(A,B,F2,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)),image2(A,B,F2,A4)),image2(A,B,F2,B3)) ) ) ) ) ).
% inj_on_image_Int
tff(fact_2806_translation__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S3: set(A),T3: 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)),S3),T3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3),S3)),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3),T3)) ) ).
% translation_Int
tff(fact_2807_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_2808_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_2809_foldr__append,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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),F2),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),F2),Ys),A3)) ).
% foldr_append
tff(fact_2810_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),L)))
=> ( aa(A,A,aa(B,fun(A,A),F2,X2),A6) = aa(A,A,aa(B,fun(A,A),G,X2),A6) ) )
=> ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),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_2811_transpose_Osimps_I1_J,axiom,
! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).
% transpose.simps(1)
tff(fact_2812_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_2813_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_2814_transpose__map__map,axiom,
! [A: $tType,B: $tType,F2: 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),F2)),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),F2)),transpose(B,Xs)) ).
% transpose_map_map
tff(fact_2815_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_2816_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_2817_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_2818_inj__on__Int,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B3: set(A)] :
( ( inj_on(A,B,F2,A4)
| inj_on(A,B,F2,B3) )
=> inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3)) ) ).
% inj_on_Int
tff(fact_2819_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_2820_map__tailrec__def,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),As2: list(A)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map_tailrec(A,B),F2),As2) = aa(list(B),list(B),rev(B),map_tailrec_rev(A,B,F2,As2,nil(B))) ).
% map_tailrec_def
tff(fact_2821_refl__on__Int,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B3: set(A),S3: set(product_prod(A,A))] :
( refl_on(A,A4,R2)
=> ( refl_on(A,B3,S3)
=> 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))),R2),S3)) ) ) ).
% refl_on_Int
tff(fact_2822_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs: list(A),F2: fun(A,B)] :
( ( Xs != nil(A) )
=> ( aa(A,B,F2,arg_min_list(A,B,F2,Xs)) = lattic643756798350308766er_Min(B,image2(A,B,F2,aa(list(A),set(A),set2(A),Xs))) ) ) ) ).
% f_arg_min_list_f
tff(fact_2823_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)))
=> ( 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_2824_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: set(list(A))] :
( inj_on(A,B,F2,complete_Sup_Sup(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),F2),A4) ) ).
% inj_on_mapI
tff(fact_2825_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_2826_Int__atLeastLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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,C2,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).
% Int_atLeastLessThan
tff(fact_2827_ex__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
& pp(aa(nat,bool,P,M4)) )
<=> ? [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_2828_all__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
=> pp(aa(nat,bool,P,M4)) )
<=> ! [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_2829_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_2830_insert__partition,axiom,
! [A: $tType,X: set(A),F: set(set(A))] :
( ~ pp(member2(set(A),X,F))
=> ( ! [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))),insert3(set(A)),X),F)))
=> ! [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))),insert3(set(A)),X),F)))
=> ( ( 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),complete_Sup_Sup(set(A),F)) = bot_bot(set(A)) ) ) ) ).
% insert_partition
tff(fact_2831_Union__disjoint,axiom,
! [A: $tType,C5: set(set(A)),A4: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_Sup_Sup(set(A),C5)),A4) = bot_bot(set(A)) )
<=> ! [X4: set(A)] :
( pp(member2(set(A),X4,C5))
=> ( 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_2832_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A3: A,As2: list(A),Bs: list(B)] : map_tailrec_rev(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),Bs) = map_tailrec_rev(A,B,F2,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F2,A3)),Bs)) ).
% map_tailrec_rev.simps(2)
tff(fact_2833_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)),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)),complete_Sup_Sup(set(A),A4)),complete_Sup_Sup(set(A),B3)))) ).
% Union_Int_subset
tff(fact_2834_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),Bs: list(B)] : map_tailrec_rev(A,B,F2,nil(A),Bs) = Bs ).
% map_tailrec_rev.simps(1)
tff(fact_2835_atLeastLessThan__upt,axiom,
! [I2: nat,J: nat] : set_or7035219750837199246ssThan(nat,I2,J) = aa(list(nat),set(nat),set2(nat),upt(I2,J)) ).
% atLeastLessThan_upt
tff(fact_2836_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),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),complete_Sup_Sup(A,A4)),complete_Sup_Sup(A,B3)))) ) ).
% Sup_inter_less_eq
tff(fact_2837_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_2838_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_2839_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_2840_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),X: A] : arg_min_list(A,B,F2,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_2841_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 ) )
=> ~ ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),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,A6)),Xb)) ) ) ) ) ).
% map_tailrec_rev.elims
tff(fact_2842_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs: list(A),F2: fun(A,B)] :
( ( Xs != nil(A) )
=> pp(member2(A,arg_min_list(A,B,F2,Xs),aa(list(A),set(A),set2(A),Xs))) ) ) ).
% arg_min_list_in
tff(fact_2843_card__partition,axiom,
! [A: $tType,C5: set(set(A)),K: nat] :
( pp(aa(set(set(A)),bool,finite_finite(set(A)),C5))
=> ( pp(aa(set(A),bool,finite_finite(A),complete_Sup_Sup(set(A),C5)))
=> ( ! [C4: set(A)] :
( pp(member2(set(A),C4,C5))
=> ( finite_card(A,C4) = K ) )
=> ( ! [C1: set(A),C22: set(A)] :
( pp(member2(set(A),C1,C5))
=> ( pp(member2(set(A),C22,C5))
=> ( ( C1 != C22 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(set(A),C5)) = finite_card(A,complete_Sup_Sup(set(A),C5)) ) ) ) ) ) ).
% card_partition
tff(fact_2844_dvd__partition,axiom,
! [A: $tType,C5: set(set(A)),K: nat] :
( pp(aa(set(A),bool,finite_finite(A),complete_Sup_Sup(set(A),C5)))
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,C5))
=> pp(dvd_dvd(nat,K,finite_card(A,X2))) )
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,C5))
=> ! [Xa4: set(A)] :
( pp(member2(set(A),Xa4,C5))
=> ( ( 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,complete_Sup_Sup(set(A),C5)))) ) ) ) ).
% dvd_partition
tff(fact_2845_map__tailrec__rev,axiom,
! [B: $tType,A: $tType,F2: fun(B,A),As2: list(B),Bs: list(A)] : map_tailrec_rev(B,A,F2,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),F2),As2))),Bs) ).
% map_tailrec_rev
tff(fact_2846_inf__Sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B3: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),complete_Sup_Sup(A,B3)) = complete_Sup_Sup(A,image2(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) ) ).
% inf_Sup
tff(fact_2847_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),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_2848_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F2: 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),F2),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),F2),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),F2),A3),Ys))) ) ).
% horner_sum_append
tff(fact_2849_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F2: 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),F2),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,F2,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),F2),A3),Xs))) ) ).
% horner_sum_simps(2)
tff(fact_2850_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_2851_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F2: 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),F2),A3),nil(B)) = zero_zero(A) ) ).
% horner_sum_simps(1)
tff(fact_2852_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_2853_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_2854_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_2855_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_2856_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,P7: 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),P7))
=> ( 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,P7))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,P7)) ) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_2857_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_2858_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_atLeastLessThan__upto,axiom,
! [I2: int,J: int] : set_or7035219750837199246ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,minus_minus(int,J,one_one(int)))) ).
% atLeastLessThan_upto
tff(fact_2868_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_2869_prod_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A),P7: 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),P7))) = 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),P7)))) ) ) ) ).
% prod.ub_add_nat
tff(fact_2870_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_finite(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)),insert3(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_2871_prod__eq__1__iff,axiom,
! [A: $tType,A4: set(A),F2: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4) = one_one(nat) )
<=> ! [X4: A] :
( pp(member2(A,X4,A4))
=> ( aa(A,nat,F2,X4) = one_one(nat) ) ) ) ) ).
% prod_eq_1_iff
tff(fact_2872_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( ~ pp(aa(set(B),bool,finite_finite(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_2873_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_2874_prod__Un,axiom,
! [A: $tType,B: $tType] :
( field(A)
=> ! [A4: set(B),B3: set(B),F2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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,F2,X2) != zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),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),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ) ).
% prod_Un
tff(fact_2875_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) )
=> ~ ! [A6: B] :
( pp(member2(B,A6,A4))
=> ( aa(B,A,G,A6) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_2876_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_2877_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F2: 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,F2,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),F2),A4))) ) ) ).
% prod_ge_1
tff(fact_2878_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F2: 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,F2,X2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,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),F2),A4)),one_one(A))) ) ) ).
% prod_le_1
tff(fact_2879_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R: fun(A,fun(A,bool)),S: set(B),H2: 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_finite(B),S))
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H2,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),H2),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S))) ) ) ) ) ) ).
% prod.related
tff(fact_2880_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_finite(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)),insert3(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)),insert3(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_2881_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T4: set(C),S: set(B),I2: fun(C,B),J: fun(B,C),T: set(C),G: fun(B,A),H2: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite(B),S2))
=> ( pp(aa(set(C),bool,finite_finite(C),T4))
=> ( ! [A6: B] :
( pp(member2(B,A6,minus_minus(set(B),S,S2)))
=> ( aa(C,B,I2,aa(B,C,J,A6)) = A6 ) )
=> ( ! [A6: B] :
( pp(member2(B,A6,minus_minus(set(B),S,S2)))
=> pp(member2(C,aa(B,C,J,A6),minus_minus(set(C),T,T4))) )
=> ( ! [B5: C] :
( pp(member2(C,B5,minus_minus(set(C),T,T4)))
=> ( aa(B,C,J,aa(C,B,I2,B5)) = B5 ) )
=> ( ! [B5: C] :
( pp(member2(C,B5,minus_minus(set(C),T,T4)))
=> pp(member2(B,aa(C,B,I2,B5),minus_minus(set(B),S,S2))) )
=> ( ! [A6: B] :
( pp(member2(B,A6,S2))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B5: C] :
( pp(member2(C,B5,T4))
=> ( aa(C,A,H2,B5) = one_one(A) ) )
=> ( ! [A6: B] :
( pp(member2(B,A6,S))
=> ( aa(C,A,H2,aa(B,C,J,A6)) = aa(B,A,G,A6) ) )
=> ( 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),H2),T) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_2882_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I: set(A),I2: A,F2: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite(A),I))
=> ( pp(member2(A,I2,I))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,I2)))
=> ( ! [I3: A] :
( pp(member2(A,I3,I))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,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),F2),I))) ) ) ) ) ) ).
% less_1_prod2
tff(fact_2883_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I: set(A),F2: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite(A),I))
=> ( ( I != bot_bot(set(A)) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,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),F2),I))) ) ) ) ) ).
% less_1_prod
tff(fact_2884_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_finite(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_2885_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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,H2,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),H2),S) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_2886_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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,H2,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H2,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),H2),T) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_2887_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_finite(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_2888_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_finite(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_2889_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(B),A4: set(B),B3: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
=> ( ! [A6: B] :
( pp(member2(B,A6,minus_minus(set(B),C5,A4)))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B5: B] :
( pp(member2(B,B5,minus_minus(set(B),C5,B3)))
=> ( aa(B,A,H2,B5) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),C5) )
=> ( 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),H2),B3) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_2890_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(B),A4: set(B),B3: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
=> ( ! [A6: B] :
( pp(member2(B,A6,minus_minus(set(B),C5,A4)))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B5: B] :
( pp(member2(B,B5,minus_minus(set(B),C5,B3)))
=> ( aa(B,A,H2,B5) = 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),H2),B3) )
<=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),C5) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_2891_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_2892_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_finite(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_2893_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),T))
=> ( pp(aa(set(B),bool,finite_finite(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H2,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,H2,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),H2),T) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_2894_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_finite(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)),insert3(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)),insert3(B),X),bot_bot(set(B)))))) ) ) ) ).
% prod.insert_remove
tff(fact_2895_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_finite(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)),insert3(B),X),bot_bot(set(B)))))) ) ) ) ) ).
% prod.remove
tff(fact_2896_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_2897_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_2898_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_2899_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [B3: set(A),A4: set(A),F2: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite(A),B3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B3))
=> ( ! [B5: A] :
( pp(member2(A,B5,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,F2,B5))) )
=> ( ! [A6: A] :
( pp(member2(A,A6,A4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A6))) )
=> 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),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3))) ) ) ) ) ) ).
% prod_mono2
tff(fact_2900_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(A)
=> ! [A4: set(B),F2: 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,F2,I3)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,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),F2),A4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).
% prod_le_power
tff(fact_2901_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_2902_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set(A),K: nat] :
( pp(aa(set(A),bool,finite_finite(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_ab(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_ac(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_2903_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_2904_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))),insert3(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))),insert3(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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(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),Ys3)))),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)),Ys3))) )
=> ~ 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),Ys3))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_2905_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_ad(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_ac(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_2906_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_ae(A,A)),X3) = X3 ).
% map_ident
tff(fact_2907_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_af(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_2908_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_ag(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_2909_bool__assn__proper_I2_J,axiom,
pp(aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),bool,proper,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),bool))) ).
% bool_assn_proper(2)
tff(fact_2910_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_2911_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_ai(B,A)),A4) = one_one(A) ) ).
% prod.neutral_const
tff(fact_2912_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_aj(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_2913_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_2914_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_finite(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_2915_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_finite(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_al(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_al(B,fun(fun(B,A),fun(B,A)),A3),B2)),S) = one_one(A) ) ) ) ) ) ).
% prod.delta'
tff(fact_2916_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_2917_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: fun(B,bool),F2: fun(B,A),G: fun(B,A),S: set(B)] : image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_am(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image2(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),collect(B,P)))),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_an(fun(B,bool),fun(B,bool),P))))) ).
% if_image_distrib
tff(fact_2918_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_ao(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_2919_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_ap(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_2920_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_2921_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_2922_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_2923_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_2924_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F2: 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_aq(fun(B,A),fun(B,list(A)),F2)),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) ).
% concat_map_singleton
tff(fact_2925_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)) = complete_Sup_Sup(set(A),image2(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).
% set_concat
tff(fact_2926_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ab(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_distinct_length_eq
tff(fact_2927_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H2: 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_ar(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),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),H2),A4)) ) ).
% prod.distrib
tff(fact_2928_sngr__prec,axiom,
! [A: $tType] :
( heap(A)
=> precise(A,ref(A),aTP_Lamp_as(A,fun(ref(A),assn))) ) ).
% sngr_prec
tff(fact_2929_snga__prec,axiom,
! [A: $tType] :
( heap(A)
=> precise(list(A),array(A),aTP_Lamp_at(list(A),fun(array(A),assn))) ) ).
% snga_prec
tff(fact_2930_sorted__wrt__true,axiom,
! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_au(A,fun(A,bool)),Xs) ).
% sorted_wrt_true
tff(fact_2931_lambda__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ( aTP_Lamp_av(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).
% lambda_zero
tff(fact_2932_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_aw(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_2933_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F2: 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),F2),Xs))
<=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_ax(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F2),Xs) ) ).
% sorted_wrt_map
tff(fact_2934_list_Omap__ident,axiom,
! [A: $tType,T3: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ae(A,A)),T3) = T3 ).
% list.map_ident
tff(fact_2935_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_2936_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_2937_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_2938_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_2939_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_ba(set(A),fun(set(A),fun(A,bool)),A4),B3)) ).
% Int_def
tff(fact_2940_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_2941_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_2942_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_bb(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_2943_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_bc(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),aTP_Lamp_bc(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_2944_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)),complete_Sup_Sup(set(A),B3)),A4) = complete_Sup_Sup(set(A),image2(set(A),set(A),aTP_Lamp_bd(set(A),fun(set(A),set(A)),A4),B3)) ).
% Int_Union2
tff(fact_2945_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),complete_Sup_Sup(set(A),B3)) = complete_Sup_Sup(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_2946_bot__assn__def,axiom,
bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),assn,abs_assn,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
% bot_assn_def
tff(fact_2947_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_2948_list_Osel_I2_J,axiom,
! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).
% list.sel(2)
tff(fact_2949_map__tl,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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),F2),Xs)) ).
% map_tl
tff(fact_2950_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_2951_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_2952_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_2953_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),X: fun(B,A),Y: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),X))))
=> ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),Y))))
=> pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_bf(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I),X),Y)))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_2954_translation__subtract__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S3: set(A),T3: set(A)] : image2(A,A,aTP_Lamp_bg(A,fun(A,A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image2(A,A,aTP_Lamp_bg(A,fun(A,A),A3),S3)),image2(A,A,aTP_Lamp_bg(A,fun(A,A),A3),T3)) ) ).
% translation_subtract_Int
tff(fact_2955_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F2: fun(B,A),B3: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Sup_Sup(A,image2(B,A,F2,B3))),A3) = complete_Sup_Sup(A,image2(B,A,aa(A,fun(B,A),aTP_Lamp_bh(fun(B,A),fun(A,fun(B,A)),F2),A3),B3)) ) ).
% SUP_inf
tff(fact_2956_Sup__inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B3: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Sup_Sup(A,B3)),A3) = complete_Sup_Sup(A,image2(A,A,aTP_Lamp_bi(A,fun(A,A),A3),B3)) ) ).
% Sup_inf
tff(fact_2957_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F2: fun(B,A),B3: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),complete_Sup_Sup(A,image2(B,A,F2,B3))) = complete_Sup_Sup(A,image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),A3),F2),B3)) ) ).
% inf_SUP
tff(fact_2958_SUP__inf__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F2: fun(B,A),A4: set(B),G: fun(C,A),B3: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_Sup_Sup(A,image2(B,A,F2,A4))),complete_Sup_Sup(A,image2(C,A,G,B3))) = complete_Sup_Sup(A,image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_bl(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B3),A4)) ) ).
% SUP_inf_distrib2
tff(fact_2959_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_finite(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_bm(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_bn(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A4) ) ) ) ).
% prod.inter_filter
tff(fact_2960_sorted__map,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),Xs))
<=> sorted_wrt(B,aTP_Lamp_bo(fun(B,A),fun(B,fun(B,bool)),F2),Xs) ) ) ).
% sorted_map
tff(fact_2961_Int__UN__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType,A4: fun(B,set(A)),I: 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)),complete_Sup_Sup(set(A),image2(B,set(A),A4,I))),complete_Sup_Sup(set(A),image2(C,set(A),B3,J4))) = complete_Sup_Sup(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_bq(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A4),B3),J4),I)) ).
% Int_UN_distrib2
tff(fact_2962_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B3: set(A),A4: fun(B,set(A)),I: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),complete_Sup_Sup(set(A),image2(B,set(A),A4,I))) = complete_Sup_Sup(set(A),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_br(set(A),fun(fun(B,set(A)),fun(B,set(A))),B3),A4),I)) ).
% Int_UN_distrib
tff(fact_2963_UN__extend__simps_I4_J,axiom,
! [H8: $tType,G3: $tType,A4: fun(G3,set(H8)),C5: set(G3),B3: set(H8)] : aa(set(H8),set(H8),aa(set(H8),fun(set(H8),set(H8)),inf_inf(set(H8)),complete_Sup_Sup(set(H8),image2(G3,set(H8),A4,C5))),B3) = complete_Sup_Sup(set(H8),image2(G3,set(H8),aa(set(H8),fun(G3,set(H8)),aTP_Lamp_bs(fun(G3,set(H8)),fun(set(H8),fun(G3,set(H8))),A4),B3),C5)) ).
% UN_extend_simps(4)
tff(fact_2964_UN__extend__simps_I5_J,axiom,
! [I6: $tType,J5: $tType,A4: set(I6),B3: fun(J5,set(I6)),C5: set(J5)] : aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),A4),complete_Sup_Sup(set(I6),image2(J5,set(I6),B3,C5))) = complete_Sup_Sup(set(I6),image2(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_bt(set(I6),fun(fun(J5,set(I6)),fun(J5,set(I6))),A4),B3),C5)) ).
% UN_extend_simps(5)
tff(fact_2965_lists__eq__set,axiom,
! [A: $tType,A4: set(A)] : aa(set(A),set(list(A)),lists(A),A4) = collect(list(A),aTP_Lamp_bu(set(A),fun(list(A),bool),A4)) ).
% lists_eq_set
tff(fact_2966_length__nths,axiom,
! [A: $tType,Xs: list(A),I: 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),I)) = finite_card(nat,collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_bv(list(A),fun(set(nat),fun(nat,bool)),Xs),I))) ).
% length_nths
tff(fact_2967_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_bw(A,product_prod(A,A))),Xs) ).
% zip_same_conv_map
tff(fact_2968_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_2969_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F2,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,F2,X)),aa(A,B,F2,arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).
% arg_min_list.simps(2)
tff(fact_2970_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)),insert3(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).
% set_insort_insert
tff(fact_2971_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_2972_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_bx(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).
% sup_assn_def
tff(fact_2973_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_by(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P),Q)) ).
% inf_assn_def
tff(fact_2974_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_bz(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).
% product_concat_map
tff(fact_2975_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_2976_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_2977_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_2978_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_finite(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_ca(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A4) ) ) ) ).
% prod.inter_restrict
tff(fact_2979_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_finite(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_cb(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_2980_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_2981_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))
=> ( ! [Ys3: 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)),Ys3))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
=> ( ! [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,Ys3: 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),Ys3)))
=> ( 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),Ys3)))
=> ( 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)),Ys3))
=> 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),Ys3))) ) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).
% shuffles.pinduct
tff(fact_2982_nths__append,axiom,
! [A: $tType,L: list(A),L6: 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),L6)),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),L6),collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_cc(list(A),fun(set(nat),fun(nat,bool)),L),A4)))) ).
% nths_append
tff(fact_2983_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_cd(list(A),fun(nat,nat))),Xs),zero_zero(nat)) ).
% length_transpose
tff(fact_2984_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_ce(set(list(A)),fun(list(A),fun(A,bool)),Kl2),Kl)) ).
% Succ_def
tff(fact_2985_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F2)) = complete_Sup_Sup(set(A),image2(B,set(A),aTP_Lamp_cf(fun(B,list(A)),fun(B,set(A)),F2),aa(list(B),set(B),set2(B),Xs))) ).
% set_list_bind
tff(fact_2986_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_2987_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_2988_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_2989_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_ch(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).
% n_lists.simps(2)
tff(fact_2990_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_2991_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list(A),F2: 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),F2),A3)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),tl(A),A3)) ) ) ).
% list.map_sel(2)
tff(fact_2992_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F2: 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),F2),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_ci(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(N,M2)) ).
% enumerate_map_upt
tff(fact_2993_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_cj(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_2994_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_2995_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_ck(list(list(A)),fun(A,list(list(A))),Xss2)),Xs)) ).
% product_lists.simps(2)
tff(fact_2996_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_2997_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_2998_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_2999_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_3000_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_3001_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),G: fun(A,B),B3: set(A)] :
( inj_on(A,B,F2,A4)
=> ( inj_on(A,B,G,B3)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),image2(A,B,F2,A4)),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_cl(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),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_3002_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_cm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_3003_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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_cn(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),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),H2),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_3004_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_cm(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).
% prod.atMost_Suc_shift
tff(fact_3005_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_cm(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.lessThan_Suc_shift
tff(fact_3006_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_3007_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_co(A,fun(A,bool)),aTP_Lamp_cp(A,fun(A,bool))) ) ).
% sup_bot.semilattice_neutr_order_axioms
tff(fact_3008_map__upt__Suc,axiom,
! [A: $tType,F2: fun(nat,A),N: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),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,F2,zero_zero(nat))),aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),N))) ).
% map_upt_Suc
tff(fact_3009_finite__lists__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_cr(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_length_eq
tff(fact_3010_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_3011_horner__sum__foldr,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F2: 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),F2),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_cs(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A3)),Xs),zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_3012_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_ct(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).
% uminus_assn_def
tff(fact_3013_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [F2: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_cu(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B2,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_3014_fact__prod,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ac(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).
% fact_prod
tff(fact_3015_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_cv(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).
% set_n_lists
tff(fact_3016_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_cw(nat,nat)),upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = upt(M2,N) ).
% map_decr_upt
tff(fact_3017_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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_cx(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),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),C2),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(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_cx(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),minus_minus(set(B),S,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).
% prod.delta_remove
tff(fact_3018_prod_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite(B),I))
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> pp(aa(set(C),bool,finite_finite(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I))
=> ( ( 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),complete_Sup_Sup(set(C),image2(B,set(C),A4,I))) = 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_cy(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A4),G)),I) ) ) ) ) ) ).
% prod.UNION_disjoint
tff(fact_3019_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_cz(set(nat),fun(nat,bool),A4)))) ).
% nths_Cons
tff(fact_3020_UN__le__eq__Un0,axiom,
! [A: $tType,M: fun(nat,set(A)),N: nat] : complete_Sup_Sup(set(A),image2(nat,set(A),M,set_ord_atMost(nat,N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),complete_Sup_Sup(set(A),image2(nat,set(A),M,set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M,zero_zero(nat))) ).
% UN_le_eq_Un0
tff(fact_3021_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_cm(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.atMost_shift
tff(fact_3022_finite__lists__length__le,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_da(set(A),fun(nat,fun(list(A),bool)),A4),N)))) ) ).
% finite_lists_length_le
tff(fact_3023_lists__of__len__fin2,axiom,
! [A: $tType,P: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),P))
=> pp(aa(set(list(A)),bool,finite_finite(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_db(nat,fun(list(A),bool),N))))) ) ).
% lists_of_len_fin2
tff(fact_3024_lists__of__len__fin1,axiom,
! [A: $tType,P: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),P))
=> pp(aa(set(list(A)),bool,finite_finite(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_dc(nat,fun(list(A),bool),N))))) ) ).
% lists_of_len_fin1
tff(fact_3025_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_3026_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,C2: 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)),C2))
=> ( image2(A,A,aTP_Lamp_dd(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( image2(A,A,aTP_Lamp_dd(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( image2(A,A,aTP_Lamp_dd(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_3027_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_de(A,fun(A,fun(A,A)),M2),C2),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))
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_de(A,fun(A,fun(A,A)),M2),C2),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)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_de(A,fun(A,fun(A,A)),M2),C2),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)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
tff(fact_3028_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_df(A,fun(A,fun(A,A)),M2),C2),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))
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_df(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
=> ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_df(A,fun(A,fun(A,A)),M2),C2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2),C2),minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_3029_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_3030_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_3031_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),A3: B,B2: fun(B,A),C2: A] :
( pp(aa(set(B),bool,finite_finite(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_dg(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),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),C2),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_dg(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),finite_card(B,S)) ) ) ) ) ) ).
% prod_gen_delta
tff(fact_3032_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_3033_pochhammer__prod__rev,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dh(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).
% pochhammer_prod_rev
tff(fact_3034_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_3035_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_3036_card__lists__length__eq,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_cr(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_3037_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
=> ( divide_divide(nat,semiring_char_0_fact(nat,M2),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ac(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M2)) ) ) ).
% fact_div_fact
tff(fact_3038_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))),insert3(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(2)
tff(fact_3039_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))),insert3(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(1)
tff(fact_3040_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_di(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).
% prod.in_pairs
tff(fact_3041_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_di(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% prod.in_pairs_0
tff(fact_3042_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_3043_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_dj(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact
tff(fact_3044_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_dj(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact'
tff(fact_3045_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))),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)))),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_3046_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_3047_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P7: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P7))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P7))
=> ( 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_dk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,P7)) = 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_dl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,minus_minus(nat,P7,aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% prod.zero_middle
tff(fact_3048_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_dm(A,fun(nat,fun(A,A)),A3),zero_zero(nat),minus_minus(nat,N,one_one(nat)),one_one(A)) ) ) ) ) ).
% pochhammer_code
tff(fact_3049_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_3050_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_do(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).
% transpose_rectangle
tff(fact_3051_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_3052_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_dp(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_3053_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_3054_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))),insert3(list(A)),nil(A)),bot_bot(set(list(A))))) = image2(A,list(A),aTP_Lamp_dq(A,list(A)),A4) ).
% set_Cons_sing_Nil
tff(fact_3055_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_3056_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_bz(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).
% product_code
tff(fact_3057_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list(list(A)),I2: 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),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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))),filter(list(A)),aTP_Lamp_dr(nat,fun(list(A),bool),I2)),Xs))))
=> ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I2) ) ) ) ) ).
% nth_nth_transpose_sorted
tff(fact_3058_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_ds(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_3059_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_3060_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_3061_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)),filter(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),Q),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_dt(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),Xs) ).
% filter_filter
tff(fact_3062_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)),filter(A),P),Xs) = Xs ) ) ).
% filter_True
tff(fact_3063_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)),filter(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)),filter(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),Ys)) ).
% filter_append
tff(fact_3064_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)),filter(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),Xs) ) ) ).
% remove1_filter_not
tff(fact_3065_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)),filter(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),Xs) ) ) ).
% removeAll_filter_not
tff(fact_3066_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)),filter(A),P),Xs)) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_du(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).
% set_filter
tff(fact_3067_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)),filter(A),P),Xs) = nil(A) ) ) ).
% filter_False
tff(fact_3068_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_dv(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_3069_sum__zero__power,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A4: set(nat),C2: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite(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_dw(fun(nat,A),fun(nat,A),C2)),A4) = aa(nat,A,C2,zero_zero(nat)) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite(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_dw(fun(nat,A),fun(nat,A),C2)),A4) = zero_zero(A) ) ) ) ) ).
% sum_zero_power
tff(fact_3070_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A4: set(B),F2: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite(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_dx(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P)),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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_3071_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A4: set(B),P: fun(B,bool),F2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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_dy(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F2)),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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_3072_sum__zero__power_H,axiom,
! [A: $tType] :
( field(A)
=> ! [A4: set(nat),C2: fun(nat,A),D3: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite(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_dz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A4) = divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite(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_dz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A4) = zero_zero(A) ) ) ) ) ).
% sum_zero_power'
tff(fact_3073_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ea(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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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)),filter(A),P),nil(A)) = nil(A) ).
% filter.simps(1)
tff(fact_3081_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)),filter(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_3082_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)),filter(A),P),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),Q),Ys) ) ) ) ).
% filter_cong
tff(fact_3083_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)),filter(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)),filter(A),P),Xs)) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(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)),filter(A),P),Xs) ) ) ) ).
% filter.simps(2)
tff(fact_3084_sorted__wrt__filter,axiom,
! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,F2,Xs)
=> sorted_wrt(A,F2,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),Xs)) ) ).
% sorted_wrt_filter
tff(fact_3085_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)),filter(A),P),Xs))) ) ).
% distinct_filter
tff(fact_3086_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)),filter(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),aa(list(A),list(A),rev(A),Xs)) ).
% rev_filter
tff(fact_3087_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)),filter(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)),filter(A),Q),Xs)) ).
% filter_remove1
tff(fact_3088_sum__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_0(B)
=> ! [F2: 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),F2),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_ec(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F2),G),B3)),A4) ) ).
% sum_product
tff(fact_3089_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),A4: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_ed(fun(B,A),fun(A,fun(B,A)),F2),R2)),A4) ) ).
% sum_distrib_right
tff(fact_3090_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [R2: A,F2: fun(B,A),A4: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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_ee(A,fun(fun(B,A),fun(B,A)),R2),F2)),A4) ) ).
% sum_distrib_left
tff(fact_3091_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)),filter(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aTP_Lamp_ef(fun(A,bool),fun(A,bool),P)),Xs)))) ).
% partition_in_shuffles
tff(fact_3092_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)),filter(A),aa(A,fun(A,bool),aTP_Lamp_eg(A,fun(A,bool)),X)) ).
% removeAll_filter_not_eq
tff(fact_3093_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)),filter(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_3094_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)),filter(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_3095_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)),filter(A),P),Xs))),aa(list(A),set(A),set2(A),Xs))) ).
% filter_is_subset
tff(fact_3096_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)),filter(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_filter_le
tff(fact_3097_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs)))) ) ).
% distinct_map_filter
tff(fact_3098_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)),filter(A),P),Xs)) = last(A,Xs) ) ) ) ).
% last_filter
tff(fact_3099_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)),filter(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)),filter(A),aTP_Lamp_ef(fun(A,bool),fun(A,bool),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).
% sum_length_filter_compl
tff(fact_3100_filter__concat,axiom,
! [A: $tType,P7: fun(A,bool),Xs: list(list(A))] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P7),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)),filter(A),P7)),Xs)) ).
% filter_concat
tff(fact_3101_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)),filter(A),aTP_Lamp_a(set(A),fun(A,bool),A4)),Xs)) ).
% inter_set_filter
tff(fact_3102_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)),filter(A),aa(list(A),fun(A,bool),aTP_Lamp_eh(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs)),Xs)) ) ).
% sorted_same
tff(fact_3103_filter__shuffles,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : image2(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(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)),filter(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),Ys)) ).
% filter_shuffles
tff(fact_3104_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))),filter(list(A)),aTP_Lamp_ei(list(A),bool)),Xs)) = aa(list(list(A)),list(A),concat(A),Xs) ).
% concat_filter_neq_Nil
tff(fact_3105_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)),filter(A),aTP_Lamp_ej(set(A),fun(A,bool),A4)),Xs)) ).
% union_coset_filter
tff(fact_3106_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)),filter(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% length_filter_less
tff(fact_3107_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)),filter(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)),filter(A),P),Vs3) ) ) ) ).
% Cons_eq_filterD
tff(fact_3108_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)),filter(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)),filter(A),P),Vs3) ) ) ) ).
% filter_eq_ConsD
tff(fact_3109_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)),filter(A),P),Ys) )
<=> ? [Us2: list(A),Vs2: list(A)] :
( ( 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),X),Vs2)) )
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Us2)))
=> ~ 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)),filter(A),P),Vs2) ) ) ) ).
% Cons_eq_filter_iff
tff(fact_3110_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)),filter(A),P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
<=> ? [Us2: list(A),Vs2: list(A)] :
( ( 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),X),Vs2)) )
& ! [X4: A] :
( pp(member2(A,X4,aa(list(A),set(A),set2(A),Us2)))
=> ~ 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)),filter(A),P),Vs2) ) ) ) ).
% filter_eq_Cons_iff
tff(fact_3111_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs))) ) ) ).
% sorted_filter
tff(fact_3112_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y),aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(A,fun(A,bool),aTP_Lamp_ek(fun(A,B),fun(A,fun(A,bool)),F2),Y)),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(A,fun(A,bool),fequal(A),Y)),Xs) ) ) ).
% inj_on_filter_key_eq
tff(fact_3113_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_el(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F2),G),Xs)),Xs))) ) ).
% sorted_map_same
tff(fact_3114_length__filter__conv__card,axiom,
! [A: $tType,P7: 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)),filter(A),P7),Xs)) = finite_card(nat,collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_em(fun(A,bool),fun(list(A),fun(nat,bool)),P7),Xs))) ).
% length_filter_conv_card
tff(fact_3115_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_finite(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_en(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A4) ) ) ) ).
% sum.inter_restrict
tff(fact_3116_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,I: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_eo(A,fun(nat,fun(nat,A)),X),M2)),I) = 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)),I)) ) ).
% sum_power_add
tff(fact_3117_filter__in__nths,axiom,
! [A: $tType,Xs: list(A),S3: set(nat)] :
( pp(aa(list(A),bool,distinct(A),Xs))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(set(nat),fun(A,bool),aTP_Lamp_ep(list(A),fun(set(nat),fun(A,bool)),Xs),S3)),Xs) = aa(set(nat),list(A),aa(list(A),fun(set(nat),list(A)),nths(A),Xs),S3) ) ) ).
% filter_in_nths
tff(fact_3118_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)),filter(A),aTP_Lamp_a(set(A),fun(A,bool),A4)),Xs)) ).
% minus_coset_filter
tff(fact_3119_filter__eq__snocD,axiom,
! [A: $tType,P: fun(A,bool),L: list(A),L6: list(A),X: A] :
( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L6),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_3120_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)),filter(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_eq(fun(A,bool),fun(A,fun(list(A),list(A))),P)),Xs),nil(A)) ).
% filter_conv_foldr
tff(fact_3121_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),F2: 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,F2,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),F2),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_3122_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),K5: A,F2: 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,F2,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),F2),A4))) ) ) ).
% sum_bounded_below
tff(fact_3123_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),T))
=> ( pp(aa(set(B),bool,finite_finite(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H2,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,H2,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),H2),T) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
tff(fact_3124_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_3125_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_finite(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_3126_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)),insert3(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)),filter(A),aTP_Lamp_er(A,fun(A,bool),Y)),Xs)) ).
% set_minus_filter_out
tff(fact_3127_listset_Osimps_I2_J,axiom,
! [A: $tType,A4: set(A),As8: 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),As8)) = 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),As8)) ).
% listset.simps(2)
tff(fact_3128_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)),filter(A),aTP_Lamp_es(list(A),fun(A,bool),Xs)),Zs) = Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
tff(fact_3129_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)),filter(A),aTP_Lamp_et(list(A),fun(A,bool),Xs)),Zs) = Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
tff(fact_3130_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)),filter(A),aTP_Lamp_es(list(A),fun(A,bool),Ys)),Zs) = Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
tff(fact_3131_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)),filter(A),aTP_Lamp_et(list(A),fun(A,bool),Ys)),Zs) = Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
tff(fact_3132_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(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_eu(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),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),H2),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_3133_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)),filter(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_em(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).
% filter_eq_nths
tff(fact_3134_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ev(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).
% sum_bounds_lt_plus1
tff(fact_3135_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ew(nat,fun(nat,nat),N)),set_ord_atMost(nat,M2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),one_one(nat))),M2) ).
% choose_rising_sum(2)
tff(fact_3136_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ew(nat,fun(nat,nat),N)),set_ord_atMost(nat,M2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).
% choose_rising_sum(1)
tff(fact_3137_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_3138_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [A4: set(B),B3: set(B),F2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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),F2),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% sum_Un
tff(fact_3139_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_3140_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_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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_3141_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A4: set(A),B3: set(A),F2: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite(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),F2),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),F2),minus_minus(set(A),A4,B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),minus_minus(set(A),B3,A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).
% sum_Un2
tff(fact_3142_sum_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M2: nat,N: nat,G: fun(nat,A),P7: 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),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P7))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P7)))) ) ) ) ).
% sum.ub_add_nat
tff(fact_3143_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_ex(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_ey(list(B),fun(nat,nat))),aa(list(list(B)),list(list(B)),aa(fun(list(B),bool),fun(list(list(B)),list(list(B))),filter(list(B)),aTP_Lamp_ez(list(B),bool)),Xss2)),zero_zero(nat)))) ).
% transpose_aux_max
tff(fact_3144_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: set(B),F2: fun(B,A),B2: A] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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_fa(fun(B,A),fun(A,fun(B,A)),F2),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_fb(fun(B,A),fun(A,fun(B,bool)),F2),B2))))),divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),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_fc(fun(B,A),fun(A,fun(B,bool)),F2),B2)))),B2)) ) ) ) ).
% sum_div_partition
tff(fact_3145_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)),filter(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_3146_sum_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite(B),I))
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> pp(aa(set(C),bool,finite_finite(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I))
=> ( ( 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),complete_Sup_Sup(set(C),image2(B,set(C),A4,I))) = 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_fd(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A4),G)),I) ) ) ) ) ) ).
% sum.UNION_disjoint
tff(fact_3147_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_3148_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_3149_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_3150_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_cd(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))),filter(list(A)),aTP_Lamp_ei(list(A),bool)),Xs)) ).
% transpose_max_length
tff(fact_3151_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_fe(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_3152_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add(A)
& semiring_1(A) )
=> ! [A4: set(B),F2: 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,F2,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),F2),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_3153_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
( ! [I3: A] :
( pp(member2(A,I3,I))
=> 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),I) = one_one(B) )
=> ( ! [I3: A] :
( pp(member2(A,I3,I))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(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),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_ff(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I),B2))),Delta)) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_3154_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
( pp(aa(set(A),bool,finite_finite(A),S))
=> ( pp(aa(set(B),bool,finite_finite(B),R))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(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_fg(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_fi(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_3155_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_3156_sum__natinterval__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M2: nat,N: nat,F2: 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),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fj(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = minus_minus(A,aa(nat,A,F2,M2),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fj(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) ) ) ) ).
% sum_natinterval_diff
tff(fact_3157_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_fk(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_3158_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_fl(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,N))) ) ).
% power_diff_sumr2
tff(fact_3159_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_3160_sum__telescope_H_H,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M2: nat,N: nat,F2: 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),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = minus_minus(A,aa(nat,A,F2,N),aa(nat,A,F2,M2)) ) ) ) ).
% sum_telescope''
tff(fact_3161_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_3162_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_fn(nat,fun(nat,bool),N))) ) ).
% mask_eq_sum_exp
tff(fact_3163_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_3164_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_fo(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq'
tff(fact_3165_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_fp(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_3166_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_fq(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% binomial_ring
tff(fact_3167_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_fr(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% pochhammer_binomial_sum
tff(fact_3168_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F2: 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),F2),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_fs(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum
tff(fact_3169_nth__transpose,axiom,
! [A: $tType,I2: nat,Xs: list(list(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_ft(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter(list(A)),aTP_Lamp_dr(nat,fun(list(A),bool),I2)),Xs)) ) ) ).
% nth_transpose
tff(fact_3170_sum_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [P7: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P7))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P7))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,P7)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,minus_minus(nat,P7,aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% sum.zero_middle
tff(fact_3171_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_fw(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_3172_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_fx(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_fy(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_3173_sum__power2,axiom,
! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K),one_one(nat)) ).
% sum_power2
tff(fact_3174_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ac(nat,nat)),set_or7035219750837199246ssThan(nat,M2,N)) = divide_divide(nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),minus_minus(nat,N,one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),minus_minus(nat,M2,one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% Sum_Ico_nat
tff(fact_3175_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))),filter(list(A)),aTP_Lamp_ei(list(A),bool)),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)) ) ) ) )
=> pp(aa(list(A),bool,distinct(A),aa(list(list(A)),list(A),concat(A),Xs))) ) ) ) ).
% distinct_concat'
tff(fact_3176_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_fz(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_3177_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_3178_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_3179_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_fx(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_ga(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_3180_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_gb(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_3181_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ac(nat,nat)),set_or1337092689740270186AtMost(nat,M2,N)) = divide_divide(nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),minus_minus(nat,M2,one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% Sum_Icc_nat
tff(fact_3182_binomial__r__part__sum,axiom,
! [M2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)),one_one(nat)))),set_ord_atMost(nat,M2)) = 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))),M2)) ).
% binomial_r_part_sum
tff(fact_3183_choose__linear__sum,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gc(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),minus_minus(nat,N,one_one(nat)))) ).
% choose_linear_sum
tff(fact_3184_card__lists__length__le,axiom,
! [A: $tType,A4: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_da(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_3185_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_3186_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_gd(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_3187_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_3188_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_3189_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_ge(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_3190_transpose__column__length,axiom,
! [A: $tType,Xs: list(list(A)),I2: 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),I2),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))),filter(list(A)),aTP_Lamp_dr(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).
% transpose_column_length
tff(fact_3191_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_3192_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [R2: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gf(A,fun(nat,A),R2)),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,R2),aa(nat,nat,suc,M2))) ) ).
% gchoose_row_sum_weighted
tff(fact_3193_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_3194_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_3195_transpose__column,axiom,
! [A: $tType,Xs: list(list(A)),I2: 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),I2),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_ft(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter(list(A)),aTP_Lamp_dr(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).
% transpose_column
tff(fact_3196_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_gf(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_3197_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_gg(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_3198_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_gh(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))),filter(list(A)),aTP_Lamp_ei(list(A),bool)),Xss2)) ).
% transpose_aux_filter_tail
tff(fact_3199_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_gi(A,A),N,zero_zero(A)) ) ).
% of_nat_code
tff(fact_3200_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_3201_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_gj(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))),filter(list(A)),aTP_Lamp_ei(list(A),bool)),Xss2)) ).
% transpose_aux_filter_head
tff(fact_3202_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_gk(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).
% divmod_algorithm_code(6)
tff(fact_3203_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_gl(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).
% divmod_algorithm_code(5)
tff(fact_3204_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: fun(B,C),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H2,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,H2,F1)),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_gm(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H2),F22)),List) ).
% list.case_distrib
tff(fact_3205_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_go(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_3206_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_3207_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_3208_card__eq__sum,axiom,
! [A: $tType,A4: set(A)] : finite_card(A,A4) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_gp(A,nat)),A4) ).
% card_eq_sum
tff(fact_3209_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_gq(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_3210_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))),filter(product_prod(A,nat)),aTP_Lamp_gr(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))),filter(product_prod(A,nat)),aTP_Lamp_gs(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_3211_sum__eq__1__iff,axiom,
! [A: $tType,A4: set(A),F2: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = one_one(nat) )
<=> ? [X4: A] :
( pp(member2(A,X4,A4))
& ( aa(A,nat,F2,X4) = one_one(nat) )
& ! [Xa2: A] :
( pp(member2(A,Xa2,A4))
=> ( ( X4 != Xa2 )
=> ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_3212_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_gt(A,fun(list(A),list(A)))),List) ).
% tl_def
tff(fact_3213_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_gu(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).
% tl_append
tff(fact_3214_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F2: 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),F2),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_gv(fun(C,A),fun(C,fun(B,product_prod(A,B))),F2))),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_3215_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F2: 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),F2),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_gw(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),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_3216_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),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_gx(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),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),F2),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_3217_map2__map__map,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,H2: fun(B,fun(C,A)),F2: 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,H2)),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),F2),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_gy(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H2),F2),G)),Xs) ).
% map2_map_map
tff(fact_3218_semilattice__map2,axiom,
! [A: $tType,F2: fun(A,fun(A,A))] :
( semilattice(A,F2)
=> semilattice(list(A),aTP_Lamp_gz(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),F2)) ) ).
% semilattice_map2
tff(fact_3219_nths__shift__lemma,axiom,
! [A: $tType,A4: set(nat),Xs: list(A),I2: 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))),filter(product_prod(A,nat)),aTP_Lamp_ha(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(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),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))),filter(product_prod(A,nat)),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_hb(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A4),I2)),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_3220_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))),filter(product_prod(A,nat)),aTP_Lamp_ha(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_3221_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F2),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_hc(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F2),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_3222_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F2),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_hd(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F2),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_3223_sum__Un__nat,axiom,
! [A: $tType,A4: set(A),B3: set(A),F2: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(A),B3))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),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),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))) ) ) ) ).
% sum_Un_nat
tff(fact_3224_sum__count__set,axiom,
! [A: $tType,Xs: list(A),X6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
=> ( pp(aa(set(A),bool,finite_finite(A),X6))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).
% sum_count_set
tff(fact_3225_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_3226_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I: set(A),A4: fun(A,set(B))] :
( pp(aa(set(A),bool,finite_finite(A),I))
=> ( ! [X2: A] :
( pp(member2(A,X2,I))
=> pp(aa(set(B),bool,finite_finite(B),aa(A,set(B),A4,X2))) )
=> ( ! [X2: A] :
( pp(member2(A,X2,I))
=> ! [Xa4: A] :
( pp(member2(A,Xa4,I))
=> ( ( 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,complete_Sup_Sup(set(B),image2(A,set(B),A4,I))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_he(fun(A,set(B)),fun(A,nat),A4)),I) ) ) ) ) ).
% card_UN_disjoint
tff(fact_3227_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_hf(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).
% min_list.simps
tff(fact_3228_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_gj(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_gh(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ) ) ) ).
% transpose.elims
tff(fact_3229_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_gj(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_gh(A,fun(list(A),list(list(A)))))),Xss2))))) ).
% transpose.simps(3)
tff(fact_3230_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_3231_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_3232_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_hg(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).
% zip_Cons1
tff(fact_3233_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_hh(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).
% zip_Cons
tff(fact_3234_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hi(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_3235_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))
=> ( ! [Ys3: 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)),Ys3))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
=> ( ! [X2: A,Xs2: list(A),Ys3: 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)),Ys3))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys3),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)),Ys3)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).
% splice.pinduct
tff(fact_3236_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hj(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_3237_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_3238_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_3239_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N))
=> ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_hk(int,int)),set_or1337092689740270186AtMost(int,M2,N)) = divide_divide(int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),M2),minus_minus(int,M2,one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).
% Sum_Icc_int
tff(fact_3240_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_hl(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_3241_card__UNION,axiom,
! [A: $tType,A4: set(set(A))] :
( pp(aa(set(set(A)),bool,finite_finite(set(A)),A4))
=> ( ! [X2: set(A)] :
( pp(member2(set(A),X2,A4))
=> pp(aa(set(A),bool,finite_finite(A),X2)) )
=> ( finite_card(A,complete_Sup_Sup(set(A),A4)) = nat2(aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_hm(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_hn(set(set(A)),fun(set(set(A)),bool),A4)))) ) ) ) ).
% card_UNION
tff(fact_3242_normalize__def,axiom,
! [P7: product_prod(int,int)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P7)))
=> ( normalize(P7) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P7),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P7)),aa(product_prod(int,int),int,product_snd(int,int),P7)))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P7),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P7)),aa(product_prod(int,int),int,product_snd(int,int),P7)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P7)))
=> ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P7) = zero_zero(int) )
=> ( normalize(P7) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ) )
& ( ( aa(product_prod(int,int),int,product_snd(int,int),P7) != zero_zero(int) )
=> ( normalize(P7) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P7),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P7)),aa(product_prod(int,int),int,product_snd(int,int),P7))))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P7),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P7)),aa(product_prod(int,int),int,product_snd(int,int),P7))))) ) ) ) ) ) ).
% normalize_def
tff(fact_3243_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,complete_Sup_Sup(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_3244_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_ei(list(A),bool)),Xs) ) ) ).
% transpose_transpose
tff(fact_3245_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType,Xs: list(A),F2: fun(A,nat)] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),nat,size_list(A,F2),Xs) = zero_zero(nat) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),nat,size_list(A,F2),Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).
% Nitpick.size_list_simp(1)
tff(fact_3246_remdups__upt,axiom,
! [M2: nat,N: nat] : aa(list(nat),list(nat),remdups(nat),upt(M2,N)) = upt(M2,N) ).
% remdups_upt
tff(fact_3247_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_3248_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_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_gcd__1__int,axiom,
! [M2: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),one_one(int)) = one_one(int) ).
% gcd_1_int
tff(fact_3256_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_3257_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_3258_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)),insert3(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_3259_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_3260_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_3261_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_3262_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_3263_size__list__append,axiom,
! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F2),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,F2),Xs)),aa(list(A),nat,size_list(A,F2),Ys)) ).
% size_list_append
tff(fact_3264_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)),image2(B,set(A),B3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(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)),image2(B,set(A),B3,A4))) ).
% INT_insert
tff(fact_3265_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set(A),A4: fun(A,set(B)),B3: set(B)] :
( ( ( C5 = bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ho(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3),C5)) = top_top(set(B)) ) )
& ( ( C5 != bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ho(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3),C5)) = 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)),image2(A,set(B),A4,C5))),B3) ) ) ) ).
% INT_simps(1)
tff(fact_3266_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set(D),A4: set(C),B3: fun(D,set(C))] :
( ( ( C5 = bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_hp(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3),C5)) = top_top(set(C)) ) )
& ( ( C5 != bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_hp(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3),C5)) = 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)),image2(D,set(C),B3,C5))) ) ) ) ).
% INT_simps(2)
tff(fact_3267_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X12: A,X24: A,Xs: 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_hq(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_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_remdups_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),remdups(A),nil(A)) = nil(A) ).
% remdups.simps(1)
tff(fact_3275_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),Xs)) ).
% remdups_map_remdups
tff(fact_3276_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_3277_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_3278_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)),filter(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),aa(list(A),list(A),remdups(A),Xs)) ).
% remdups_filter
tff(fact_3279_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: fun(bool,A)] : aa(set(A),A,complete_Inf_Inf(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_3280_inj__split__Cons,axiom,
! [A: $tType,X6: 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_cg(list(A),fun(A,list(A)))),X6) ).
% inj_split_Cons
tff(fact_3281_INT__bool__eq,axiom,
! [A: $tType,A4: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_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_3290_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_hr(A,fun(list(A),bool))),List)) ) ).
% list.disc_eq_case(2)
tff(fact_3291_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_hs(A,fun(list(A),bool))),List)) ) ).
% list.disc_eq_case(1)
tff(fact_3292_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_3293_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_3294_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [K: B,I: set(B),A4: fun(B,A)] :
( pp(member2(B,K,I))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,A4,K)),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,A4,I))) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,A4,I)) ) ) ) ).
% INF_absorb
tff(fact_3295_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: 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),image2(B,A,F2,A4))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G,A4))) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ht(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A4)) ) ).
% INF_inf_distrib
tff(fact_3296_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_3297_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))),insert3(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_3298_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_3299_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I: set(A),A4: fun(A,set(B))] :
( pp(member2(A,K,I))
=> ( 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)),image2(A,set(B),A4,I))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),image2(A,set(B),A4,I)) ) ) ).
% INT_absorb
tff(fact_3300_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),I: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_hu(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3),I)) = 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)),image2(B,set(A),A4,I))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(B,set(A),B3,I))) ).
% INT_Int_distrib
tff(fact_3301_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A4: fun(B,set(A)),B3: fun(B,set(A)),C5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_hu(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B3),C5)) = 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)),image2(B,set(A),A4,C5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(B,set(A),B3,C5))) ).
% Int_Inter_image
tff(fact_3302_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B9: set(set(A)),A4: set(A)] :
( ( ( B9 = 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)),B9)) = A4 ) )
& ( ( B9 != 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)),B9)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B9)) ) ) ) ).
% Int_Inter_eq(1)
tff(fact_3303_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B9: set(set(A)),A4: set(A)] :
( ( ( B9 = 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)),B9)),A4) = A4 ) )
& ( ( B9 != 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)),B9)),A4) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(set(A),set(A),aTP_Lamp_bd(set(A),fun(set(A),set(A)),A4),B9)) ) ) ) ).
% Int_Inter_eq(2)
tff(fact_3304_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_3305_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_mult_unit1
tff(fact_3306_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_mult_unit2
tff(fact_3307_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),divide_divide(A,C2,A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_div_unit2
tff(fact_3308_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_div_unit1
tff(fact_3309_finite__Inf__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_3310_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_3311_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_3312_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F2: 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,F2,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(list(A),nat,size_list(A,F2),Xs))) ) ) ).
% size_list_estimation
tff(fact_3313_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F2: 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,F2,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F2),Xs))) ) ) ).
% size_list_estimation'
tff(fact_3314_size__list__pointwise,axiom,
! [A: $tType,Xs: list(A),F2: 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,F2,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,F2),Xs)),aa(list(A),nat,size_list(A,G),Xs))) ) ).
% size_list_pointwise
tff(fact_3315_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_3316_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_3317_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: set(B),F2: fun(B,A),X: A] :
( ( I != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_hv(fun(B,A),fun(A,fun(B,A)),F2),X),I)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,I))),X) ) ) ) ).
% INF_inf_const2
tff(fact_3318_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: set(B),X: A,F2: fun(B,A)] :
( ( I != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hw(A,fun(fun(B,A),fun(B,A)),X),F2),I)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,I))) ) ) ) ).
% INF_inf_const1
tff(fact_3319_INF__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,A4))) ) ).
% INF_insert
tff(fact_3320_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),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),image2(B,A,M,A4))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,M,B3))) ) ).
% INF_union
tff(fact_3321_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set(A),A4: fun(A,set(B)),B3: set(B)] :
( ( ( C5 = 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)),image2(A,set(B),A4,C5))),B3) = B3 ) )
& ( ( C5 != 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)),image2(A,set(B),A4,C5))),B3) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ho(fun(A,set(B)),fun(set(B),fun(A,set(B))),A4),B3),C5)) ) ) ) ).
% INT_extend_simps(1)
tff(fact_3322_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set(D),A4: set(C),B3: fun(D,set(C))] :
( ( ( C5 = 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)),image2(D,set(C),B3,C5))) = A4 ) )
& ( ( C5 != 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)),image2(D,set(C),B3,C5))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_hp(set(C),fun(fun(D,set(C)),fun(D,set(C))),A4),B3),C5)) ) ) ) ).
% INT_extend_simps(2)
tff(fact_3323_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)),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)),image2(B,set(A),M,A4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image2(B,set(A),M,B3))) ).
% INT_Un
tff(fact_3324_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_hx(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_3325_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_3326_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_3327_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_hy(list(A),fun(A,bool),Xs)),Xs) = aa(list(A),list(A),butlast(A),Xs) ) ) ).
% takeWhile_not_last
tff(fact_3328_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_3329_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_3330_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F2: 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_hz(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F2))),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),F2),Ys),B2) ) ) ).
% foldr_snd_zip
tff(fact_3331_lenlex__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_ia(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).
% lenlex_conv
tff(fact_3332_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_3333_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),Xs: list(B),T3: 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),F2),Xs)))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),aa(A,fun(B,bool),aTP_Lamp_ib(fun(B,A),fun(A,fun(B,bool)),F2),T3)),Xs) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),aa(A,fun(B,bool),aTP_Lamp_ib(fun(B,A),fun(A,fun(B,bool)),F2),T3)),Xs) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
tff(fact_3334_List_Olexordp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( lexordp(A,R2,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,R2))))) ) ).
% List.lexordp_def
tff(fact_3335_listrel1p__def,axiom,
! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( listrel1p(A,R2,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,R2))))) ) ).
% listrel1p_def
tff(fact_3336_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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_ig(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),F2),Xs)),Xs) ) ).
% sort_key_by_quicksort_code
tff(fact_3337_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)),insert3(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_3338_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_gj(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_gh(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ).
% transpose.psimps(3)
tff(fact_3339_gcd__1__nat,axiom,
! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),one_one(nat)) = one_one(nat) ).
% gcd_1_nat
tff(fact_3340_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_ac(nat,nat)),upt(M2,N)) = upt(M2,N) ).
% sort_upt
tff(fact_3341_sort__upto,axiom,
! [I2: int,J: int] : aa(list(int),list(int),aa(fun(int,int),fun(list(int),list(int)),linorder_sort_key(int,int),aTP_Lamp_hk(int,int)),upto(I2,J)) = upto(I2,J) ).
% sort_upto
tff(fact_3342_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),nil(B)) = nil(B) ) ).
% sort_key_simps(1)
tff(fact_3343_set__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),Xs)) = aa(list(B),set(B),set2(B),Xs) ) ).
% set_sort
tff(fact_3344_length__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).
% length_sort
tff(fact_3345_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),Xs)))
<=> pp(aa(list(B),bool,distinct(B),Xs)) ) ) ).
% distinct_sort
tff(fact_3346_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [C2: 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_ih(B,fun(A,B),C2)),Xs) = Xs ) ).
% sort_key_const
tff(fact_3347_filter__sort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [P: fun(B,bool),F2: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) = aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs)) ) ).
% filter_sort
tff(fact_3348_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_3349_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F2: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(B,fun(A,bool),aTP_Lamp_ii(fun(A,B),fun(B,fun(A,bool)),F2),K)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(B,fun(A,bool),aTP_Lamp_ii(fun(A,B),fun(B,fun(A,bool)),F2),K)),Xs) ) ).
% sort_key_stable
tff(fact_3350_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_3351_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_3352_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_3353_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs))) ) ).
% sorted_sort_key
tff(fact_3354_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_3355_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_gh(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_3356_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_gj(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_gh(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_3357_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_ij(A,fun(list(A),A))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(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_3358_rat__inverse__code,axiom,
! [P7: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P7)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ik(int,fun(int,product_prod(int,int)))),quotient_of(P7)) ).
% rat_inverse_code
tff(fact_3359_quicksort_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Y: 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),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aTP_Lamp_il(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))),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(A,fun(A,bool),ord_less_eq(A),X2)),Xs2)))) ) ) ) ) ) ).
% quicksort.elims
tff(fact_3360_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: 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),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aTP_Lamp_il(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))),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aa(A,fun(A,bool),ord_less_eq(A),X)),Xs)))) ) ).
% quicksort.simps(2)
tff(fact_3361_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_in(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).
% set_relcomp
tff(fact_3362_inverse__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : aa(A,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),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) ) ).
% inverse_mult_distrib
tff(fact_3363_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_3364_inverse__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A] :
( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
<=> ( X = one_one(A) ) ) ) ).
% inverse_eq_1_iff
tff(fact_3365_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),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).
% right_inverse
tff(fact_3366_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),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).
% left_inverse
tff(fact_3367_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : aa(A,A,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_3368_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : aa(A,A,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_3369_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),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_3370_quicksort_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_quicksort(A,nil(A)) = nil(A) ) ) ).
% quicksort.simps(1)
tff(fact_3371_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,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),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_3372_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) )
=> ( aa(A,A,inverse_inverse(A),A3) = B2 ) ) ) ).
% inverse_unique
tff(fact_3373_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),aa(A,A,inverse_inverse(A),B2)) ) ).
% field_class.field_divide_inverse
tff(fact_3374_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),aa(A,A,inverse_inverse(A),B2)) ) ).
% divide_inverse
tff(fact_3375_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),aa(A,A,inverse_inverse(A),B2)),A3) ) ).
% divide_inverse_commute
tff(fact_3376_inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = divide_divide(A,one_one(A),A3) ) ).
% inverse_eq_divide
tff(fact_3377_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),aa(A,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),aa(A,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_3378_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)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).
% power_mult_inverse_distrib
tff(fact_3379_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),aa(A,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),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).
% mult_inverse_of_nat_commute
tff(fact_3380_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),aa(A,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),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).
% mult_inverse_of_int_commute
tff(fact_3381_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).
% inverse_le_1_iff
tff(fact_3382_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).
% one_less_inverse_iff
tff(fact_3383_one__less__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% one_less_inverse
tff(fact_3384_division__ring__inverse__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% division_ring_inverse_add
tff(fact_3385_inverse__add,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% inverse_add
tff(fact_3386_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),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).
% field_class.field_inverse
tff(fact_3387_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,aa(A,A,inverse_inverse(A),A3),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),minus_minus(A,B2,A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% division_ring_inverse_diff
tff(fact_3388_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(A),A3) = divide_divide(A,one_one(A),A3) ) ) ) ).
% nonzero_inverse_eq_divide
tff(fact_3389_trancl__Int__subset,axiom,
! [A: $tType,R2: 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))),R2),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))),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,R2)),S3),R2)),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))),transitive_trancl(A,R2)),S3)) ) ) ).
% trancl_Int_subset
tff(fact_3390_inverse__less__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).
% inverse_less_iff
tff(fact_3391_inverse__le__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).
% inverse_le_iff
tff(fact_3392_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).
% one_le_inverse_iff
tff(fact_3393_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).
% inverse_less_1_iff
tff(fact_3394_one__le__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% one_le_inverse
tff(fact_3395_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),aa(A,A,inverse_inverse(A),X)),M2)) ) ) ) ) ).
% power_diff_conv_inverse
tff(fact_3396_quicksort_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Y: 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),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aTP_Lamp_il(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))),linorder_quicksort(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(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_3397_inverse__rat_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),X)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_3398_divmod__step__integer__def,axiom,
! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_io(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_3399_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [A3: B,Xs: list(B),F2: 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),F2),Xs))
=> ( ( aa(list(B),B,hd(B),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ip(B,fun(fun(B,A),fun(B,bool)),A3),F2)),Xs)) = A3 )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),remove1(B),A3),Xs)) = Xs ) ) ) ) ) ).
% insort_key_remove1
tff(fact_3400_inverse__rat_Otransfer,axiom,
pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_iq(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).
% inverse_rat.transfer
tff(fact_3401_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [X: B,F2: 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,F2),X),Xs)) = Xs ) ).
% remove1_insort_key
tff(fact_3402_length__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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,F2),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).
% length_insort
tff(fact_3403_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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,F2),X),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) ) ).
% sort_key_simps(2)
tff(fact_3404_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_finite(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)),insert3(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_3405_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_finite(A),A4))
=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(A),X),bot_bot(set(A)))))) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_3406_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_3407_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_3408_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_3409_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_3410_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),X: B,Y: B,Xs: list(B)] :
( ( aa(B,A,F2,X) != aa(B,A,F2,Y) )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),Xs)) ) ) ) ).
% insort_key_left_comm
tff(fact_3411_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_3412_sgn__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = zero_zero(code_integer) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = one_one(code_integer) ) ) ) ) ) ).
% sgn_integer_code
tff(fact_3413_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_3414_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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,F2),A3),Xs) != nil(B) ) ).
% insort_not_Nil
tff(fact_3415_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [P: fun(B,bool),X: B,F2: 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)),filter(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs) ) ) ) ).
% filter_insort_triv
tff(fact_3416_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_3417_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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,F2,X)),aa(B,A,F2,Y)))
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),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,F2,X)),aa(B,A,F2,Y)))
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),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,F2),X),Ys)) ) ) ) ) ).
% insort_key.simps(2)
tff(fact_3418_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),X: B] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),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_3419_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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,F2),X),Xs)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),X),aa(list(B),set(B),set2(B),Xs)) ) ).
% set_insort_key
tff(fact_3420_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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,F2),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_3421_one__rat_Orsp,axiom,
pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)))) ).
% one_rat.rsp
tff(fact_3422_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_3423_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_3424_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),F2: 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,F2,A3)),aa(B,A,F2,X2))) )
=> ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),A3),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xs) ) ) ) ).
% insort_is_Cons
tff(fact_3425_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),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),F2),Xs)) ) ) ).
% sorted_insort_key
tff(fact_3426_zero__rat_Orsp,axiom,
pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)))) ).
% zero_rat.rsp
tff(fact_3427_one__integer_Orsp,axiom,
one_one(int) = one_one(int) ).
% one_integer.rsp
tff(fact_3428_one__natural_Orsp,axiom,
one_one(nat) = one_one(nat) ).
% one_natural.rsp
tff(fact_3429_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),X: B,Xs: list(B)] :
( ( pp(member2(A,aa(B,A,F2,X),image2(B,A,F2,aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F2,X,Xs) = Xs ) )
& ( ~ pp(member2(A,aa(B,A,F2,X),image2(B,A,F2,aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F2,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs) ) ) ) ) ).
% insort_insert_key_def
tff(fact_3430_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),X: B,Xs: list(B)] :
( ~ pp(member2(A,aa(B,A,F2,X),image2(B,A,F2,aa(list(B),set(B),set2(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F2,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs) ) ) ) ).
% insort_insert_insort_key
tff(fact_3431_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),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,F2)),Xs),nil(B)) ) ).
% sort_key_def
tff(fact_3432_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs))))
<=> ( ~ pp(member2(A,aa(B,A,F2,X),image2(B,A,F2,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),F2),Xs))) ) ) ) ).
% distinct_insort_key
tff(fact_3433_filter__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: 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),F2),Xs))
=> ( pp(aa(B,bool,P,X))
=> ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs)) ) ) ) ) ).
% filter_insort
tff(fact_3434_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_3435_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(A),X),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
tff(fact_3436_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_3437_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_3438_integer__of__int__code,axiom,
! [K: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( code_integer_of_int(K) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( ( ( K = zero_zero(int) )
=> ( code_integer_of_int(K) = zero_zero(code_integer) ) )
& ( ( K != zero_zero(int) )
=> ( code_integer_of_int(K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))))),one_one(code_integer))) ) ) ) ) ) ).
% integer_of_int_code
tff(fact_3439_integer__of__num_I3_J,axiom,
! [N: num] : code_integer_of_num(bit1(N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N))),one_one(code_integer)) ).
% integer_of_num(3)
tff(fact_3440_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_ir(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).
% set_zip
tff(fact_3441_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: 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,R2)))
<=> ( ( 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_bc(set(product_prod(A,B)),fun(A,fun(B,bool)),R2)),X4)) ) ) ) ).
% listrel_iff_zip
tff(fact_3442_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_is(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_3443_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_it(set(A),fun(set(list(A)),fun(list(A),bool)),A4),XS)) ).
% set_Cons_def
tff(fact_3444_Fract_Orsp,axiom,
pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_iu(int,fun(int,product_prod(int,int)))),aTP_Lamp_iu(int,fun(int,product_prod(int,int))))) ).
% Fract.rsp
tff(fact_3445_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_3446_one__integer__def,axiom,
one_one(code_integer) = code_integer_of_int(one_one(int)) ).
% one_integer_def
tff(fact_3447_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),complete_Sup_Sup(A,A4)) = complete_Sup_Sup(A,collect(A,aa(set(A),fun(A,bool),aTP_Lamp_iv(A,fun(set(A),fun(A,bool)),A3),A4))) ) ).
% finite_inf_Sup
tff(fact_3448_listrel1__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_iw(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).
% listrel1_def
tff(fact_3449_lexord__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_ix(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).
% lexord_def
tff(fact_3450_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ sorted_wrt(A,X,Xa)
=> ~ ! [X2: A,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( ! [Xa4: A] :
( pp(member2(A,Xa4,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa4)) )
& sorted_wrt(A,X,Ys3) ) ) ) ).
% sorted_wrt.elims(3)
tff(fact_3451_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_3452_lex__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_iy(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).
% lex_conv
tff(fact_3453_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_3454_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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ~ ( ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa3)) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ).
% sorted_wrt.elims(2)
tff(fact_3455_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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( pp(Y)
<=> ~ ( ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa2)) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ) ).
% sorted_wrt.elims(1)
tff(fact_3456_set__conv__nth,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_iz(list(A),fun(A,bool),Xs)) ).
% set_conv_nth
tff(fact_3457_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_3458_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_ja(list(A),fun(list(A),bool))),X2)) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
tff(fact_3459_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_ja(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_3460_integer__of__num__triv_I1_J,axiom,
code_integer_of_num(one2) = one_one(code_integer) ).
% integer_of_num_triv(1)
tff(fact_3461_set__nths,axiom,
! [A: $tType,Xs: list(A),I: 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),I)) = collect(A,aa(set(nat),fun(A,bool),aTP_Lamp_jb(list(A),fun(set(nat),fun(A,bool)),Xs),I)) ).
% set_nths
tff(fact_3462_FI__def,axiom,
! [M2: list(product_prod(assn,assn)),P7: assn,Q3: assn,Up: assn,Uq: assn,F2: assn] :
( fi(M2,P7,Q3,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)),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))),P7)),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))),Q3)),Uq)),F2))) ) ) ).
% FI_def
tff(fact_3463_inverse__rat_Orsp,axiom,
pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_iq(product_prod(int,int),product_prod(int,int))),aTP_Lamp_iq(product_prod(int,int),product_prod(int,int)))) ).
% inverse_rat.rsp
tff(fact_3464_FI__RESULT__def,axiom,
! [M: list(product_prod(assn,assn)),UP: assn,UQ: assn,F: assn] :
( fI_RESULT(M,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)),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)),F))) ) ) ).
% FI_RESULT_def
tff(fact_3465_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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( 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),Ys3)))
=> ~ ( ! [Xa3: A] :
( pp(member2(A,Xa3,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa3)) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
tff(fact_3466_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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( ( pp(Y)
<=> ( ! [Xa2: A] :
( pp(member2(A,Xa2,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa2)) )
& sorted_wrt(A,X,Ys3) ) )
=> ~ 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),Ys3))) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
tff(fact_3467_lexn__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),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_jc(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R2),N))) ).
% lexn_conv
tff(fact_3468_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,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( 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),Ys3)))
=> ( ! [Xa4: A] :
( pp(member2(A,Xa4,aa(list(A),set(A),set2(A),Ys3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X,X2),Xa4)) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
tff(fact_3469_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).
% lexn.simps(1)
tff(fact_3470_lex__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = complete_Sup_Sup(set(product_prod(list(A),list(A))),image2(nat,set(product_prod(list(A),list(A))),lexn(A,R2),top_top(set(nat)))) ).
% lex_def
tff(fact_3471_lexn__length,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2),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_3472_num__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( code_num_of_integer(K) = one2 ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( code_num_of_integer(K) = aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_jd(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).
% num_of_integer_code
tff(fact_3473_int__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( code_int_of_integer(K) = aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( ( ( K = zero_zero(code_integer) )
=> ( code_int_of_integer(K) = zero_zero(int) ) )
& ( ( K != zero_zero(code_integer) )
=> ( code_int_of_integer(K) = aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_je(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_3474_bit__cut__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
& ( ( K != zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool),aTP_Lamp_jf(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_3475_nat__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( code_nat_of_integer(K) = zero_zero(nat) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( code_nat_of_integer(K) = aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_jg(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).
% nat_of_integer_code
tff(fact_3476_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F2: 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),F2),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_jh(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum_funpow
tff(fact_3477_relpowp__1,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : compow(fun(A,fun(A,bool)),one_one(nat),P) = P ).
% relpowp_1
tff(fact_3478_one__integer_Orep__eq,axiom,
code_int_of_integer(one_one(code_integer)) = one_one(int) ).
% one_integer.rep_eq
tff(fact_3479_funpow__times__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [F2: fun(A,nat),X: A] : compow(fun(A,A),aa(A,nat,F2,X),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F2,X))) ) ).
% funpow_times_power
tff(fact_3480_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_3481_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_3482_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_3483_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_3484_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_3485_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ji(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_jj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).
% divmod_integer_code
tff(fact_3486_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_hf(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) ) )
=> ~ ( ( X = nil(A) )
=> ( Y != undefined(A) ) ) ) ) ) ).
% min_list.elims
tff(fact_3487_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),P7: fun(B,A),I2: B] :
( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),P7))))
=> ( ( pp(member2(B,I2,I))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),I2),I)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),I) ) )
& ( ~ pp(member2(B,I2,I))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),I2),I)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P7,I2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),I)) ) ) ) ) ) ).
% prod.insert'
tff(fact_3488_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_3489_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_jk(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_3490_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_3491_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_3492_concat__replicate__trivial,axiom,
! [A: $tType,I2: 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)),I2),nil(A))) = nil(A) ).
% concat_replicate_trivial
tff(fact_3493_map__replicate,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),N: nat,X: B] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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,F2,X)) ).
% map_replicate
tff(fact_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_nth__replicate,axiom,
! [A: $tType,I2: nat,N: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
=> ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),I2) = X ) ) ).
% nth_replicate
tff(fact_3501_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P7: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty'
tff(fact_3502_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_3503_drop__replicate,axiom,
! [A: $tType,I2: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I2),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,I2)),X) ).
% drop_replicate
tff(fact_3504_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_3505_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_3506_take__replicate,axiom,
! [A: $tType,I2: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),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),I2),K)),X) ).
% take_replicate
tff(fact_3507_foldr__replicate,axiom,
! [A: $tType,B: $tType,F2: 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),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = compow(fun(A,A),N,aa(B,fun(A,A),F2,X)) ).
% foldr_replicate
tff(fact_3508_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_3509_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_3510_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: 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),I2),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),I2),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_3511_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext(product_unit),P: assn] :
( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H2)
=> ( 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)),H2),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_3512_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)),insert3(A),X),bot_bot(set(A))) ) ) ).
% set_replicate
tff(fact_3513_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_3514_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_jl(fun(B,A),fun(set(B),fun(B,bool)),G),I))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I) ) ).
% prod.non_neutral'
tff(fact_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_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_3522_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)),filter(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)),filter(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) ) ) ).
% filter_replicate
tff(fact_3523_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) )
=> ? [M5: nat,N3: nat,Zs2: 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)),M5),Zs2)) = 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)),N3),Zs2)) = Ys ) ) ) ).
% comm_append_are_replicate
tff(fact_3524_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_jm(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_3525_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)),filter(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)),filter(A),aa(A,fun(A,bool),fequal(A),X)),Xs) ).
% replicate_length_filter
tff(fact_3526_replicate__append__same,axiom,
! [A: $tType,I2: 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),I2),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),I2),X)) ).
% replicate_append_same
tff(fact_3527_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),I))
=> ( 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_ar(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),I) = 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),I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H2),I)) ) ) ) ).
% prod.distrib_triv'
tff(fact_3528_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_jn(A,fun(list(A),A))),List) ).
% hd_def
tff(fact_3529_map__replicate__trivial,axiom,
! [A: $tType,X: A,I2: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_jo(A,fun(nat,A),X)),upt(zero_zero(nat),I2)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I2),X) ).
% map_replicate_trivial
tff(fact_3530_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)),insert3(A),X),bot_bot(set(A))) ) ) ) ).
% set_replicate_conv_if
tff(fact_3531_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)),insert3(A),X),bot_bot(set(A))) ).
% set_replicate_Suc
tff(fact_3532_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_3533_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_3534_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_3535_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S: set(B),T: set(B),H2: 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,H2,I3) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member2(B,X2,S))
=> ( aa(B,A,G,X2) = aa(B,A,H2,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),H2),T) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_3536_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),H2: 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,H2,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),H2),S) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_3537_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) ) ) )
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),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,A6),B5)),zipf(A,B,C,X,As,Bs2)) ) ) )
=> ( ( ? [V: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)
=> ( ( Xb = nil(B) )
=> ( Y != undefined(list(C)) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V: B,Va: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va)
=> ( Y != undefined(list(C)) ) ) ) ) ) ) ) ).
% zipf.elims
tff(fact_3538_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_3539_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),G))))
=> ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),H2))))
=> ( 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_ar(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),I) = 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),I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H2),I)) ) ) ) ) ).
% prod.distrib'
tff(fact_3540_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [I: set(B),P7: fun(B,A)] :
( ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),P7))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),I) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P7),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),P7))) ) )
& ( ~ pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_be(set(B),fun(fun(B,A),fun(B,bool)),I),P7))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P7),I) = one_one(A) ) ) ) ) ).
% prod.G_def
tff(fact_3541_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_gq(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_3542_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_3543_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_gq(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_3544_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_3545_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) )
=> ? [N3: nat,Zs2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N3))
& ( 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)),N3),Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).
% comm_append_is_replicate
tff(fact_3546_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,Zs2: 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),Zs2)) )
=> ( 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),Zs2)))),X2,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2))) ) )
=> ~ ( ( Xa = nil(A) )
=> ( Y != undefined(A) ) ) ) ) ) ) ).
% arg_min_list.elims
tff(fact_3547_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_hf(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_3548_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,Zs2: 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),Zs2)) )
=> ( ( 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),Zs2)))),X2,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2))) )
=> ~ 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),Zs2)))) ) )
=> ~ ( ( 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_3549_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer),times_times(code_integer)),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_jp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).
% divmod_integer_eq_cases
tff(fact_3550_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_bc(set(product_prod(A,B)),fun(A,fun(B,bool)),X3)))) ).
% listrel_def
tff(fact_3551_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A)] : linord144544945434240204of_set(B,A,F2) = finite_folding_F(B,list(B),linorder_insort_key(B,A,F2),nil(B)) ) ).
% sorted_key_list_of_set_def
tff(fact_3552_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: 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),F2),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,F2,G)) ).
% List.map.comp
tff(fact_3553_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F2: 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),F2),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,F2,G)) ).
% map_comp_map
tff(fact_3554_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F2: fun(A,B),V2: 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),F2),V2)) = aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),comp(B,C,A,G,F2)),V2) ).
% list.map_comp
tff(fact_3555_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: 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),F2),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,F2,G)),List) ).
% List.map.compositionality
tff(fact_3556_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F2: 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),F2),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,F2,G)),Xs) ).
% map_map
tff(fact_3557_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,X),Y))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs),Ys))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),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_3558_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),nil(A)),nil(B))) ).
% listrelp.Nil
tff(fact_3559_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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),F2,X),aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xs)) ).
% foldr.foldr_Cons
tff(fact_3560_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: 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),F2),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,F2)),Xs),A3) ).
% foldr_map
tff(fact_3561_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: 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,R2),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) )
=> ! [Ys3: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X2),Y2))
=> ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs2),Ys3)) ) ) ) ) ) ).
% listrelp.cases
tff(fact_3562_listrelp_Osimps,axiom,
! [A: $tType,B: $tType,R2: 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,R2),A1),A22))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs4: list(A),Ys4: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys4) )
& pp(aa(B,bool,aa(A,fun(B,bool),R2,X4),Y3))
& pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs4),Ys4)) ) ) ) ).
% listrelp.simps
tff(fact_3563_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),H2: fun(B,C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( ! [X2: B,Y2: B] :
( pp(member2(B,X2,A4))
=> ( pp(member2(B,Y2,A4))
=> ( ( X2 != Y2 )
=> ( ( aa(B,C,H2,X2) = aa(B,C,H2,Y2) )
=> ( aa(C,A,G,aa(B,C,H2,X2)) = one_one(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),image2(B,C,H2,A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),comp(C,A,B,G,H2)),A4) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_3564_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: 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_bc(set(product_prod(A,B)),fun(A,fun(B,bool)),R2)),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,R2))) ) ).
% listrelp_listrel_eq
tff(fact_3565_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_bw(A,product_prod(A,A))),Xs)) ).
% Id_on_set
tff(fact_3566_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_3567_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))),image2(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),X),pow(B,aa(list(B),set(B),set2(B),Xs)))) ).
% Pow_set(2)
tff(fact_3568_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_jq(set(A),fun(nat,fun(list(A),bool)),A4),N)) = image2(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_cg(list(A),fun(A,list(A)))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_cr(set(A),fun(nat,fun(list(A),bool)),A4),N)),aTP_Lamp_jr(set(A),fun(list(A),set(A)),A4))) ).
% lists_length_Suc_eq
tff(fact_3569_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,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),F2)),Xs),top_top(A)) ) ).
% INF_set_fold
tff(fact_3570_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_3571_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_3572_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_3573_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_3574_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_3575_size__list__map,axiom,
! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F2),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,F2,G)),Xs) ).
% size_list_map
tff(fact_3576_length__filter__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F2: 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)),filter(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),comp(A,bool,B,P,F2)),Xs)) ).
% length_filter_map
tff(fact_3577_fold__append,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),Ys),aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xs)) ).
% fold_append
tff(fact_3578_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_3579_fold__replicate,axiom,
! [A: $tType,B: $tType,F2: 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),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = compow(fun(A,A),N,aa(B,fun(A,A),F2,X)) ).
% fold_replicate
tff(fact_3580_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_3581_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_js(list(B),fun(A,set(B)),Ys)) ).
% set_product
tff(fact_3582_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : comp(list(B),nat,list(A),size_list(B,F2),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = size_list(A,comp(B,nat,A,F2,G)) ).
% list.size_gen_o_map
tff(fact_3583_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_3584_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_3585_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),S3: 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),F2),nil(B)),S3) = S3 ).
% fold_simps(1)
tff(fact_3586_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),X: B,Xs: list(B),S3: 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),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),S3) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xs),aa(A,A,aa(B,fun(A,A),F2,X),S3)) ).
% fold_simps(2)
tff(fact_3587_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_3588_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,bool),P: fun(B,bool),S3: B,F2: 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,S3))
=> ( ! [X2: A,S4: B] :
( pp(aa(A,bool,Q,X2))
=> ( pp(aa(B,bool,P,S4))
=> pp(aa(B,bool,P,aa(B,B,aa(A,fun(B,B),F2,X2),S4))) ) )
=> 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),F2),Xs),S3))) ) ) ) ).
% fold_invariant
tff(fact_3589_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,Xs: list(B),Ys: list(B),F2: 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),F2,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),F2),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_3590_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_3591_filter__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),comp(A,bool,B,P,F2)),Xs)) ).
% filter_map
tff(fact_3592_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),comp(A,bool,B,P,F2)),Xs)) ).
% takeWhile_map
tff(fact_3593_fold__Cons,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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),F2),Xs),aa(A,fun(B,B),F2,X)) ).
% fold_Cons
tff(fact_3594_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H2: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C)),S3: B] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,C,B,H2,aa(A,fun(B,B),G,X2)) = comp(C,C,B,aa(A,fun(C,C),F2,X2),H2) ) )
=> ( aa(B,C,H2,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),S3)) = aa(C,C,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),fold(A,C),F2),Xs),aa(B,C,H2,S3)) ) ) ).
% fold_commute_apply
tff(fact_3595_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H2: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( comp(B,C,B,H2,aa(A,fun(B,B),G,X2)) = comp(C,C,B,aa(A,fun(C,C),F2,X2),H2) ) )
=> ( comp(B,C,B,H2,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),F2),Xs),H2) ) ) ).
% fold_commute
tff(fact_3596_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_3597_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_3598_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_3599_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_3600_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_3601_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F2: 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),F2),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,F2)),Xs) ).
% fold_map
tff(fact_3602_listrel__subset,axiom,
! [A: $tType,R2: 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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,R2)),product_Sigma(list(A),list(A),aa(set(A),set(list(A)),lists(A),A4),aTP_Lamp_ju(set(A),fun(list(A),set(list(A))),A4)))) ) ).
% listrel_subset
tff(fact_3603_foldr__conv__fold,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),rev(B),Xs)) ).
% foldr_conv_fold
tff(fact_3604_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),F2: fun(B,A),X: B,Y: B] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> ( 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),F2),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),F2),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),F2),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),F2),Y)) ) ) ) ) ).
% folding_insort_key.insort_key_commute
tff(fact_3605_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)),insert3(A)),Xs),A4) ).
% union_set_fold
tff(fact_3606_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_3607_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_3608_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_3609_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_3610_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_3611_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: 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),F2,Y2),aa(A,fun(B,B),F2,X2)) = comp(B,B,B,aa(A,fun(B,B),F2,X2),aa(A,fun(B,B),F2,Y2)) ) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),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),F2),Xs) ) ) ).
% fold_rev
tff(fact_3612_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
( inj_on(A,B,F2,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),F2),Xs)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),remdups_adj(A),Xs)) ) ) ).
% remdups_adj_map_injective
tff(fact_3613_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_3614_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: 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),F2,Y2),aa(A,fun(B,B),F2,X2)) = comp(B,B,B,aa(A,fun(B,B),F2,X2),aa(A,fun(B,B),F2,Y2)) ) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xs) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs) ) ) ).
% foldr_fold
tff(fact_3615_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: 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),F2,X2),aa(A,fun(B,B),F2,Y2)) = comp(B,B,B,aa(A,fun(B,B),F2,Y2),aa(A,fun(B,B),F2,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),F2),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),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),remove1(A),X),Xs)),aa(A,fun(B,B),F2,X)) ) ) ) ).
% fold_remove1_split
tff(fact_3616_Restr__trancl__mono,axiom,
! [A: $tType,V2: A,W: A,E4: 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),V2),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))),E4),product_Sigma(A,A,U2,aTP_Lamp_jt(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),V2),W),transitive_trancl(A,E4))) ) ).
% Restr_trancl_mono
tff(fact_3617_set__zip__cart,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L6: 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),L6))))
=> pp(member2(product_prod(A,B),X,product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_js(list(B),fun(A,set(B)),L6)))) ) ).
% set_zip_cart
tff(fact_3618_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_3619_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_finite(B),X2)) )
=> ( ! [A12: set(B)] :
( pp(member2(set(B),A12,B3))
=> ! [A23: set(B)] :
( pp(member2(set(B),A23,B3))
=> ( ( A12 != A23 )
=> ! [X2: B] :
( pp(member2(B,X2,A12))
=> ( 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),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_3620_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_3621_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_jv(A,fun(A,fun(list(A),list(A))),X)),aa(list(A),list(A),remdups_adj(A),Xs)) ).
% remdups_adj_Cons
tff(fact_3622_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_3623_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_3624_Sup__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(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_3625_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_3626_remdups__adj__adjacent,axiom,
! [A: $tType,I2: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups_adj(A),Xs))))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xs)),I2) != aa(nat,A,nth(A,aa(list(A),list(A),remdups_adj(A),Xs)),aa(nat,nat,suc,I2)) ) ) ).
% remdups_adj_adjacent
tff(fact_3627_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_3628_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_3629_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_3630_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_3631_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_3632_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> pp(aa(set(B),bool,finite_finite(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C5))
=> ( ( 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),complete_Sup_Sup(set(B),C5)) = 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),C5) ) ) ) ) ).
% prod.Union_disjoint
tff(fact_3633_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_3634_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_3635_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_3636_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F2: fun(B,A),Xs: list(B)] :
( inj_on(B,A,F2,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),F2),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,F2)),Xs),nil(B)) ) ) ) ).
% sort_key_conv_fold
tff(fact_3637_subseqs__powset,axiom,
! [A: $tType,Xs: list(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_3638_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [C5: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> pp(aa(set(B),bool,finite_finite(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C5))
=> ( ( 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),complete_Sup_Sup(set(B),C5)) = 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),C5) ) ) ) ) ).
% sum.Union_disjoint
tff(fact_3639_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))),insert3(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_set(1)
tff(fact_3640_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(B,A),Xs: list(B)] : complete_Sup_Sup(A,image2(B,A,F2,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),F2)),Xs),bot_bot(A)) ) ).
% SUP_set_fold
tff(fact_3641_Times__Int__Times,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(B),C5: 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_jw(set(B),fun(A,set(B)),B3))),product_Sigma(A,B,C5,aTP_Lamp_jw(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),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_jx(set(B),fun(set(B),fun(A,set(B))),B3),D2)) ).
% Times_Int_Times
tff(fact_3642_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I: set(A),A4: fun(A,set(B)),B3: fun(A,set(B))] : product_Sigma(A,B,I,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_jy(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,I,A4)),product_Sigma(A,B,I,B3)) ).
% Sigma_Int_distrib2
tff(fact_3643_Times__Int__distrib1,axiom,
! [A: $tType,B: $tType,A4: set(A),B3: set(A),C5: 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_jw(set(B),fun(A,set(B)),C5)) = 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_jw(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B3,aTP_Lamp_jw(set(B),fun(A,set(B)),C5))) ).
% Times_Int_distrib1
tff(fact_3644_Restr__subset,axiom,
! [A: $tType,A4: set(A),B3: set(A),R2: 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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4))) ) ) ).
% Restr_subset
tff(fact_3645_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_3646_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Int_distrib1
tff(fact_3647_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(list(A),list(A),remdups_adj(A),Xs) = Ys )
<=> ? [F9: fun(nat,nat)] :
( order_mono(nat,nat,F9)
& ( image2(nat,nat,F9,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)) )
& ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),aa(nat,nat,F9,I4)) ) )
& ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
<=> ( aa(nat,nat,F9,I4) = aa(nat,nat,F9,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).
% remdups_adj_altdef
tff(fact_3648_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_finite(A))) ).
% Fpow_Pow_finite
tff(fact_3649_rtrancl__last__visit__node,axiom,
! [A: $tType,S3: 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),S3),S5),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R)))
=> ( ( ( S3 != 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),S3),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_jz(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),S3),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_jz(A,fun(A,set(A)),Sh)))))) ) ) ) ).
% rtrancl_last_visit_node
tff(fact_3650_sum__mult__sum__if__inj,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [F2: 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_eb(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F2),G)),product_Sigma(A,C,A4,aTP_Lamp_ka(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),F2),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_kb(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F2),G),A4),B3))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_3651_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_3652_rotate0,axiom,
! [A: $tType] : aa(nat,fun(list(A),list(A)),rotate(A),zero_zero(nat)) = id(list(A)) ).
% rotate0
tff(fact_3653_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list(A),R2: 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),R2)))) ).
% listrel_rtrancl_refl
tff(fact_3654_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: 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),R2))),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)))) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_3655_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : listrel(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)) ).
% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_3656_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F2)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),M2),N)) ) ) ) ).
% max_of_mono
tff(fact_3657_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F2)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,M2)),aa(A,B,F2,N)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)) ) ) ) ).
% min_of_mono
tff(fact_3658_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_3659_list_Omap__id,axiom,
! [A: $tType,T3: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),id(A)),T3) = T3 ).
% list.map_id
tff(fact_3660_List_Omap_Oidentity,axiom,
! [A: $tType] : aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ae(A,A)) = id(list(A)) ).
% List.map.identity
tff(fact_3661_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: 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),R2)))
=> ( 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,R2))))
=> 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,R2)))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_3662_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf(A)
& semilattice_inf(B) )
=> ! [F2: fun(A,B),A4: A,B3: A] :
( order_mono(A,B,F2)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,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,F2,A4)),aa(A,B,F2,B3)))) ) ) ).
% mono_inf
tff(fact_3663_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup(A)
& semilattice_sup(B) )
=> ! [F2: fun(A,B),A4: A,B3: A] :
( order_mono(A,B,F2)
=> 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,F2,A4)),aa(A,B,F2,B3))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3)))) ) ) ).
% mono_sup
tff(fact_3664_fold__Nil,axiom,
! [A: $tType,B: $tType,F2: 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),F2),nil(A)) = id(B) ).
% fold_Nil
tff(fact_3665_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
( ! [X2: A] :
( pp(member2(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,fun(B,B),F2,X2) = id(B) ) )
=> ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs) = id(B) ) ) ).
% fold_id
tff(fact_3666_foldr__Nil,axiom,
! [A: $tType,B: $tType,F2: 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),F2),nil(A)) = id(B) ).
% foldr_Nil
tff(fact_3667_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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),R2))))
=> ( 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),R2))))
=> 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),R2)))) ) ) ).
% listrel_rtrancl_trans
tff(fact_3668_fold__filter,axiom,
! [A: $tType,B: $tType,F2: 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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(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_kc(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F2),P)),Xs) ).
% fold_filter
tff(fact_3669_foldr__filter,axiom,
! [A: $tType,B: $tType,F2: 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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(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_kc(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F2),P)),Xs) ).
% foldr_filter
tff(fact_3670_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_3671_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2))))
=> 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,R2)))) ) ).
% rtrancl_listrel1_ConsI1
tff(fact_3672_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2))))
=> ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).
% rtrancl_listrel1_eq_len
tff(fact_3673_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> 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),R2)))) ) ).
% listrel_reflcl_if_listrel1
tff(fact_3674_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: 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,R2)))
=> 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,R2)))) ) ).
% rtrancl_listrel1_if_listrel
tff(fact_3675_Restr__rtrancl__mono,axiom,
! [A: $tType,V2: A,W: A,E4: 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),V2),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))),E4),product_Sigma(A,A,U2,aTP_Lamp_jt(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),V2),W),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),E4))) ) ).
% Restr_rtrancl_mono
tff(fact_3676_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: 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,R2)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),transitive_rtrancl(list(A)),listrel1(A,R2)))) ).
% listrel_subset_rtrancl_listrel1
tff(fact_3677_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))
<=> ? [Z4: list(product_prod(A,B))] :
( pp(member2(list(product_prod(A,B)),Z4,collect(list(product_prod(A,B)),aTP_Lamp_kd(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)),Z4) = 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)),Z4) = B2 ) ) ) ).
% list.in_rel
tff(fact_3678_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),F2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,A4)) = finite_fold(B,A,comp(A,fun(A,A),B,inf_inf(A),F2),top_top(A),A4) ) ) ) ).
% INF_fold_inf
tff(fact_3679_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),aTP_Lamp_er(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),aTP_Lamp_er(A,fun(A,bool),X)),Xs))) ) ) ) ).
% takeWhile_neq_rev
tff(fact_3680_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)
<=> ! [F9: fun(B,A),Xs4: list(B),K4: A] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),aa(A,fun(B,bool),aTP_Lamp_ke(fun(B,A),fun(A,fun(B,bool)),F9),K4)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F9),Xs4)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),aa(A,fun(B,bool),aTP_Lamp_ke(fun(B,A),fun(A,fun(B,bool)),F9),K4)),Xs4) ) ) ).
% stable_sort_key_def
tff(fact_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_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_3688_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_3689_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_3690_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_3691_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))),filter(A)),filter(B))) ).
% List.filter_transfer
tff(fact_3692_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_3693_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_3694_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_3695_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_3696_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_3697_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_3698_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_3699_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_3700_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_3701_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_3702_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_3703_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_3704_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_3705_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_3706_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_3707_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_3708_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 )
=> ( ! [Z3: A,Yb: B] :
( pp(member2(A,Z3,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,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),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_3709_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))
=> ( ! [Z3: A,Yb: B] :
( pp(member2(A,Z3,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,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),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_3710_list_Orel__refl__strong,axiom,
! [A: $tType,X: list(A),Ra: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z3),Z3)) )
=> 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_3711_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_3712_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_3713_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_3714_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_3715_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_3716_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_3717_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_3718_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_3719_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))
<=> ? [Z4: B,Zs3: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z4),Zs3) )
& pp(aa(B,bool,aa(A,fun(B,bool),P,X),Z4))
& 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),Zs3)) ) ) ).
% list_all2_Cons1
tff(fact_3720_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)))
<=> ? [Z4: A,Zs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Zs3) )
& pp(aa(B,bool,aa(A,fun(B,bool),P,Z4),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),Zs3),Ys)) ) ) ).
% list_all2_Cons2
tff(fact_3721_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: list(A),B2: list(B),C2: 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),C2),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),C2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B2),D3))) ) ) ).
% list_all2_appendI
tff(fact_3722_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_3723_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_3724_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_3725_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_3726_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,Ys3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,Xs2),Ys3))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,Xs2),Ys3)) )
=> 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_3727_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_3728_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),Cs: list(C)] :
( ! [A6: A,B5: B,C4: C] :
( pp(aa(B,bool,aa(A,fun(B,bool),P1,A6),B5))
=> ( pp(aa(C,bool,aa(B,fun(C,bool),P22,B5),C4))
=> pp(aa(C,bool,aa(A,fun(C,bool),P32,A6),C4)) ) )
=> ( 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),Cs))
=> 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),Cs)) ) ) ) ).
% list_all2_trans
tff(fact_3729_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_3730_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_3731_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,I2: 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),I2),X)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I2),Y))) ) ) ).
% list_all2_update_cong
tff(fact_3732_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_3733_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_3734_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_3735_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_3736_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_3737_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_3738_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_3739_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_3740_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_3741_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_ei(list(A),bool)),aTP_Lamp_ez(list(B),bool))) ).
% list.disc_transfer(2)
tff(fact_3742_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_kf(list(A),bool)),aTP_Lamp_kg(list(B),bool))) ).
% list.disc_transfer(1)
tff(fact_3743_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As2: list(A),F2: 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),F2),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_kh(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F2)),As2),Bs)) ) ).
% list_all2_map2
tff(fact_3744_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F2: 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),F2),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_ki(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F2)),As2),Bs)) ) ).
% list_all2_map1
tff(fact_3745_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: list(A),Y: list(B)] :
( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),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),I2),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_kj(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).
% list.rel_map(1)
tff(fact_3746_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_kk(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y)) ) ).
% list.rel_map(2)
tff(fact_3747_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_3748_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_3749_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_3750_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_3751_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,B22: 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),B22))
=> 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),B22))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,X),Y)) ) ) ) ).
% list.rel_induct
tff(fact_3752_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,Ys3: 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),Ys3))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs2),Ys3))
=> 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),Ys3))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs),Ys)) ) ) ) ).
% List.list_all2_induct
tff(fact_3753_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),L: list(A),L6: 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),L6))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
=> ( ! [X2: A,X7: B,Ls2: list(A),Ls3: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),X7))
=> ( 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),Ls3))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,Ls2),Ls3))
=> 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),X7),Ls3))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,L),L6)) ) ) ) ).
% Misc.list_all2_induct
tff(fact_3754_list_Omap__transfer,axiom,
! [A: $tType,B: $tType,F10: $tType,E5: $tType,Rb: fun(A,fun(E5,bool)),Sd: fun(B,fun(F10,bool))] : pp(aa(fun(fun(E5,F10),fun(list(E5),list(F10))),bool,aa(fun(fun(A,B),fun(list(A),list(B))),fun(fun(fun(E5,F10),fun(list(E5),list(F10))),bool),bNF_rel_fun(fun(A,B),fun(E5,F10),fun(list(A),list(B)),fun(list(E5),list(F10)),bNF_rel_fun(A,E5,B,F10,Rb,Sd),bNF_rel_fun(list(A),list(E5),list(B),list(F10),aa(fun(A,fun(E5,bool)),fun(list(A),fun(list(E5),bool)),list_all2(A,E5),Rb),aa(fun(B,fun(F10,bool)),fun(list(B),fun(list(F10),bool)),list_all2(B,F10),Sd))),map(A,B)),map(E5,F10))) ).
% list.map_transfer
tff(fact_3755_mono__Int,axiom,
! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A4: set(A),B3: set(A)] :
( order_mono(set(A),set(B),F2)
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A4)),aa(set(A),set(B),F2,B3)))) ) ).
% mono_Int
tff(fact_3756_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)))
<=> ? [Us2: list(A),Vs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Vs2) )
& ( aa(list(A),nat,size_size(list(A)),Us2) = 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),Us2),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_3757_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))
<=> ? [Us2: list(B),Vs2: list(B)] :
( ( Zs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
& ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(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),Us2))
& 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_3758_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(A,fun(B,bool)),Us: 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),Us)),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),Us),Vs)) ) ) ) ).
% list_all2_append
tff(fact_3759_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_3760_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_3761_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_3762_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_3763_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_3764_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_3765_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),F2: 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),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),dropWhile(B),comp(A,bool,B,P,F2)),Xs)) ).
% dropWhile_map
tff(fact_3766_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_kl(A,fun(A,bool),X)),Xs))) ).
% remdups_adj_Cons'
tff(fact_3767_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P7: 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),P7),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),P7)),aa(nat,B,nth(B,Ys),P7))) ) ) ).
% list_all2_nthD
tff(fact_3768_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P7: 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),P7),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),P7)),aa(nat,B,nth(B,Ys),P7))) ) ) ).
% list_all2_nthD2
tff(fact_3769_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) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),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),N3)),aa(nat,B,nth(B,B2),N3))) )
=> 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_3770_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) )
& ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),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),I4)),aa(nat,B,nth(B,Ys),I4))) ) ) ) ).
% list_all2_conv_all_nth
tff(fact_3771_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_3772_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_3773_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_3774_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A),B3: A] :
( pp(aa(set(A),bool,finite_finite(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_3775_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_3776_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)),filter(A),P),Xs) ) ) ).
% takeWhile_eq_filter
tff(fact_3777_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_3778_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_3779_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_kn(list(list(A)),fun(list(A),bool),Xss2)) ).
% product_lists_set
tff(fact_3780_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_3781_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_3782_Inf__fold__inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_3783_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_3784_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_ko(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_3785_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_kl(A,fun(A,bool),Y)),Ys))) ).
% remdups_adj_append_dropWhile
tff(fact_3786_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_kp(list(A),fun(A,bool),Xs)),Ys))) ) ) ).
% remdups_adj_append''
tff(fact_3787_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_kq(list(A),fun(A,bool),Ys)),aa(list(A),list(A),tl(A),Ys))) ) ) ).
% tl_remdups_adj
tff(fact_3788_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_kr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ).
% lexordp.mono
tff(fact_3789_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_3790_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(B),B3: A,F2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,A4))) = finite_fold(B,A,comp(A,fun(A,A),B,inf_inf(A),F2),B3,A4) ) ) ) ).
% inf_INF_fold_inf
tff(fact_3791_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_3792_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_3793_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_3794_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),aTP_Lamp_er(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),aTP_Lamp_er(A,fun(A,bool),X)),Xs))) ) ) ) ).
% dropWhile_neq_rev
tff(fact_3795_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),B3: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( 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_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(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,F2,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B3)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z2,A4),B3) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
tff(fact_3796_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_3797_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
( finite673082921795544331dem_on(A,B,S,F2)
=> ( 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,F2,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),F2),Xs),Y) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
tff(fact_3798_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
( finite4664212375090638736ute_on(A,B,S,F2)
=> ( 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,F2,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),F2),aa(list(A),list(A),remdups(A),Xs)),Y) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_3799_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_3800_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_3801_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_3802_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_3803_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_3804_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_3805_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( pp(aa(set(A),bool,finite_finite(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_finite(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ) ).
% Gcd_fin.eq_fold
tff(fact_3806_card__length__sum__list,axiom,
! [M2: nat,N2: nat] : finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_ks(nat,fun(nat,fun(list(nat),bool)),M2),N2))) = aa(nat,nat,binomial(minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M2),one_one(nat))),N2) ).
% card_length_sum_list
tff(fact_3807_foldl__append,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),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),F2),A3),Xs)),Ys) ).
% foldl_append
tff(fact_3808_sum__list_ONil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).
% sum_list.Nil
tff(fact_3809_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_3810_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_3811_foldl__cong,axiom,
! [A: $tType,B: $tType,A3: A,B2: A,L: list(B),K: list(B),F2: fun(A,fun(B,A)),G: fun(A,fun(B,A))] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X2: B] :
( pp(member2(B,X2,aa(list(B),set(B),set2(B),L)))
=> ( aa(B,A,aa(A,fun(B,A),F2,A6),X2) = aa(B,A,aa(A,fun(B,A),G,A6),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),F2),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_3812_foldl__Nil,axiom,
! [A: $tType,B: $tType,F2: 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),F2),A3),nil(A)) = A3 ).
% foldl_Nil
tff(fact_3813_foldl__Cons,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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),F2),aa(A,B,aa(B,fun(A,B),F2,A3),X)),Xs) ).
% foldl_Cons
tff(fact_3814_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A3: A,F2: 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),F2),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_kt(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F2)),A3),Xs) ).
% foldl_map
tff(fact_3815_foldl__conv__fold,axiom,
! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),S3: 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),F2),S3),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_ku(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),Xs),S3) ).
% foldl_conv_fold
tff(fact_3816_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_3817_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_3818_foldl__conc__empty__eq,axiom,
! [A: $tType,I2: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),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)),I2),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I2),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),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_3819_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_3820_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F2: 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_kw(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F2)),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),F2),B2),Ys) ) ) ).
% foldl_snd_zip
tff(fact_3821_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F2: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ee(A,fun(fun(B,A),fun(B,A)),C2),F2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))) ) ).
% sum_list_const_mult
tff(fact_3822_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),C2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(A,fun(B,A),aTP_Lamp_ed(fun(B,A),fun(A,fun(B,A)),F2),C2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))),C2) ) ).
% sum_list_mult_const
tff(fact_3823_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F2: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
( ( F2 = 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,F2,nil(A)) = F1 ) ) ).
% rec_list_Nil_imp
tff(fact_3824_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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_kx(fun(B,fun(A,A)),fun(A,fun(B,A)),F2)),A3),aa(list(B),list(B),rev(B),Xs)) ).
% foldr_conv_foldl
tff(fact_3825_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType,F2: 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),F2),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_ku(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),aa(list(B),list(B),rev(B),Xs)),A3) ).
% foldl_conv_foldr
tff(fact_3826_sum__list__replicate,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,C2: A] : groups8242544230860333062m_list(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C2) ) ).
% sum_list_replicate
tff(fact_3827_foldl__rule__aux__P,axiom,
! [Sigma: $tType,A: $tType,I: fun(Sigma,fun(list(A),bool)),Sigma_0: Sigma,L0: list(A),F2: fun(Sigma,fun(A,Sigma)),P: fun(Sigma,bool)] :
( pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,Sigma_0),L0))
=> ( ! [L1: list(A),L2: list(A),X2: A,Sigma2: Sigma] :
( ( 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),L2)) )
=> ( pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,Sigma2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L2)))
=> pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,aa(A,Sigma,aa(Sigma,fun(A,Sigma),F2,Sigma2),X2)),L2)) ) )
=> ( ! [Sigma2: Sigma] :
( pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,Sigma2),nil(A)))
=> pp(aa(Sigma,bool,P,Sigma2)) )
=> pp(aa(Sigma,bool,P,aa(list(A),Sigma,aa(Sigma,fun(list(A),Sigma),aa(fun(Sigma,fun(A,Sigma)),fun(Sigma,fun(list(A),Sigma)),foldl(Sigma,A),F2),Sigma_0),L0))) ) ) ) ).
% foldl_rule_aux_P
tff(fact_3828_foldl__rule__aux,axiom,
! [Sigma: $tType,A: $tType,I: fun(Sigma,fun(list(A),bool)),Sigma_0: Sigma,L0: list(A),F2: fun(Sigma,fun(A,Sigma))] :
( pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,Sigma_0),L0))
=> ( ! [L1: list(A),L2: list(A),X2: A,Sigma2: Sigma] :
( ( 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),L2)) )
=> ( pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,Sigma2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L2)))
=> pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,aa(A,Sigma,aa(Sigma,fun(A,Sigma),F2,Sigma2),X2)),L2)) ) )
=> pp(aa(list(A),bool,aa(Sigma,fun(list(A),bool),I,aa(list(A),Sigma,aa(Sigma,fun(list(A),Sigma),aa(fun(Sigma,fun(A,Sigma)),fun(Sigma,fun(list(A),Sigma)),foldl(Sigma,A),F2),Sigma_0),L0)),nil(A))) ) ) ).
% foldl_rule_aux
tff(fact_3829_foldl__rule__P,axiom,
! [Sigma: $tType,A: $tType,I: fun(Sigma,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma,L0: list(A),F2: fun(Sigma,fun(A,Sigma)),P: fun(Sigma,bool)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,Sigma_0),nil(A)),L0))
=> ( ! [L1: list(A),L2: list(A),X2: A,Sigma2: Sigma] :
( ( 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),L2)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,Sigma2),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L2)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,aa(A,Sigma,aa(Sigma,fun(A,Sigma),F2,Sigma2),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)))),L2)) ) )
=> ( ! [Sigma2: Sigma] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,Sigma2),L0),nil(A)))
=> pp(aa(Sigma,bool,P,Sigma2)) )
=> pp(aa(Sigma,bool,P,aa(list(A),Sigma,aa(Sigma,fun(list(A),Sigma),aa(fun(Sigma,fun(A,Sigma)),fun(Sigma,fun(list(A),Sigma)),foldl(Sigma,A),F2),Sigma_0),L0))) ) ) ) ).
% foldl_rule_P
tff(fact_3830_foldl__rule,axiom,
! [Sigma: $tType,A: $tType,I: fun(Sigma,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma,L0: list(A),F2: fun(Sigma,fun(A,Sigma))] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,Sigma_0),nil(A)),L0))
=> ( ! [L1: list(A),L2: list(A),X2: A,Sigma2: Sigma] :
( ( 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),L2)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,Sigma2),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L2)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,aa(A,Sigma,aa(Sigma,fun(A,Sigma),F2,Sigma2),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)))),L2)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma,fun(list(A),fun(list(A),bool)),I,aa(list(A),Sigma,aa(Sigma,fun(list(A),Sigma),aa(fun(Sigma,fun(A,Sigma)),fun(Sigma,fun(list(A),Sigma)),foldl(Sigma,A),F2),Sigma_0),L0)),L0),nil(A))) ) ) ).
% foldl_rule
tff(fact_3831_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& strict9044650504122735259up_add(B) )
=> ! [Xs: list(A),F2: 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,F2,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),F2),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_3832_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [R2: 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_ky(B,fun(C,B),R2)),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))),R2) ) ).
% sum_list_triv
tff(fact_3833_map__by__foldl,axiom,
! [B: $tType,A: $tType,F2: 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_kz(fun(A,B),fun(list(B),fun(A,list(B))),F2)),nil(B)),L) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),L) ).
% map_by_foldl
tff(fact_3834_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,fun(list(B),fun(C,C))),F2: 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),F2)) = 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_la(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F2)) ).
% list.rec_o_map
tff(fact_3835_card__length__sum__list__rec,axiom,
! [M2: nat,N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
=> ( finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_ks(nat,fun(nat,fun(list(nat),bool)),M2),N2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_lb(nat,fun(nat,fun(list(nat),bool)),M2),N2)))),finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_lc(nat,fun(nat,fun(list(nat),bool)),M2),N2)))) ) ) ).
% card_length_sum_list_rec
tff(fact_3836_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_ld(A,fun(list(A),fun(set(A),set(A))))),Xs) ).
% set_rec
tff(fact_3837_map__rec,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),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_le(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F2)),Xs) ).
% map_rec
tff(fact_3838_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_3839_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_3840_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A4: set(C)] : image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_lf(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G),A4) = product_Sigma(A,B,image2(C,A,F2,A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_lg(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A4)) ).
% image_split_eq_Sigma
tff(fact_3841_rat__number__expand_I5_J,axiom,
! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).
% rat_number_expand(5)
tff(fact_3842_vimage__Int,axiom,
! [A: $tType,B: $tType,F2: 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),F2),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),F2),A4)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F2),B3)) ).
% vimage_Int
tff(fact_3843_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_3844_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_3845_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_3846_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_3847_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_3848_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: set(A)] : image2(B,A,F2,aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F2),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),image2(B,A,F2,top_top(set(B)))) ).
% image_vimage_eq
tff(fact_3849_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_lh(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_3850_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_3851_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))
=> ( ! [Z3: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),X)))
=> ( pp(aa(A,bool,P,Z3))
=> pp(aa(A,bool,Pa,Z3)) ) )
=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),Pa),X)) ) ) ).
% list.pred_mono_strong
tff(fact_3852_list__all__cong,axiom,
! [A: $tType,X: list(A),Ya: list(A),P: fun(A,bool),Pa: fun(A,bool)] :
( ( X = Ya )
=> ( ! [Z3: A] :
( pp(member2(A,Z3,aa(list(A),set(A),set2(A),Ya)))
=> ( pp(aa(A,bool,P,Z3))
<=> pp(aa(A,bool,Pa,Z3)) ) )
=> ( 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_3853_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set(A),F2: fun(A,B),G: fun(A,B),Y: set(B)] :
( ! [W3: A] :
( pp(member2(A,W3,S))
=> ( aa(A,B,F2,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),F2),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_3854_Fract__of__int__eq,axiom,
! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ).
% Fract_of_int_eq
tff(fact_3855_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_3856_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_li(A,bool)),X3)) ).
% list.pred_True
tff(fact_3857_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list(A),Ya: list(A),F2: 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_lj(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G)),Ya))
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),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_3858_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F2: 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)),F2),product_Sigma(B,C,A4,aTP_Lamp_lk(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),F2)),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),F2)),B3)) ).
% vimage_Times
tff(fact_3859_eq__rat_I2_J,axiom,
! [A3: int] : aa(int,rat,aa(int,fun(int,rat),fract,A3),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
% eq_rat(2)
tff(fact_3860_Fract__of__nat__eq,axiom,
! [K: nat] : aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ).
% Fract_of_nat_eq
tff(fact_3861_One__rat__def,axiom,
one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).
% One_rat_def
tff(fact_3862_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_3863_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_3864_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_3865_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: fun(B,bool),F2: 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),F2),X)))
<=> pp(aa(list(A),bool,aa(fun(A,bool),fun(list(A),bool),list_all(A),comp(B,bool,A,Q,F2)),X)) ) ).
% list.pred_map
tff(fact_3866_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F: set(A),H2: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite(A),F))
=> ( inj_on(B,A,H2,A4)
=> pp(aa(set(B),bool,finite_finite(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),H2),F)),A4))) ) ) ).
% finite_vimage_IntI
tff(fact_3867_Zero__rat__def,axiom,
zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
% Zero_rat_def
tff(fact_3868_Fract__add__one,axiom,
! [N: int,M2: int] :
( ( N != zero_zero(int) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M2),N)),one_one(rat)) ) ) ).
% Fract_add_one
tff(fact_3869_rat__number__collapse_I3_J,axiom,
! [W: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W) ).
% rat_number_collapse(3)
tff(fact_3870_rat__number__expand_I3_J,axiom,
! [K: num] : aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ).
% rat_number_expand(3)
tff(fact_3871_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))
<=> ! [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N4))) ) ) ).
% list_all_length
tff(fact_3872_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F: set(A),H2: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite(A),F))
=> ( ! [Y2: A] :
( pp(member2(A,Y2,F))
=> pp(aa(set(B),bool,finite_finite(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),H2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))),A4))) )
=> pp(aa(set(B),bool,finite_finite(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),H2),F)),A4))) ) ) ).
% finite_finite_vimage_IntI
tff(fact_3873_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3)) ) ) ).
% one_le_Fract_iff
tff(fact_3874_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B2)) ) ) ).
% Fract_le_one_iff
tff(fact_3875_Fract_Oabs__eq,axiom,
! [Xa: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa),X) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),X))) ).
% Fract.abs_eq
tff(fact_3876_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A3,B2))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
& pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,B2)),one_one(int))))) ) ).
% rat_floor_lemma
tff(fact_3877_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A3)) ) ) ).
% one_less_Fract_iff
tff(fact_3878_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B2)) ) ) ).
% Fract_less_one_iff
tff(fact_3879_rat__number__collapse_I5_J,axiom,
aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).
% rat_number_collapse(5)
tff(fact_3880_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite(A),image2(B,A,F2,A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite(B),A4))
=> ~ ! [Y2: A] :
( pp(member2(A,Y2,image2(B,A,F2,A4)))
=> pp(aa(set(B),bool,finite_finite(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),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))),A4))) ) ) ) ).
% inf_img_fin_domE'
tff(fact_3881_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,finite_finite(A),image2(B,A,F2,A4)))
=> ( ~ pp(aa(set(B),bool,finite_finite(B),A4))
=> ? [X2: A] :
( pp(member2(A,X2,image2(B,A,F2,A4)))
& ~ pp(aa(set(B),bool,finite_finite(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),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X2),bot_bot(set(A))))),A4))) ) ) ) ).
% inf_img_fin_dom'
tff(fact_3882_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),D2: set(A),A4: set(B)] :
( inj_on(A,B,F2,D2)
=> ( pp(aa(set(B),bool,finite_finite(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),F2),A4)),D2))),finite_card(B,A4))) ) ) ).
% card_vimage_inj_on_le
tff(fact_3883_Fract_Otransfer,axiom,
pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_iu(int,fun(int,product_prod(int,int)))),fract)) ).
% Fract.transfer
tff(fact_3884_rat__number__collapse_I4_J,axiom,
! [W: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W)) ).
% rat_number_collapse(4)
tff(fact_3885_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)))),insert3(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_3886_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A3: B] :
( inj_on(A,B,F2,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),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),bot_bot(set(B))))),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ll(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F2),A4),A3))),bot_bot(set(A))))) ) ).
% inj_on_vimage_singleton
tff(fact_3887_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_3888_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_3889_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_3890_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_3891_euclidean__size__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).
% euclidean_size_1
tff(fact_3892_partition__filter__conv,axiom,
! [A: $tType,F2: 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),F2),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)),filter(A),F2),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),comp(bool,bool,A,fNot,F2)),Xs)) ).
% partition_filter_conv
tff(fact_3893_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_3894_distinct__adj__Nil,axiom,
! [A: $tType] : pp(aa(list(A),bool,distinct_adj(A),nil(A))) ).
% distinct_adj_Nil
tff(fact_3895_distinct__adj__mapD,axiom,
! [A: $tType,B: $tType,F2: 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),F2),Xs)))
=> pp(aa(list(B),bool,distinct_adj(B),Xs)) ) ).
% distinct_adj_mapD
tff(fact_3896_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_3897_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_3898_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_3899_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_3900_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_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_distinct__adj__map__iff,axiom,
! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
( inj_on(A,B,F2,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),F2),Xs)))
<=> pp(aa(list(A),bool,distinct_adj(A),Xs)) ) ) ).
% distinct_adj_map_iff
tff(fact_3910_distinct__adj__mapI,axiom,
! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
( pp(aa(list(A),bool,distinct_adj(A),Xs))
=> ( inj_on(A,B,F2,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),F2),Xs))) ) ) ).
% distinct_adj_mapI
tff(fact_3911_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_3912_floor__rat__def,axiom,
! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_lm(rat,fun(int,bool),X)) ).
% floor_rat_def
tff(fact_3913_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_ln(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_3914_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)),filter(A),P),Xs) ).
% partition_filter1
tff(fact_3915_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)),filter(A),comp(bool,bool,A,fNot,P)),Xs) ).
% partition_filter2
tff(fact_3916_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_3917_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) )
=> ! [Q5: A,R3: 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)))
=> ( ( R3 != zero_zero(A) )
=> ( ( divide_divide(A,A3,B2) = Q5 )
=> ( ( modulo_modulo(A,A3,B2) = R3 )
=> ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R3) ) ) ) ) ) ) )
=> ( B2 = zero_zero(A) ) ) ) ) ).
% divmod_cases
tff(fact_3918_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R2)),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),Q3),B2)),R2) = A3 )
=> ( modulo_modulo(A,A3,B2) = R2 ) ) ) ) ) ) ).
% mod_eqI
tff(fact_3919_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R2)),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),Q3),B2)),R2),B2) = Q3 ) ) ) ) ) ).
% div_bounded
tff(fact_3920_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R2)),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),Q3),B2)),R2) = A3 )
=> ( divide_divide(A,A3,B2) = Q3 ) ) ) ) ) ) ).
% div_eqI
tff(fact_3921_abs__division__segment,axiom,
! [K: int] : abs_abs(int,euclid7384307370059645450egment(int,K)) = one_one(int) ).
% abs_division_segment
tff(fact_3922_division__segment__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).
% division_segment_1
tff(fact_3923_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_3924_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_3925_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_3926_division__segment__nat__def,axiom,
! [N: nat] : euclid7384307370059645450egment(nat,N) = one_one(nat) ).
% division_segment_nat_def
tff(fact_3927_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_3928_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_3929_division__segment__int__def,axiom,
! [K: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( euclid7384307370059645450egment(int,K) = one_one(int) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( euclid7384307370059645450egment(int,K) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% division_segment_int_def
tff(fact_3930_of__rat__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R2)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R2),one_one(rat))) ) ) ).
% of_rat_le_1_iff
tff(fact_3931_one__le__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R2)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R2)) ) ) ).
% one_le_of_rat_iff
tff(fact_3932_of__rat__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),field_char_0_of_rat(A,R2)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R2),one_one(rat))) ) ) ).
% of_rat_less_1_iff
tff(fact_3933_one__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),field_char_0_of_rat(A,R2)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R2)) ) ) ).
% one_less_of_rat_iff
tff(fact_3934_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_3935_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_3936_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_3937_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_3938_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_3939_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(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))),insert3(list(A)),nil(A)),bot_bot(set(list(A)))) ).
% listrel_Nil
tff(fact_3940_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_3941_inverse__rat__def,axiom,
inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_iq(product_prod(int,int),product_prod(int,int))) ).
% inverse_rat_def
tff(fact_3942_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_3943_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_3944_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_3945_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_3946_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_3947_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_3948_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_3949_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_3950_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_3951_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_3952_lexordp__append__leftI,axiom,
! [A: $tType] :
( ord(A)
=> ! [Us: list(A),Vs: list(A),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),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),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ) ).
% lexordp_append_leftI
tff(fact_3953_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_3954_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_3955_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_3956_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_3957_lexordp__append__leftD,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A),Us: list(A),Vs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
=> ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),A6))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).
% lexordp_append_leftD
tff(fact_3958_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_3959_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,Ys3: 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),Ys3)))
=> ( ! [X2: A,Xs2: list(A),Y2: A,Ys3: 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),Ys3))) )
=> ( ! [X2: A,Xs2: list(A),Ys3: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys3))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys3))
=> 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),Ys3))) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys)) ) ) ) ) ) ).
% lexordp_induct
tff(fact_3960_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,Ys2: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
=> ( ! [X2: A] :
( ? [Xs3: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3)
=> ! [Y2: A] :
( ? [Ys2: list(A)] : Ys = 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,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
=> ! [Ys2: list(A)] :
( ( Ys = 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),ord_lexordp(A),Xs3),Ys2)) ) ) ) ) ) ) ).
% lexordp_cases
tff(fact_3961_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,Ys4: list(A)] :
( ( A1 = nil(A) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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),Xs4),Ys4)) ) ) ) ) ).
% lexordp.simps
tff(fact_3962_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,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
=> ~ 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) )
=> ! [Ys3: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ~ 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),Ys3)) ) ) ) ) ) ) ) ) ).
% lexordp.cases
tff(fact_3963_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Us: 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),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ) ) ).
% lexordp_append_left_rightI
tff(fact_3964_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))
| ? [Us2: list(A),A5: A,B4: A,Vs2: list(A),Ws3: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B4))
& ( 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),A5),Vs2)) )
& ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),Ws3)) ) ) ) ) ) ).
% lexordp_iff
tff(fact_3965_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B,Xs: list(B)] : image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(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,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),X),bot_bot(set(B))))),image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert3(list(B)),Xs),bot_bot(set(list(B)))))) ).
% listrel_Cons
tff(fact_3966_num__of__nat__One,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
=> ( num_of_nat(N) = one2 ) ) ).
% num_of_nat_One
tff(fact_3967_Sigma__Image,axiom,
! [A: $tType,B: $tType,A4: set(B),B3: fun(B,set(A)),X6: set(B)] : image(B,A,product_Sigma(B,A,A4,B3),X6) = complete_Sup_Sup(set(A),image2(B,set(A),B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X6),A4))) ).
% Sigma_Image
tff(fact_3968_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V2: A,E4: 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),V2),transitive_trancl(A,E4)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,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),V2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E4),product_Sigma(A,A,S,aTP_Lamp_jt(set(A),fun(A,set(A)),S)))))) ) ) ) ).
% trancl_restrict_reachable
tff(fact_3969_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ord(A)
=> ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_kr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).
% ord_class.lexordp_def
tff(fact_3970_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_3971_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : uniq(list(A),aTP_Lamp_lo(set(A),fun(list(A),bool),A4)) ) ).
% strict_sorted_equal_Uniq
tff(fact_3972_one__int_Otransfer,axiom,
pcr_int(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)),one_one(int)) ).
% one_int.transfer
tff(fact_3973_lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F2: fun(A,A),P: A] :
( order_mono(A,A,F2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F2)),P))),P))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F2)),P)) ) ) ) ).
% lfp_induct
tff(fact_3974_def__lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: A,F2: fun(A,A),P: A] :
( ( A4 = complete_lattice_lfp(A,F2) )
=> ( order_mono(A,A,F2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,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_3975_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_ko(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_3976_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: fun(C,assn),H2: 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)),H2),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)),H2),bot_bot(set(nat))))) ) ).
% mod_h_bot_iff(8)
tff(fact_3977_ex__assn__const,axiom,
! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_lp(assn,fun(A,assn),C2)) = C2 ).
% ex_assn_const
tff(fact_3978_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_3979_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_3980_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_3981_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_lq(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).
% norm_assertion_simps(16)
tff(fact_3982_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_lr(assn,fun(fun(B,assn),fun(B,assn)),R),Q)) ).
% norm_assertion_simps(17)
tff(fact_3983_mod__ex__dist,axiom,
! [A: $tType,P: fun(A,assn),H2: 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)),H2))
<=> ? [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)),H2)) ) ).
% mod_ex_dist
tff(fact_3984_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_3985_norm__assertion__simps_I21_J,axiom,
! [F10: $tType,Q: assn,P: fun(F10,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Q),ex_assn(F10,P)) = ex_assn(F10,aa(fun(F10,assn),fun(F10,assn),aTP_Lamp_ls(assn,fun(fun(F10,assn),fun(F10,assn)),Q),P)) ).
% norm_assertion_simps(21)
tff(fact_3986_norm__assertion__simps_I20_J,axiom,
! [E5: $tType,Q: fun(E5,assn),P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(E5,Q)),P) = ex_assn(E5,aa(assn,fun(E5,assn),aTP_Lamp_lt(fun(E5,assn),fun(assn,fun(E5,assn)),Q),P)) ).
% norm_assertion_simps(20)
tff(fact_3987_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_lu(assn,fun(fun(D,assn),fun(D,assn)),Q),P)) ).
% norm_assertion_simps(19)
tff(fact_3988_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_lv(fun(C,assn),fun(assn,fun(C,assn)),Q),P)) ).
% norm_assertion_simps(18)
tff(fact_3989_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_3990_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_3991_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_3992_ord_Olexordp_Ocong,axiom,
! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = lexordp2(A,Less) ).
% ord.lexordp.cong
tff(fact_3993_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Us: list(A),Vs: list(A),Xs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us),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),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ).
% ord.lexordp_append_leftI
tff(fact_3994_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A),Us: 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),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
=> ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),A6))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us),Vs)) ) ) ).
% ord.lexordp_append_leftD
tff(fact_3995_ex__distrib__star,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_lq(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_3996_mod__exE,axiom,
! [A: $tType,P: fun(A,assn),H2: 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)),H2))
=> ~ ! [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)),H2)) ) ).
% mod_exE
tff(fact_3997_mod__exI,axiom,
! [A: $tType,P: fun(A,assn),H2: 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)),H2))
=> 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)),H2)) ) ).
% mod_exI
tff(fact_3998_ex__one__point__gen,axiom,
! [A: $tType,P: fun(A,assn),V2: 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 = V2 ) )
=> ( ex_assn(A,P) = aa(A,assn,P,V2) ) ) ).
% ex_one_point_gen
tff(fact_3999_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_4000_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_4001_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_4002_ex__distrib__or,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_lw(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_4003_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_lx(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ly(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).
% ex_join_or
tff(fact_4004_ex__distrib__and,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_lz(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_4005_lfp__induct__set,axiom,
! [A: $tType,A3: A,F2: fun(set(A),set(A)),P: fun(A,bool)] :
( pp(member2(A,A3,complete_lattice_lfp(set(A),F2)))
=> ( order_mono(set(A),set(A),F2)
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(set(A),set(A),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F2)),collect(A,P)))))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% lfp_induct_set
tff(fact_4006_def__lfp__induct__set,axiom,
! [A: $tType,A4: set(A),F2: fun(set(A),set(A)),A3: A,P: fun(A,bool)] :
( ( A4 = complete_lattice_lfp(set(A),F2) )
=> ( order_mono(set(A),set(A),F2)
=> ( pp(member2(A,A3,A4))
=> ( ! [X2: A] :
( pp(member2(A,X2,aa(set(A),set(A),F2,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_4007_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,Ys4: list(A)] :
( ( A1 = nil(A) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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),Xs4),Ys4)) ) ) ) ).
% ord.lexordp.simps
tff(fact_4008_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,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)
=> ! [Y2: A] :
( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
=> ~ 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) )
=> ! [Ys3: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ~ 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),Ys3)) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
tff(fact_4009_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_4010_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Us: 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),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ) ).
% ord.lexordp_append_left_rightI
tff(fact_4011_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_4012_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F2: 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)),F2)))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F2)
=> ( ! [A6: A,B5: 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),A6),B5),aa(set(product_prod(A,B)),set(product_prod(A,B)),F2,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F2)),collect(product_prod(A,B),product_case_prod(A,B,bool,P))))))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A6),B5)) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) ) ) ) ).
% lfp_induct2
tff(fact_4013_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_ma(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P)) ).
% ex_assn_def
tff(fact_4014_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_mb(A,fun(list(A),fun(nat,list(A))),X),Xs),N) ).
% take_Cons
tff(fact_4015_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)))) ) ) )
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),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,A6),B5)),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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))) ) ) )
=> ( ! [V: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( 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),V),Va)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va) )
=> ( ( 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),V),Va)))) ) ) ) ) ) ) ) ) ).
% zipf.pelims
tff(fact_4016_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)
<=> ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),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),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4)))) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
tff(fact_4017_one__int__def,axiom,
one_one(int) = abs_Integ(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).
% one_int_def
tff(fact_4018_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_4019_transp__inf,axiom,
! [A: $tType,R2: fun(A,fun(A,bool)),S3: fun(A,fun(A,bool))] :
( transp(A,R2)
=> ( transp(A,S3)
=> 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))),R2),S3)) ) ) ).
% transp_inf
tff(fact_4020_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_4021_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_mc(list(A),fun(nat,list(A)),Xs),N) ).
% drop_Cons
tff(fact_4022_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),I2: nat,V2: 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)),I2),V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_md(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V2),I2) ).
% list_update.simps(2)
tff(fact_4023_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_4024_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType,N: nat,X: A,F2: fun(nat,A)] :
( ( ( N = zero_zero(nat) )
=> ( case_nat(A,X,F2,N) = X ) )
& ( ( N != zero_zero(nat) )
=> ( case_nat(A,X,F2,N) = aa(nat,A,F2,minus_minus(nat,N,one_one(nat))) ) ) ) ).
% Nitpick.case_nat_unfold
tff(fact_4025_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))) ) )
=> ~ ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),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,A6)),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),A6),As)),Xb))) ) ) ) ) ) ).
% map_tailrec_rev.pelims
tff(fact_4026_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,Ys3: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ( ( 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),Ys3))) ) )
& ( ~ 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),Ys3)) ) ) )
=> ~ 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),Ys3)))) ) ) )
=> ( ( ( 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) )
=> ! [V: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ~ 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),V),Va)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
tff(fact_4027_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)) ) )
=> ~ ! [A6: A,As: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ( ( Y = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),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),A6),As)),Xa)) ) ) ) ) ) ).
% revg.pelims
tff(fact_4028_irrefl__tranclI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)) = 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,R2))) ) ).
% irrefl_tranclI
tff(fact_4029_in__listrel1__converse,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2))))
<=> 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,R2)))) ) ).
% in_listrel1_converse
tff(fact_4030_listrel1__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,converse(A,A,R2)) = converse(list(A),list(A),listrel1(A,R2)) ).
% listrel1_converse
tff(fact_4031_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),R2),S3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S3)) ).
% converse_Int
tff(fact_4032_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list(A)] : revg(A,nil(A),B2) = B2 ).
% revg.simps(1)
tff(fact_4033_revg_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [A6: A,As: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ( Y != revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),Xa)) ) ) ) ) ).
% revg.elims
tff(fact_4034_is__singleton__altdef,axiom,
! [A: $tType,A4: set(A)] :
( is_singleton(A,A4)
<=> ( finite_card(A,A4) = one_one(nat) ) ) ).
% is_singleton_altdef
tff(fact_4035_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_4036_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_4037_lexn_Osimps_I2_J,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),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)))),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),R2,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)))),collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_me(nat,fun(list(A),fun(list(A),bool)),N)))) ).
% lexn.simps(2)
tff(fact_4038_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_4039_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_4040_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_4041_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_4042_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_4043_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_4044_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_4045_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_4046_successively__map,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(A,bool)),F2: 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),F2),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_ax(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),P),F2)),Xs)) ) ).
% successively_map
tff(fact_4047_distinct__adj__def,axiom,
! [A: $tType] : distinct_adj(A) = aa(fun(A,fun(A,bool)),fun(list(A),bool),successively(A),aTP_Lamp_eg(A,fun(A,bool))) ).
% distinct_adj_def
tff(fact_4048_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_4049_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_4050_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_4051_successively__iff__sorted__wrt__strong,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
( ! [X2: A,Y2: A,Z3: 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,Z3,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),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Z3)) ) ) ) ) )
=> ( 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_4052_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_4053_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_4054_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_mg(fun(A,fun(A,bool)),fun(A,fun(list(A),fun(bool,bool))),X3)) ).
% successively_altdef
tff(fact_4055_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_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_bit__2__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( pp(bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(one2)),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),one_one(nat)))
& ( N = one_one(nat) ) ) ) ) ).
% bit_2_iff
tff(fact_4064_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_4065_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_4066_Refl__Field__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R2),R2)
=> ( 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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,R2)),A4) ) ) ).
% Refl_Field_Restr
tff(fact_4067_Refl__Field__Restr2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R2),R2)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R2)))
=> ( 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).
% Refl_Field_Restr2
tff(fact_4068_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_4069_rtrancl__Int__subset,axiom,
! [A: $tType,S3: set(product_prod(A,A)),R2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id2(A)),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))),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),R2)),S3),R2)),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))),aa(set(product_prod(A,A)),set(product_prod(A,A)),transitive_rtrancl(A),R2)),S3)) ) ) ).
% rtrancl_Int_subset
tff(fact_4070_Restr__Field,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,field2(A,R2),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,set(A)),R2))) = R2 ).
% Restr_Field
tff(fact_4071_Field__Restr__subset,axiom,
! [A: $tType,R2: 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4))))),A4)) ).
% Field_Restr_subset
tff(fact_4072_Refl__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( refl_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Refl_Restr
tff(fact_4073_inf__Sup__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A4: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite(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_4074_inf__Sup1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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_mi(set(A),fun(A,fun(A,bool)),A4),X))) ) ) ) ) ).
% inf_Sup1_distrib
tff(fact_4075_inf__Sup2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite(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_mj(set(A),fun(set(A),fun(A,bool)),A4),B3))) ) ) ) ) ) ) ).
% inf_Sup2_distrib
tff(fact_4076_Preorder__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_preorder_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Preorder_Restr
tff(fact_4077_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(A),X),A4)) = X ) )
& ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(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)),insert3(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).
% Inf_fin.insert_remove
tff(fact_4078_Inf__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(member2(A,X,A4))
=> ( ( ( minus_minus(set(A),A4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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)),insert3(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)),insert3(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Inf_fin.remove
tff(fact_4079_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(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_4080_Partial__order__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_7125193373082350890der_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Partial_order_Restr
tff(fact_4081_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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_4082_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [H2: fun(A,A),N2: set(A)] :
( ! [X2: A,Y2: A] : aa(A,A,H2,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,H2,X2)),aa(A,A,H2,Y2))
=> ( pp(aa(set(A),bool,finite_finite(A),N2))
=> ( ( N2 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic7752659483105999362nf_fin(A),N2)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),image2(A,A,H2,N2)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_4083_Inf__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(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_4084_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),Y2),bot_bot(set(A))))))
=> pp(member2(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A4),A4)) ) ) ) ) ).
% Inf_fin.closed
tff(fact_4085_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(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)),insert3(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_4086_Inf__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( ( A4 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite(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_4087_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),A4)) = finite_fold(A,A,inf_inf(A),X,A4) ) ) ) ).
% Inf_fin.eq_fold
tff(fact_4088_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_4089_Linear__order__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_679001287576687338der_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Linear_order_Restr
tff(fact_4090_inter__Set__filter,axiom,
! [A: $tType,B3: set(A),A4: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),B3))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = filter2(A,aTP_Lamp_a(set(A),fun(A,bool),A4),B3) ) ) ).
% inter_Set_filter
tff(fact_4091_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_4092_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_4093_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),C5: set(A),B3: set(B),D2: set(B)] :
( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D2))
=> ( bij_betw(A,B,F2,C5,D2)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C5) = 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,F2,A4,B3) ) ) ) ) ).
% bij_betw_partition
tff(fact_4094_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B3: set(B),C5: set(A),D2: set(B)] :
( bij_betw(A,B,F2,A4,B3)
=> ( bij_betw(A,B,F2,C5,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,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D2)) ) ) ) ).
% bij_betw_combine
tff(fact_4095_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),C5: set(B),G: fun(A,B),B3: set(A),D2: set(B)] :
( bij_betw(A,B,F2,A4,C5)
=> ( 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)),C5),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_cl(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),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)),C5),D2)) ) ) ) ) ).
% bij_betw_disjoint_Un
tff(fact_4096_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),M))
=> ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or1337092689740270186AtMost(nat,one_one(nat),finite_card(A,M)),M) ) ).
% ex_bij_betw_nat_finite_1
tff(fact_4097_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T4: set(C),H2: fun(B,C),S: set(B),T: set(C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite(B),S2))
=> ( pp(aa(set(C),bool,finite_finite(C),T4))
=> ( bij_betw(B,C,H2,minus_minus(set(B),S,S2),minus_minus(set(C),T,T4))
=> ( ! [A6: B] :
( pp(member2(B,A6,S2))
=> ( aa(C,A,G,aa(B,C,H2,A6)) = one_one(A) ) )
=> ( ! [B5: C] :
( pp(member2(C,B5,T4))
=> ( aa(C,A,G,B5) = 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_mk(fun(B,C),fun(fun(C,A),fun(B,A)),H2),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_4098_filter__set,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter2(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)),filter(A),P),Xs)) ).
% filter_set
tff(fact_4099_linear__order__on__Restr,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A] :
( order_679001287576687338der_on(A,A4,R2)
=> 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,R2,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))),R2),product_Sigma(A,A,order_above(A,R2,X),aa(A,fun(A,set(A)),aTP_Lamp_ml(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X)))) ) ).
% linear_order_on_Restr
tff(fact_4100_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_4101_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_4102_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_4103_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_4104_power__int__1__right,axiom,
! [A: $tType] :
( ( inverse(A)
& monoid_mult(A) )
=> ! [Y: A] : power_int(A,Y,one_one(int)) = Y ) ).
% power_int_1_right
tff(fact_4105_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_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_power__int__minus1__right,axiom,
! [A: $tType] :
( ( inverse(A)
& monoid_mult(A) )
=> ! [Y: A] : power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ).
% power_int_minus1_right
tff(fact_4113_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_4114_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_4115_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_4116_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_4117_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N2: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N2))
=> ( 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,N2))) ) ) ) ).
% power_int_increasing
tff(fact_4118_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N2: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N2))
=> ( 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,N2))) ) ) ) ).
% power_int_strict_increasing
tff(fact_4119_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_4120_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_4121_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N2: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N2))
=> ( 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,N2)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_4122_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_4123_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_4124_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_4125_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_4126_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N2: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N2))
=> ( 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) )
| ( N2 != zero_zero(int) )
| ( N = zero_zero(int) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N2)),power_int(A,A3,N))) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_4127_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_4128_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_4129_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_4130_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_4131_mset__set__Union,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(aa(set(A),bool,finite_finite(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_4132_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_4133_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)) ) )
=> ( ! [V: A,Va: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ~ 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),V),Va)),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,L2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L2) )
=> ( ( ( 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),L2))) ) )
& ( ~ 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,L2)) ) )
& ( ( 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),L2)) ) ) ) ) )
=> ~ 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),L2))) ) ) ) ) ) ) ) ) ).
% merge.pelims
tff(fact_4134_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F2,A3),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).
% group.inverse_distrib_swap
tff(fact_4135_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,Z2),A3) = A3 ) ) ).
% group.group_left_neutral
tff(fact_4136_group_Oinverse__neutral,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,Z2) = Z2 ) ) ).
% group.inverse_neutral
tff(fact_4137_group_Oinverse__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).
% group.inverse_inverse
tff(fact_4138_group_Oinverse__unique,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = Z2 )
=> ( aa(A,A,Inverse,A3) = B2 ) ) ) ).
% group.inverse_unique
tff(fact_4139_group_Oright__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).
% group.right_inverse
tff(fact_4140_group_Oright__cancel,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,B2),A3) = aa(A,A,aa(A,fun(A,A),F2,C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% group.right_cancel
tff(fact_4141_group_Oleft__inverse,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F2,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).
% group.left_inverse
tff(fact_4142_group_Oleft__cancel,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
( group(A,F2,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = aa(A,A,aa(A,fun(A,A),F2,A3),C2) )
<=> ( B2 = C2 ) ) ) ).
% group.left_cancel
tff(fact_4143_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_4144_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L22: list(A)] : merge(A,nil(A),L22) = L22 ) ).
% merge.simps(1)
tff(fact_4145_group_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> semigroup(A,F2) ) ).
% group.axioms(1)
tff(fact_4146_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [V2: A,Va2: list(A)] : merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ).
% merge.simps(2)
tff(fact_4147_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_4148_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_4149_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_4150_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_4151_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 ) )
=> ( ! [V: A,Va: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ) ) )
=> ~ ! [X1: A,L1: list(A)] :
( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1) )
=> ! [X23: A,L2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L2) )
=> ~ ( ( 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),L2))) ) )
& ( ~ 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,L2)) ) )
& ( ( 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),L2)) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
tff(fact_4152_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_4153_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_4154_group_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( semigroup(A,F2)
=> ( group_axioms(A,F2,Z2,Inverse)
=> group(A,F2,Z2,Inverse) ) ) ).
% group.intro
tff(fact_4155_group__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
<=> ( semigroup(A,F2)
& group_axioms(A,F2,Z2,Inverse) ) ) ).
% group_def
tff(fact_4156_merge__list_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X,Xa) = Y )
=> ( ( ( X = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != nil(A) ) ) )
=> ( ( ( X = nil(list(A)) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( Y != L3 ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) ) ) )
=> ~ ! [L1: list(A),L2: 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)),L2),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,L2)),X),Ls2) ) ) ) ) ) ) ) ) ).
% merge_list.elims
tff(fact_4157_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)),insert3(A),N),bot_bot(set(A))) ) ).
% atMost_Int_atLeast
tff(fact_4158_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: 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,C2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),B2) ) ).
% Int_atLeastAtMostL2
tff(fact_4159_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: 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,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2),D3) ) ).
% Int_atLeastAtMostR2
tff(fact_4160_group__axioms_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A6) = A6
=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A6)),A6) = Z2
=> group_axioms(A,F2,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_4161_group__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F2,Z2,Inverse)
<=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,Z2),A5) = A5
& ! [A5: A] : aa(A,A,aa(A,fun(A,A),F2,aa(A,A,Inverse,A5)),A5) = Z2 ) ) ).
% group_axioms_def
tff(fact_4162_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_4163_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_4164_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_4165_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_4166_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_4167_group_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F2,Z2,Inverse)
=> group_axioms(A,F2,Z2,Inverse) ) ).
% group.axioms(2)
tff(fact_4168_mergesort__remdups__def,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: 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_mm(A,list(A))),Xs)) ) ).
% mergesort_remdups_def
tff(fact_4169_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)) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( ( Y = L3 )
=> ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)) )
=> ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A)))) ) ) )
=> ( ! [La: list(A),Acc2: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2))) )
=> ~ accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) ) )
=> ~ ! [L1: list(A),L2: 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)),L2),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,L2)),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)),L2),Ls2)))) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
tff(fact_4170_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_4171_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),F2: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F2)
=> foldin3648464208017769352axioms(B,A,S,F2) ) ).
% folding_insort_key.axioms(2)
tff(fact_4172_folding__insort__key__axioms__def,axiom,
! [A: $tType,B: $tType,S: set(B),F2: fun(B,A)] :
( foldin3648464208017769352axioms(B,A,S,F2)
<=> inj_on(B,A,F2,S) ) ).
% folding_insort_key_axioms_def
tff(fact_4173_folding__insort__key__axioms_Ointro,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),S: set(B)] :
( inj_on(B,A,F2,S)
=> foldin3648464208017769352axioms(B,A,S,F2) ) ).
% folding_insort_key_axioms.intro
tff(fact_4174_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_mn(set(A),fun(A,set(A)),A4))) ).
% rel_restrict_alt_def
tff(fact_4175_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_4176_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_4177_total__on__imp__Total__Restr,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
( total_on(A,A4,R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% total_on_imp_Total_Restr
tff(fact_4178_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_4179_total__lexord,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( total_on(A,top_top(set(A)),R2)
=> total_on(list(A),top_top(set(list(A))),lexord(A,R2)) ) ).
% total_lexord
tff(fact_4180_total__lenlex,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( total_on(A,top_top(set(A)),R2)
=> total_on(list(A),top_top(set(list(A))),lenlex(A,R2)) ) ).
% total_lenlex
tff(fact_4181_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_4182_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_4183_sorted__wrt__quicksort__by__rel,axiom,
! [X8: $tType,R: fun(X8,fun(X8,bool)),Xs: list(X8)] :
( ! [X2: X8,Y2: X8] :
( pp(aa(X8,bool,aa(X8,fun(X8,bool),R,X2),Y2))
| pp(aa(X8,bool,aa(X8,fun(X8,bool),R,Y2),X2)) )
=> ( ! [X2: X8,Y2: X8,Z3: X8] :
( pp(aa(X8,bool,aa(X8,fun(X8,bool),R,X2),Y2))
=> ( pp(aa(X8,bool,aa(X8,fun(X8,bool),R,Y2),Z3))
=> pp(aa(X8,bool,aa(X8,fun(X8,bool),R,X2),Z3)) ) )
=> sorted_wrt(X8,R,aa(list(X8),list(X8),quicksort_by_rel(X8,R,nil(X8)),Xs)) ) ) ).
% sorted_wrt_quicksort_by_rel
tff(fact_4184_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_4185_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_mo(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_4186_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_mo(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_4187_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_4188_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_4189_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_4190_Total__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( total_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Total_Restr
tff(fact_4191_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_mo(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_4192_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_mo(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_4193_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_4194_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_4195_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_4196_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))
=> ( ! [Xs1: 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)),Xs1),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)),Xs1),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)),Xs1),Xs22)),nil(A))) ) ) )
=> ( ! [Xs1: 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)),Xs1),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),Xs1)),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) ) ) )
=> ~ ! [Xs1: 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)),Xs1),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),Xs1)),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
tff(fact_4197_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_4198_integer__of__nat__1,axiom,
code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).
% integer_of_nat_1
tff(fact_4199_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_4200_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat,S3: A] :
( pp(aa(set(A),bool,finite_finite(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,S3)))))
=> ( infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),set_ord_lessThan(A,S3)),N) = infini527867602293511546merate(A,S,N) ) ) ) ) ).
% finite_enumerate_initial_segment
tff(fact_4201_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F2)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).
% max_of_antimono
tff(fact_4202_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F2: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F2)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).
% min_of_antimono
tff(fact_4203_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_4204_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_4205_well__order__on__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R2)))
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ) ).
% well_order_on_Restr
tff(fact_4206_char__of__def,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: A] : unique5772411509450598832har_of(A,N) = char2(aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),N)),bit_se5641148757651400278ts_bit(A,N,one_one(nat)),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit0(one2))),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit1(one2))),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),bit_se5641148757651400278ts_bit(A,N,aa(num,nat,numeral_numeral(nat),bit1(bit1(one2))))) ) ).
% char_of_def
tff(fact_4207_list__all__iff__all__interval__int,axiom,
! [P: fun(int,bool),I2: int,J: int] :
( pp(aa(list(int),bool,aa(fun(int,bool),fun(list(int),bool),list_all(int),P),upto(I2,J)))
<=> all_interval_int(P,I2,J) ) ).
% list_all_iff_all_interval_int
tff(fact_4208_list__ex__iff__not__all__inverval__int,axiom,
! [P: fun(int,bool),I2: int,J: int] :
( pp(aa(list(int),bool,aa(fun(int,bool),fun(list(int),bool),list_ex(int),P),upto(I2,J)))
<=> ~ all_interval_int(comp(bool,bool,int,fNot,P),I2,J) ) ).
% list_ex_iff_not_all_inverval_int
tff(fact_4209_Well__order__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% Well_order_Restr
tff(fact_4210_all__interval__int__def,axiom,
! [P: fun(int,bool),I2: int,J: int] :
( all_interval_int(P,I2,J)
<=> ! [X4: int] :
( pp(member2(int,X4,set_or1337092689740270186AtMost(int,I2,J)))
=> pp(aa(int,bool,P,X4)) ) ) ).
% all_interval_int_def
tff(fact_4211_ofilter__Restr__Int,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(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_4212_ofilter__Restr__subset,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3))),A4) ) ) ) ).
% ofilter_Restr_subset
tff(fact_4213_Field__Restr__ofilter,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).
% Field_Restr_ofilter
tff(fact_4214_ofilter__subset__embedS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ).
% ofilter_subset_embedS
tff(fact_4215_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
tff(fact_4216_ofilter__subset__embed,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3))),id(A)) ) ) ) ) ).
% ofilter_subset_embed
tff(fact_4217_ofilter__embed,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R2)))
& 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4))),R2,id(A)) ) ) ) ).
% ofilter_embed
tff(fact_4218_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).
% ofilter_subset_ordLess
tff(fact_4219_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B)),F2: fun(B,A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R5),R5)
=> ( bNF_Wellorder_embed(B,A,R5,R2,F2)
=> bNF_Wellorder_iso(B,A,R5,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,image2(B,A,F2,field2(B,R5)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_mp(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R5),F2))),F2) ) ) ) ).
% embed_implies_iso_Restr
tff(fact_4220_ofilter__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),field2(A,R2)))
<=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))),R2),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% ofilter_ordLess
tff(fact_4221_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( ( field2(A,R2) != 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))),R2),product_Sigma(A,A,order_underS(A,R2,A3),aa(A,fun(A,set(A)),aTP_Lamp_mq(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A3)))),R2),bNF_We4044943003108391690rdLess(A,A))) ) ) ).
% underS_Restr_ordLess
tff(fact_4222_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,A4)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(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))),R2),product_Sigma(A,A,B3,aTP_Lamp_jt(set(A),fun(A,set(A)),B3)))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).
% ofilter_subset_ordLeq
tff(fact_4223_ofilter__Restr__under,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),A3: A] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4))),A3) = order_under(A,R2,A3) ) ) ) ) ).
% ofilter_Restr_under
tff(fact_4224_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_4225_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R5),R5)
=> ( 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))),R2),R5),bNF_Wellorder_ordLeq(A,B)))
<=> ! [X4: A] :
( pp(member2(A,X4,field2(A,R2)))
=> 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))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_mq(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),R5),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
tff(fact_4226_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_4227_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_4228_ordLess__iff__ordIso__Restr,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R5),R5)
=> ( 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))),R5),R2),bNF_We4044943003108391690rdLess(B,A)))
<=> ? [X4: A] :
( pp(member2(A,X4,field2(A,R2)))
& 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))),R5),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_mq(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
tff(fact_4229_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_4230_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_4231_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F2,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,F2,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_mr(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xs),aa(B,option(A),F2,X)) ).
% map_filter_simps(1)
tff(fact_4232_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_4233_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_4234_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_4235_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_4236_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_4237_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_4238_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_4239_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_4240_unit__factor__simps_I2_J,axiom,
! [N: nat] : unit_f5069060285200089521factor(nat,aa(nat,nat,suc,N)) = one_one(nat) ).
% unit_factor_simps(2)
tff(fact_4241_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_4242_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_4243_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_4244_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_4245_unit__factor__nat__def,axiom,
! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( unit_f5069060285200089521factor(nat,N) = zero_zero(nat) ) )
& ( ( N != zero_zero(nat) )
=> ( unit_f5069060285200089521factor(nat,N) = one_one(nat) ) ) ) ).
% unit_factor_nat_def
tff(fact_4246_mult__gcd__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),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,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).
% mult_gcd_left
tff(fact_4247_mult__gcd__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).
% mult_gcd_right
tff(fact_4248_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_4249_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_4250_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_4251_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_ms(A,bool)))) ) ).
% Lcm_no_units
tff(fact_4252_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_4253_list__all__iff__all__interval__nat,axiom,
! [P: fun(nat,bool),I2: nat,J: nat] :
( pp(aa(list(nat),bool,aa(fun(nat,bool),fun(list(nat),bool),list_all(nat),P),upt(I2,J)))
<=> all_interval_nat(P,I2,J) ) ).
% list_all_iff_all_interval_nat
tff(fact_4254_list__ex__iff__not__all__inverval__nat,axiom,
! [P: fun(nat,bool),I2: nat,J: nat] :
( pp(aa(list(nat),bool,aa(fun(nat,bool),fun(list(nat),bool),list_ex(nat),P),upt(I2,J)))
<=> ~ all_interval_nat(comp(bool,bool,nat,fNot,P),I2,J) ) ).
% list_ex_iff_not_all_inverval_nat
tff(fact_4255_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_4256_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_4257_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_4258_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_4259_Gcd__nat__eq__one,axiom,
! [N2: set(nat)] :
( pp(member2(nat,one_one(nat),N2))
=> ( gcd_Gcd(nat,N2) = one_one(nat) ) ) ).
% Gcd_nat_eq_one
tff(fact_4260_Lcm__nat__empty,axiom,
gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).
% Lcm_nat_empty
tff(fact_4261_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_4262_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_4263_all__interval__nat__def,axiom,
! [P: fun(nat,bool),I2: nat,J: nat] :
( all_interval_nat(P,I2,J)
<=> ! [X4: nat] :
( pp(member2(nat,X4,set_or7035219750837199246ssThan(nat,I2,J)))
=> pp(aa(nat,bool,P,X4)) ) ) ).
% all_interval_nat_def
tff(fact_4264_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_4265_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_4266_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_4267_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_4268_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_bb(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_4269_lcm__1__iff__int,axiom,
! [M2: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M2),N) = one_one(int) )
<=> ( ( ( M2 = one_one(int) )
| ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) ) )
& ( ( N = one_one(int) )
| ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
% lcm_1_iff_int
tff(fact_4270_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_4271_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_4272_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_4273_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_4274_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_4275_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_4276_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_4277_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_4278_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_4279_listsp_ONil,axiom,
! [A: $tType,A4: fun(A,bool)] : pp(aa(list(A),bool,listsp(A,A4),nil(A))) ).
% listsp.Nil
tff(fact_4280_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_4281_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_4282_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_4283_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_4284_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_4285_mult__lcm__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),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,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).
% mult_lcm_left
tff(fact_4286_mult__lcm__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).
% mult_lcm_right
tff(fact_4287_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_4288_Lcm__nat__set__eq__fold,axiom,
! [Xs: list(nat)] : gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),fold(nat,nat),gcd_lcm(nat)),Xs),one_one(nat)) ).
% Lcm_nat_set_eq_fold
tff(fact_4289_listsp_Osimps,axiom,
! [A: $tType,A4: fun(A,bool),A3: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),A3))
<=> ( ( A3 = nil(A) )
| ? [A5: A,L4: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),L4) )
& pp(aa(A,bool,A4,A5))
& pp(aa(list(A),bool,listsp(A,A4),L4)) ) ) ) ).
% listsp.simps
tff(fact_4290_listsp_Ocases,axiom,
! [A: $tType,A4: fun(A,bool),A3: list(A)] :
( pp(aa(list(A),bool,listsp(A,A4),A3))
=> ( ( A3 != nil(A) )
=> ~ ! [A6: A,L3: list(A)] :
( ( A3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),L3) )
=> ( pp(aa(A,bool,A4,A6))
=> ~ pp(aa(list(A),bool,listsp(A,A4),L3)) ) ) ) ) ).
% listsp.cases
tff(fact_4291_Lcm__int__set__eq__fold,axiom,
! [Xs: list(int)] : gcd_Lcm(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,aa(list(int),fun(int,int),aa(fun(int,fun(int,int)),fun(list(int),fun(int,int)),fold(int,int),gcd_lcm(int)),Xs),one_one(int)) ).
% Lcm_int_set_eq_fold
tff(fact_4292_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_mult_unit2
tff(fact_4293_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_mult_unit1
tff(fact_4294_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: 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)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_div_unit1
tff(fact_4295_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),divide_divide(A,C2,A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_div_unit2
tff(fact_4296_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( pp(aa(set(A),bool,finite_finite(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_finite(A),A4))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) ) ) ) ) ).
% Lcm_fin.eq_fold
tff(fact_4297_Lcm__nat__def,axiom,
! [M: set(nat)] :
( ( pp(aa(set(nat),bool,finite_finite(nat),M))
=> ( gcd_Lcm(nat,M) = lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M) ) )
& ( ~ pp(aa(set(nat),bool,finite_finite(nat),M))
=> ( gcd_Lcm(nat,M) = zero_zero(nat) ) ) ) ).
% Lcm_nat_def
tff(fact_4298_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_4299_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_4300_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_4301_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_4302_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_finite(A),A4)) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_4303_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_4304_count__mset__set_H,axiom,
! [A: $tType,A4: set(A),X: A] :
( ( ( pp(aa(set(A),bool,finite_finite(A),A4))
& pp(member2(A,X,A4)) )
=> ( aa(A,nat,count(A,mset_set(A,A4)),X) = one_one(nat) ) )
& ( ~ ( pp(aa(set(A),bool,finite_finite(A),A4))
& pp(member2(A,X,A4)) )
=> ( aa(A,nat,count(A,mset_set(A,A4)),X) = zero_zero(nat) ) ) ) ).
% count_mset_set'
tff(fact_4305_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite(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_4306_monoid__list_ONil,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A] :
( groups_monoid_list(A,F2,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F2,Z2),nil(A)) = Z2 ) ) ).
% monoid_list.Nil
tff(fact_4307_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_4308_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_4309_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_4310_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_4311_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_4312_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_4313_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_4314_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_4315_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_4316_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_4317_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_4318_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_4319_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_4320_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_4321_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_4322_count__mset__set_I1_J,axiom,
! [A: $tType,A4: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( pp(member2(A,X,A4))
=> ( aa(A,nat,count(A,mset_set(A,A4)),X) = one_one(nat) ) ) ) ).
% count_mset_set(1)
tff(fact_4323_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_4324_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_4325_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_4326_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_4327_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_4328_gcd__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).
% gcd_mult_left
tff(fact_4329_gcd__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A3)),C2)) ) ).
% gcd_mult_right
tff(fact_4330_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).
% gcd_mult_distrib'
tff(fact_4331_lcm__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: 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),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).
% lcm_mult_left
tff(fact_4332_lcm__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: 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),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A3)),C2)) ) ).
% lcm_mult_right
tff(fact_4333_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).
% lcm_mult_distrib'
tff(fact_4334_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_4335_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C2: A,A4: set(A)] : gcd_Gcd(A,image2(A,A,aa(A,fun(A,A),times_times(A),C2),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A4))) ) ).
% Gcd_mult
tff(fact_4336_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_4337_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A),C2: A] :
( ( A4 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,image2(A,A,aa(A,fun(A,A),times_times(A),C2),A4)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A4))) ) ) ) ).
% Lcm_mult
tff(fact_4338_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_4339_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),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_4340_Gcd__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A),B2: A] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(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_4341_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_4342_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_4343_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_4344_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_4345_distinct__count__atmost__1,axiom,
! [A: $tType,X: list(A)] :
( pp(aa(list(A),bool,distinct(A),X))
<=> ! [A5: A] :
( ( pp(member2(A,A5,aa(list(A),set(A),set2(A),X)))
=> ( aa(A,nat,count(A,mset(A,X)),A5) = one_one(nat) ) )
& ( ~ pp(member2(A,A5,aa(list(A),set(A),set2(A),X)))
=> ( aa(A,nat,count(A,mset(A,X)),A5) = zero_zero(nat) ) ) ) ) ).
% distinct_count_atmost_1
tff(fact_4346_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
=> ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) )
=> ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
tff(fact_4347_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc: A] :
( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc))))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = Acc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F2,A3,B2,Acc) = set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
tff(fact_4348_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_max(A),aTP_Lamp_mt(A,fun(A,bool)),aTP_Lamp_mu(A,fun(A,bool))) ) ).
% max.semilattice_order_axioms
tff(fact_4349_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_4350_mset__zero__iff,axiom,
! [A: $tType,X: list(A)] :
( ( mset(A,X) = zero_zero(multiset(A)) )
<=> ( X = nil(A) ) ) ).
% mset_zero_iff
tff(fact_4351_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F2,Less_eq,Less)
=> semilattice(A,F2) ) ).
% semilattice_order.axioms(1)
tff(fact_4352_mset_Osimps_I1_J,axiom,
! [A: $tType] : mset(A,nil(A)) = zero_zero(multiset(A)) ).
% mset.simps(1)
tff(fact_4353_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).
% semilattice_order.strict_coboundedI2
tff(fact_4354_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).
% semilattice_order.strict_coboundedI1
tff(fact_4355_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
tff(fact_4356_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).
% semilattice_order.strict_boundedE
tff(fact_4357_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).
% semilattice_order.coboundedI2
tff(fact_4358_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),C2)) ) ) ).
% semilattice_order.coboundedI1
tff(fact_4359_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
<=> ( 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),C2)) ) ) ) ).
% semilattice_order.bounded_iff
tff(fact_4360_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb_iff2
tff(fact_4361_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb_iff1
tff(fact_4362_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),B2)) ) ).
% semilattice_order.cobounded2
tff(fact_4363_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F2,A3),B2)),A3)) ) ).
% semilattice_order.cobounded1
tff(fact_4364_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F2: 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,F2,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),F2,A3),B2) ) ) ) ).
% semilattice_order.order_iff
tff(fact_4365_semilattice__order_OboundedI,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F2,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),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2))) ) ) ) ).
% semilattice_order.boundedI
tff(fact_4366_semilattice__order_OboundedE,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F2,B2),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% semilattice_order.boundedE
tff(fact_4367_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb4
tff(fact_4368_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb3
tff(fact_4369_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb2
tff(fact_4370_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F2,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb1
tff(fact_4371_semilattice__order_OorderI,axiom,
! [A: $tType,F2: 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,F2,Less_eq,Less)
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F2,A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).
% semilattice_order.orderI
tff(fact_4372_semilattice__order_OorderE,axiom,
! [A: $tType,F2: 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,F2,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),F2,A3),B2) ) ) ) ).
% semilattice_order.orderE
tff(fact_4373_semilattice__order_Omono,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A,D3: A] :
( semilattice_order(A,F2,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
=> ( 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),F2,A3),B2)),aa(A,A,aa(A,fun(A,A),F2,C2),D3))) ) ) ) ).
% semilattice_order.mono
tff(fact_4374_ex__mset__zip__left,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xs5: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( mset(A,Xs5) = mset(A,Xs) )
=> ? [Ys2: list(B)] :
( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(A),nat,size_size(list(A)),Xs5) )
& ( 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),Xs5),Ys2)) = 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)) ) ) ) ) ).
% ex_mset_zip_left
tff(fact_4375_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
=> semilattice_order(A,F2,Less_eq,Less) ) ).
% semilattice_neutr_order.axioms(2)
tff(fact_4376_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
=> ( ! [F8: fun(nat,fun(A,A)),A6: nat,B5: nat,Acc3: A] :
( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F8),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A6),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B5),Acc3))))
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B5),A6))
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F8),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),one_one(nat))),B5),aa(A,A,aa(nat,fun(A,A),F8,A6),Acc3))) )
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F8),A6),B5),Acc3)) ) )
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).
% fold_atLeastAtMost_nat.pinduct
tff(fact_4377_semilattice__neutr__order__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F2,Z2,Less_eq,Less)
<=> ( semilattice_neutr(A,F2,Z2)
& semilattice_order(A,F2,Less_eq,Less) ) ) ).
% semilattice_neutr_order_def
tff(fact_4378_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_neutr(A,F2,Z2)
=> ( semilattice_order(A,F2,Less_eq,Less)
=> semila1105856199041335345_order(A,F2,Z2,Less_eq,Less) ) ) ).
% semilattice_neutr_order.intro
tff(fact_4379_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_4380_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_4381_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice_order(A,sup_sup(A),aTP_Lamp_mv(A,fun(A,bool)),aTP_Lamp_mw(A,fun(A,bool))) ) ).
% sup.semilattice_order_axioms
tff(fact_4382_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_4383_semilattice__order__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F2,Less_eq,Less)
<=> ( semilattice(A,F2)
& semila6385135966242565138axioms(A,F2,Less_eq,Less) ) ) ).
% semilattice_order_def
tff(fact_4384_semilattice__order_Ointro,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice(A,F2)
=> ( semila6385135966242565138axioms(A,F2,Less_eq,Less)
=> semilattice_order(A,F2,Less_eq,Less) ) ) ).
% semilattice_order.intro
tff(fact_4385_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_4386_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_4387_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_4388_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_4389_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_4390_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_4391_lexordp__eq__pref,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: list(A),V2: list(A)] : ord_lexordp_eq(A,U,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)) ) ).
% lexordp_eq_pref
tff(fact_4392_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ys: list(A)] : ord_lexordp_eq(A,nil(A),Ys) ) ).
% lexordp_eq.Nil
tff(fact_4393_lexordp__eq__refl,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] : ord_lexordp_eq(A,Xs,Xs) ) ).
% lexordp_eq_refl
tff(fact_4394_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_4395_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_4396_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_4397_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),F2: fun(A,fun(A,A)),Less: fun(A,fun(A,bool))] :
( ! [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),F2,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),F2,A6),B5) )
& ( A6 != B5 ) ) )
=> semila6385135966242565138axioms(A,F2,Less_eq,Less) ) ) ).
% semilattice_order_axioms.intro
tff(fact_4398_semilattice__order__axioms__def,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila6385135966242565138axioms(A,F2,Less_eq,Less)
<=> ( ! [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),F2,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),F2,A5),B4) )
& ( A5 != B4 ) ) ) ) ) ).
% semilattice_order_axioms_def
tff(fact_4399_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_4400_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_4401_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_4402_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_4403_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_4404_count__single,axiom,
! [A: $tType,B2: A,A3: A] :
( ( ( B2 = A3 )
=> ( aa(A,nat,count(A,add_mset(A,B2,zero_zero(multiset(A)))),A3) = one_one(nat) ) )
& ( ( B2 != A3 )
=> ( aa(A,nat,count(A,add_mset(A,B2,zero_zero(multiset(A)))),A3) = zero_zero(nat) ) ) ) ).
% count_single
tff(fact_4405_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F2,Less_eq,Less)
=> semila6385135966242565138axioms(A,F2,Less_eq,Less) ) ).
% semilattice_order.axioms(2)
tff(fact_4406_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] :
( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
=> ~ 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) )
=> ! [Ys3: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ~ 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,Ys3) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
tff(fact_4407_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( ord_lexordp_eq(A,A1,A22)
<=> ( ? [Ys4: list(A)] :
( ( A1 = nil(A) )
& ( A22 = Ys4 ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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,Xs4,Ys4) ) ) ) ) ).
% lexordp_eq.simps
tff(fact_4408_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_4409_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_4410_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_4411_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> algebr8660921524188924756oprime(nat,N,minus_minus(nat,N,one_one(nat))) ) ).
% coprime_diff_one_right_nat
tff(fact_4412_coprime__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
<=> ( algebr8660921524188924756oprime(A,A3,C2)
& algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).
% coprime_mult_left_iff
tff(fact_4413_coprime__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( algebr8660921524188924756oprime(A,C2,A3)
& algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).
% coprime_mult_right_iff
tff(fact_4414_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_4415_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_4416_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_4417_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_4418_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( pp(dvd_dvd(A,C2,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_4419_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( pp(dvd_dvd(A,C2,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_4420_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_4421_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).
% coprime_1_left
tff(fact_4422_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).
% coprime_1_right
tff(fact_4423_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_4424_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_4425_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_4426_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_4427_divides__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,A3,C2))
=> ( pp(dvd_dvd(A,B2,C2))
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)) ) ) ) ) ).
% divides_mult
tff(fact_4428_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% coprime_dvd_mult_left_iff
tff(fact_4429_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% coprime_dvd_mult_right_iff
tff(fact_4430_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ! [C4: A] :
( pp(dvd_dvd(A,C4,A3))
=> ( pp(dvd_dvd(A,C4,B2))
=> pp(dvd_dvd(A,C4,one_one(A))) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprimeI
tff(fact_4431_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ! [C3: A] :
( pp(dvd_dvd(A,C3,A3))
=> ( pp(dvd_dvd(A,C3,B2))
=> pp(dvd_dvd(A,C3,one_one(A))) ) ) ) ) ).
% coprime_def
tff(fact_4432_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ~ algebr8660921524188924756oprime(A,A3,B2)
=> ~ ! [C4: A] :
( pp(dvd_dvd(A,C4,A3))
=> ( pp(dvd_dvd(A,C4,B2))
=> pp(dvd_dvd(A,C4,one_one(A))) ) ) ) ) ).
% not_coprimeE
tff(fact_4433_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: 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) ) ) ) ) ).
% not_coprimeI
tff(fact_4434_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_4435_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,D3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,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,C2)) ) ) )
=> ( ! [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_4436_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_4437_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> pp(dvd_dvd(A,C2,one_one(A))) ) ) ) ) ).
% coprime_common_divisor
tff(fact_4438_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_4439_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_4440_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_4441_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_4442_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( algebr8660921524188924756oprime(nat,A3,B2)
=> ( pp(dvd_dvd(nat,X,A3))
=> ( pp(dvd_dvd(nat,X,B2))
=> ( X = one_one(nat) ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_4443_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C2: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_left_cancel
tff(fact_4444_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C2: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_right_cancel
tff(fact_4445_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_left_cancel
tff(fact_4446_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_right_cancel
tff(fact_4447_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_4448_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_4449_coprime__crossproduct,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,D3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,D3)
=> ( algebr8660921524188924756oprime(A,B2,C2)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C2)) = 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),C2) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).
% coprime_crossproduct
tff(fact_4450_invertible__coprime,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) = one_one(A) )
=> algebr8660921524188924756oprime(A,A3,C2) ) ) ).
% invertible_coprime
tff(fact_4451_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) )
=> ? [A8: A,B7: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A8),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),B7),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
& algebr8660921524188924756oprime(A,A8,B7) ) ) ) ).
% gcd_coprime_exists
tff(fact_4452_gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,A9: A,B6: 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),A9),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),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> algebr8660921524188924756oprime(A,A9,B6) ) ) ) ) ).
% gcd_coprime
tff(fact_4453_coprime__crossproduct_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [B2: A,D3: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> ( algebr8660921524188924756oprime(A,C2,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),C2) )
<=> ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
tff(fact_4454_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_4455_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> algebr8660921524188924756oprime(nat,minus_minus(nat,N,one_one(nat)),N) ) ).
% coprime_diff_one_left_nat
tff(fact_4456_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_4457_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X: A,A3: A,B2: A] :
( equiv_equiv(A,A4,R2)
=> ( pp(member2(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(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),R2)) ) ) ).
% equiv_class_nondisjoint
tff(fact_4458_count__greater__eq__one__iff,axiom,
! [A: $tType,M: multiset(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),aa(A,nat,count(A,M),X)))
<=> pp(member2(A,X,set_mset(A,M))) ) ).
% count_greater_eq_one_iff
tff(fact_4459_in__range_Osimps,axiom,
! [H2: 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)),H2),As2)))
<=> ! [X4: nat] :
( pp(member2(nat,X4,As2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H2))) ) ) ).
% in_range.simps
tff(fact_4460_equiv__listrel,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
( equiv_equiv(A,A4,R2)
=> equiv_equiv(list(A),aa(set(A),set(list(A)),lists(A),A4),listrel(A,A,R2)) ) ).
% equiv_listrel
tff(fact_4461_coprime__common__divisor__int,axiom,
! [A3: int,B2: int,X: int] :
( algebr8660921524188924756oprime(int,A3,B2)
=> ( pp(dvd_dvd(int,X,A3))
=> ( pp(dvd_dvd(int,X,B2))
=> ( abs_abs(int,X) = one_one(int) ) ) ) ) ).
% coprime_common_divisor_int
tff(fact_4462_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_4463_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_4464_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_4465_sum__wcount__Int,axiom,
! [A: $tType,A4: set(A),F2: fun(A,nat),N2: multiset(A)] :
( pp(aa(set(A),bool,finite_finite(A),A4))
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F2,N2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),set_mset(A,N2))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),wcount(A,F2,N2)),A4) ) ) ).
% sum_wcount_Int
tff(fact_4466_count__image__mset,axiom,
! [A: $tType,B: $tType,F2: fun(B,A),A4: multiset(B),X: A] : aa(A,nat,count(A,image_mset(B,A,F2,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),F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),X),bot_bot(set(A))))),set_mset(B,A4))) ).
% count_image_mset
tff(fact_4467_size__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: multiset(A)] :
( ( pp(member2(A,X,set_mset(A,A4)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),minus_minus(multiset(A),A4,add_mset(A,X,zero_zero(multiset(A))))) = minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),A4),one_one(nat)) ) )
& ( ~ pp(member2(A,X,set_mset(A,A4)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),minus_minus(multiset(A),A4,add_mset(A,X,zero_zero(multiset(A))))) = aa(multiset(A),nat,size_size(multiset(A)),A4) ) ) ) ).
% size_Diff_singleton_if
tff(fact_4468_size__Diff__singleton,axiom,
! [A: $tType,X: A,M: multiset(A)] :
( pp(member2(A,X,set_mset(A,M)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),minus_minus(multiset(A),M,add_mset(A,X,zero_zero(multiset(A))))) = minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M),one_one(nat)) ) ) ).
% size_Diff_singleton
tff(fact_4469_size__single,axiom,
! [A: $tType,B2: A] : aa(multiset(A),nat,size_size(multiset(A)),add_mset(A,B2,zero_zero(multiset(A)))) = one_one(nat) ).
% size_single
tff(fact_4470_size__1__singleton__mset,axiom,
! [A: $tType,M: multiset(A)] :
( ( aa(multiset(A),nat,size_size(multiset(A)),M) = one_one(nat) )
=> ? [A6: A] : M = add_mset(A,A6,zero_zero(multiset(A))) ) ).
% size_1_singleton_mset
tff(fact_4471_mset__size1elem,axiom,
! [A: $tType,P: multiset(A),Q3: 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,Q3,set_mset(A,P)))
=> ( P = add_mset(A,Q3,zero_zero(multiset(A))) ) ) ) ).
% mset_size1elem
tff(fact_4472_size__diff__se,axiom,
! [A: $tType,T3: A,S: multiset(A)] :
( pp(member2(A,T3,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,T3,zero_zero(multiset(A)))))),one_one(nat)) ) ) ).
% size_diff_se
tff(fact_4473_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_mx(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_4474_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_my(B,fun(A,fun(B,A)),Y),C2),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,count(B,A4),Y))) ) ).
% sum_mset_delta
tff(fact_4475_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_mz(B,fun(A,fun(B,A)),Y),C2),A4)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,count(B,A4),Y))) ) ).
% sum_mset_delta'
tff(fact_4476_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_4477_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_4478_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,N2: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),add_mset(A,X,N2)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N2)) ) ).
% prod_mset.add_mset
tff(fact_4479_prod__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: multiset(A),N2: 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),N2)) = 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),N2)) ) ).
% prod_mset.union
tff(fact_4480_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_4481_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_ai(B,A),A4)) = one_one(A) ) ).
% prod_mset.neutral_const
tff(fact_4482_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_4483_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_4484_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H2: 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_ar(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2),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,H2,A4))) ) ).
% prod_mset.distrib
tff(fact_4485_sum__mset__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& times(A)
& semiring_0(B) )
=> ! [F2: 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,F2,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_nb(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),F2),G),B3),A4)) ) ).
% sum_mset_product
tff(fact_4486_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F2: fun(B,A),M: multiset(B),C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F2,M))),C2) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ed(fun(B,A),fun(A,fun(B,A)),F2),C2),M)) ) ).
% sum_mset_distrib_right
tff(fact_4487_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F2: fun(B,A),M: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,image_mset(B,A,F2,M))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ee(A,fun(fun(B,A),fun(B,A)),C2),F2),M)) ) ).
% sum_mset_distrib_left
tff(fact_4488_size__eq__sum__mset,axiom,
! [A: $tType,M: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),M) = comm_m7189776963980413722m_mset(nat,image_mset(A,nat,aTP_Lamp_gp(A,nat),M)) ).
% size_eq_sum_mset
tff(fact_4489_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_4490_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_4491_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_nc(B,fun(A,fun(B,A)),Y),C2),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,count(B,A4),Y)) ) ).
% prod_mset_delta
tff(fact_4492_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_nd(B,fun(A,fun(B,A)),Y),C2),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,count(B,A4),Y)) ) ).
% prod_mset_delta'
tff(fact_4493_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_4494_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_4495_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_4496_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)),insert3(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_4497_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_4498_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_4499_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_4500_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_4501_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_4502_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_4503_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_4504_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_4505_relH__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: ref(A)] :
( relH(As2,H2,H5)
=> ( pp(member2(nat,addr_of_ref(A,R2),As2))
=> ( ref_get(A,H2,R2) = ref_get(A,H5,R2) ) ) ) ) ).
% relH_ref
tff(fact_4506_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),U: list(A),V2: 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),V2))) ).
% ord.lexordp_eq_pref
tff(fact_4507_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_4508_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))
<=> ( ? [Ys4: list(A)] :
( ( A1 = nil(A) )
& ( A22 = Ys4 ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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),Xs4),Ys4)) ) ) ) ).
% ord.lexordp_eq.simps
tff(fact_4509_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] :
( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
=> ~ 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) )
=> ! [Ys3: list(A)] :
( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
=> ( ~ 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),Ys3)) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
tff(fact_4510_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A,H2: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
<=> ( ( ref_get(A,H2,R2) = X )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_ref(A,R2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R2)),lim(product_unit,H2))) ) ) ) ).
% sngr_assn_raw.simps
tff(fact_4511_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)),insert3(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_4512_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)),insert3(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_4513_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)),insert3(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_4514_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)),insert3(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_4515_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)),insert3(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_4516_relH__set__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),As2: set(nat),H2: heap_ext(product_unit),X: A] :
( ~ pp(member2(nat,addr_of_ref(A,R2),As2))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
=> relH(As2,H2,ref_set(A,R2,X,H2)) ) ) ) ).
% relH_set_ref
tff(fact_4517_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)),insert3(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_4518_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)),insert3(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_4519_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)),insert3(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_4520_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A),H2: heap_ext(product_unit),As2: set(nat)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
<=> ( ( array_get(A,H2,R2) = X )
& ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert3(nat),addr_of_array(A,R2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,R2)),lim(product_unit,H2))) ) ) ) ).
% snga_assn_raw.simps
tff(fact_4521_relH__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [As2: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A)] :
( relH(As2,H2,H5)
=> ( pp(member2(nat,addr_of_array(A,R2),As2))
=> ( array_get(A,H2,R2) = array_get(A,H5,R2) ) ) ) ) ).
% relH_array
tff(fact_4522_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)),insert3(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_4523_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)),insert3(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_4524_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)),insert3(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_4525_relH__set__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),As2: set(nat),H2: heap_ext(product_unit),X: list(A)] :
( ~ pp(member2(nat,addr_of_array(A,R2),As2))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
=> relH(As2,H2,array_set(A,R2,X,H2)) ) ) ) ).
% relH_set_array
tff(fact_4526_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_4527_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_4528_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_4529_cINF__union,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F2: fun(B,A),B3: set(B)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,image2(B,A,F2,A4))
=> ( ( B3 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,image2(B,A,F2,B3))
=> ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,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),image2(B,A,F2,A4))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,B3))) ) ) ) ) ) ) ).
% cINF_union
tff(fact_4530_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [F2: fun(B,A),G: fun(B,A),A4: set(B)] :
( condit1013018076250108175_below(A,image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ne(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A4))
<=> ( condit1013018076250108175_below(A,image2(B,A,F2,A4))
& condit1013018076250108175_below(A,image2(B,A,G,A4)) ) ) ) ).
% bdd_below_image_inf
tff(fact_4531_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_4532_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_4533_cInf__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),A3: A] :
( condit1013018076250108175_below(A,X6)
=> ( ( ( X6 = bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X6)) = A3 ) )
& ( ( X6 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ) ).
% cInf_insert_If
tff(fact_4534_cInf__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),A3: A] :
( ( X6 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X6)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).
% cInf_insert
tff(fact_4535_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_4536_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F2: fun(B,A),G: fun(B,A)] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,image2(B,A,F2,A4))
=> ( condit1013018076250108175_below(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),image2(B,A,F2,A4))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G,A4))) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_nf(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A4)) ) ) ) ) ) ).
% cINF_inf_distrib
tff(fact_4537_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_4538_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A4: set(B),F2: fun(B,A),A3: B] :
( ( A4 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,image2(B,A,F2,A4))
=> ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert3(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F2,A4))) ) ) ) ) ).
% cINF_insert
tff(fact_4539_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),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),complete_Sup_Sup(A,A4)),complete_Sup_Sup(A,B3)))) ) ) ) ) ).
% cSup_inter_less_eq
tff(fact_4540_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_4541_one__int_Orsp,axiom,
intrel(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).
% one_int.rsp
tff(fact_4542_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_4543_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_4544_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_4545_asym__lex,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lex(A,R)) ) ).
% asym_lex
tff(fact_4546_lexord__asym,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lexord(A,R)) ) ).
% lexord_asym
tff(fact_4547_asym__lenlex,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
=> asym(list(A),lenlex(A,R)) ) ).
% asym_lenlex
tff(fact_4548_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)))) ) ) )
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
=> ( ( pp(Y)
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& 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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2)))) ) ) )
=> ( ! [V: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( 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),V),Va)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va) )
=> ( ~ 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),V),Va)))) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
tff(fact_4549_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)))) ) )
=> ~ ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2))))
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& list_all_zip(A,B,X,As,Bs2) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
tff(fact_4550_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)))
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),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),A6),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& list_all_zip(A,B,X,As,Bs2) ) ) ) )
=> ( ! [V: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
=> ( ( 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),V),Va)),nil(B)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va) )
=> ~ 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),V),Va)))) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
tff(fact_4551_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_nh(set(nat),fun(set(nat),fun(nat,bool)),A4),B3))) ).
% nths_nths
tff(fact_4552_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_4553_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_4554_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)
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& list_all_zip(A,B,X,As,Bs2) ) ) )
=> ( ( ? [V: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)
=> ( Xb != nil(B) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V: B,Va: list(B)] : Xb != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va) ) ) ) ) ).
% list_all_zip.elims(3)
tff(fact_4555_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) ) )
=> ~ ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& list_all_zip(A,B,X,As,Bs2) ) ) ) ) ) ).
% list_all_zip.elims(2)
tff(fact_4556_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) ) )
=> ( ! [A6: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),As) )
=> ! [B5: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B5),Bs2) )
=> ( pp(Y)
<=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X,A6),B5))
& list_all_zip(A,B,X,As,Bs2) ) ) ) )
=> ( ( ? [V: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)
=> ( ( Xb = nil(B) )
=> pp(Y) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V: B,Va: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va)
=> pp(Y) ) ) ) ) ) ) ).
% list_all_zip.elims(1)
tff(fact_4557_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V2: B,Va2: list(B)] : ~ list_all_zip(A,B,P,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V2),Va2)) ).
% list_all_zip.simps(4)
tff(fact_4558_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V2: A,Va2: list(A)] : ~ list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(B)) ).
% list_all_zip.simps(3)
tff(fact_4559_quotient__disj,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X6: set(A),Y5: set(A)] :
( equiv_equiv(A,A4,R2)
=> ( pp(member2(set(A),X6,equiv_quotient(A,A4,R2)))
=> ( pp(member2(set(A),Y5,equiv_quotient(A,A4,R2)))
=> ( ( X6 = Y5 )
| ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X6),Y5) = bot_bot(set(A)) ) ) ) ) ) ).
% quotient_disj
tff(fact_4560_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)),range2(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),A4),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),range2(B,A,A4)),range2(B,A,B3)))) ).
% Range_Int_subset
tff(fact_4561_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_ni(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).
% Bleast_def
tff(fact_4562_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_ni(set(A),fun(fun(A,bool),fun(A,bool)),S),P)) ) ).
% abort_Bleast_def
tff(fact_4563_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)),range2(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_4564_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_4565_nth__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,Xs: list(A),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> hoare_hoare_triple(A,snga_assn(A,A3,Xs),array_nth(A,A3,I2),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_nj(nat,fun(list(A),fun(array(A),fun(A,assn))),I2),Xs),A3)) ) ) ).
% nth_rule
tff(fact_4566_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B2: bool,F2: 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)),F2,Q)
<=> ( pp(B2)
=> hoare_hoare_triple(A,P,F2,Q) ) ) ).
% norm_pre_pure_iff
tff(fact_4567_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B2: bool,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,pure_assn(B2),F2,Q)
<=> ( pp(B2)
=> hoare_hoare_triple(A,one_one(assn),F2,Q) ) ) ).
% norm_pre_pure_iff_sng
tff(fact_4568_norm__pre__pure__rule2,axiom,
! [A: $tType,B2: bool,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
( ( pp(B2)
=> hoare_hoare_triple(A,one_one(assn),F2,Q) )
=> hoare_hoare_triple(A,pure_assn(B2),F2,Q) ) ).
% norm_pre_pure_rule2
tff(fact_4569_is__hoare__triple,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,C2,Q)
=> hoare_hoare_triple(A,P,C2,Q) ) ).
% is_hoare_triple
tff(fact_4570_cons__post__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Q2: fun(A,assn)] :
( hoare_hoare_triple(A,P,C2,Q)
=> ( ! [X2: A] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(A,assn,Q,X2)),aa(A,assn,Q2,X2)))
=> hoare_hoare_triple(A,P,C2,Q2) ) ) ).
% cons_post_rule
tff(fact_4571_cons__rule,axiom,
! [A: $tType,P: assn,P2: assn,Q: fun(A,assn),Q2: fun(A,assn),C2: heap_Time_Heap(A)] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P2))
=> ( ! [X2: A] : pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,aa(A,assn,Q,X2)),aa(A,assn,Q2,X2)))
=> ( hoare_hoare_triple(A,P2,C2,Q)
=> hoare_hoare_triple(A,P,C2,Q2) ) ) ) ).
% cons_rule
tff(fact_4572_hoare__triple__preI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,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))
=> hoare_hoare_triple(A,P,C2,Q) )
=> hoare_hoare_triple(A,P,C2,Q) ) ).
% hoare_triple_preI
tff(fact_4573_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_4574_cons__pre__rule,axiom,
! [A: $tType,P: assn,P2: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( pp(aa(assn,bool,aa(assn,fun(assn,bool),entails,P),P2))
=> ( hoare_hoare_triple(A,P2,C2,Q)
=> hoare_hoare_triple(A,P,C2,Q) ) ) ).
% cons_pre_rule
tff(fact_4575_norm__pre__pure__rule1,axiom,
! [A: $tType,B2: bool,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
( ( pp(B2)
=> hoare_hoare_triple(A,P,F2,Q) )
=> hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn(B2)),F2,Q) ) ).
% norm_pre_pure_rule1
tff(fact_4576_frame__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
( hoare_hoare_triple(A,P,C2,Q)
=> hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R),C2,aa(assn,fun(A,assn),aTP_Lamp_lq(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).
% frame_rule
tff(fact_4577_cons__rulet,axiom,
! [A: $tType,P: assn,P2: assn,Q: fun(A,assn),Q2: fun(A,assn),C2: heap_Time_Heap(A)] :
( entailst(P,P2)
=> ( ! [X2: A] : entailst(aa(A,assn,Q,X2),aa(A,assn,Q2,X2))
=> ( hoare_hoare_triple(A,P2,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q))
=> hoare_hoare_triple(A,P,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q2)) ) ) ) ).
% cons_rulet
tff(fact_4578_cons__pre__rulet,axiom,
! [A: $tType,P: assn,P2: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( entailst(P,P2)
=> ( hoare_hoare_triple(A,P2,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q))
=> hoare_hoare_triple(A,P,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q)) ) ) ).
% cons_pre_rulet
tff(fact_4579_cons__post__rulet,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Q2: fun(A,assn)] :
( hoare_hoare_triple(A,P,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q))
=> ( ! [X2: A] : entailst(aa(A,assn,Q,X2),aa(A,assn,Q2,X2))
=> hoare_hoare_triple(A,P,C2,aTP_Lamp_nk(fun(A,assn),fun(A,assn),Q2)) ) ) ).
% cons_post_rulet
tff(fact_4580_if__rule__split,axiom,
! [A: $tType,B2: bool,P: assn,F2: 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,F2,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,F2,G),Q) ) ) ) ).
% if_rule_split
tff(fact_4581_upd__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,Xs: list(A),A3: array(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> hoare_hoare_triple(array(A),snga_assn(A,A3,Xs),array_upd(A,I2,X,A3),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_nl(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I2),Xs),A3),X)) ) ) ).
% upd_rule
tff(fact_4582_make__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F2: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,F2),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_nm(nat,fun(fun(nat,A),fun(array(A),assn)),N),F2)) ) ).
% make_rule
tff(fact_4583_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_nn(array(A),fun(list(A),fun(nat,assn)),A3),Xs)) ) ).
% length_rule
tff(fact_4584_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_no(nat,fun(A,fun(array(A),assn)),N),X)) ) ).
% new_rule
tff(fact_4585_lookup__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [P7: ref(A),X: A] : hoare_hoare_triple(A,sngr_assn(A,P7,X),ref_lookup(A,P7),aa(A,fun(A,assn),aTP_Lamp_np(ref(A),fun(A,fun(A,assn)),P7),X)) ) ).
% lookup_rule
tff(fact_4586_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_at(list(A),fun(array(A),assn)),Xs)) ) ).
% of_list_rule
tff(fact_4587_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_nq(array(A),fun(list(A),fun(list(A),assn)),A3),Xs)) ) ).
% freeze_rule
tff(fact_4588_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_as(A,fun(ref(A),assn)),X)) ) ).
% ref_rule
tff(fact_4589_return__sp__rule,axiom,
! [A: $tType,P: assn,X: A] : hoare_hoare_triple(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),aa(A,fun(A,assn),aTP_Lamp_nr(assn,fun(A,fun(A,assn)),P),X)) ).
% return_sp_rule
tff(fact_4590_wf__UN,axiom,
! [B: $tType,A: $tType,I: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [I3: A] :
( pp(member2(A,I3,I))
=> wf(B,aa(A,set(product_prod(B,B)),R2,I3)) )
=> ( ! [I3: A,J2: A] :
( pp(member2(A,I3,I))
=> ( pp(member2(A,J2,I))
=> ( ( aa(A,set(product_prod(B,B)),R2,I3) != aa(A,set(product_prod(B,B)),R2,J2) )
=> ( 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)),R2,I3))),range2(B,B,aa(A,set(product_prod(B,B)),R2,J2))) = bot_bot(set(B)) ) ) ) )
=> wf(B,complete_Sup_Sup(set(product_prod(B,B)),image2(A,set(product_prod(B,B)),R2,I))) ) ) ).
% wf_UN
tff(fact_4591_hoare__triple__success,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,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),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
=> heap_Time_success(A,C2,H2) ) ) ).
% hoare_triple_success
tff(fact_4592_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)),range2(A,A,R)) = bot_bot(set(A)) )
=> wf(A,R) ) ).
% wf_no_path
tff(fact_4593_wf__listrel1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(list(A),listrel1(A,R2))
<=> wf(A,R2) ) ).
% wf_listrel1_iff
tff(fact_4594_wf__lex,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
=> wf(list(A),lex(A,R2)) ) ).
% wf_lex
tff(fact_4595_wf__lenlex,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
=> wf(list(A),lenlex(A,R2)) ) ).
% wf_lenlex
tff(fact_4596_wf__Int2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
( wf(A,R2)
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R5),R2)) ) ).
% wf_Int2
tff(fact_4597_wf__Int1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),R5: set(product_prod(A,A))] :
( wf(A,R2)
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),R5)) ) ).
% wf_Int1
tff(fact_4598_wf__measures,axiom,
! [A: $tType,Fs: list(fun(A,nat))] : wf(A,measures(A,Fs)) ).
% wf_measures
tff(fact_4599_wf__lexn,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
( wf(A,R2)
=> wf(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)) ) ).
% wf_lexn
tff(fact_4600_wf__Un,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( wf(A,R2)
=> ( wf(A,S3)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,R2)),range2(A,A,S3)) = bot_bot(set(A)) )
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S3)) ) ) ) ).
% wf_Un
tff(fact_4601_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)),range2(A,A,Xa4)) = bot_bot(set(A)) ) ) ) )
=> wf(A,complete_Sup_Sup(set(product_prod(A,A)),R)) ) ) ).
% wf_Union
tff(fact_4602_trans__wf__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> ( wf(A,R2)
<=> ! [A5: A] : wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,image(A,A,converse(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert3(A),A5),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_ns(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A5)))) ) ) ).
% trans_wf_iff
tff(fact_4603_Array__Time_Omake__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F2: fun(nat,A)] : array_make(A,N,F2) = heap_Time_heap(array(A),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nu(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),F2)) ) ).
% Array_Time.make_def
tff(fact_4604_wait__rule,axiom,
! [N: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(N),aTP_Lamp_nv(product_unit,assn)) ).
% wait_rule
tff(fact_4605_lenlex__transI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> trans(list(A),lenlex(A,R2)) ) ).
% lenlex_transI
tff(fact_4606_lexn__transI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
( trans(A,R2)
=> trans(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)) ) ).
% lexn_transI
tff(fact_4607_listrel__trans,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> trans(list(A),listrel(A,A,R2)) ) ).
% listrel_trans
tff(fact_4608_lexord__transI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> trans(list(A),lexord(A,R2)) ) ).
% lexord_transI
tff(fact_4609_lex__transI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
=> trans(list(A),lex(A,R2)) ) ).
% lex_transI
tff(fact_4610_trans__Int,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( trans(A,R2)
=> ( trans(A,S3)
=> trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S3)) ) ) ).
% trans_Int
tff(fact_4611_lexord__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2)))
=> ( 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,R2)))
=> ( trans(A,R2)
=> 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,R2))) ) ) ) ).
% lexord_trans
tff(fact_4612_lenlex__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: 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,R2)))
=> ( 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,R2)))
=> ( trans(A,R2)
=> 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,R2))) ) ) ) ).
% lenlex_trans
tff(fact_4613_trans__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( trans(A,R2)
=> trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% trans_Restr
tff(fact_4614_of__list__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_nx(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).
% of_list_def
tff(fact_4615_new__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A] : array_new(A,N,X) = heap_Time_heap(array(A),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ny(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),X)) ) ).
% new_def
tff(fact_4616_return__def,axiom,
! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X) = heap_Time_heap(A,aTP_Lamp_nz(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% return_def
tff(fact_4617_effect__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H5: heap_ext(product_unit),F2: fun(nat,A),N: nat,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2) )
=> heap_Time_effect(array(A),array_make(A,N,F2),H2,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_makeI
tff(fact_4618_execute__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F2: fun(nat,A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_make(A,N,F2)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2))) ) ).
% execute_make
tff(fact_4619_effect__returnI,axiom,
! [A: $tType,H2: heap_ext(product_unit),H5: heap_ext(product_unit),X: A] :
( ( H2 = H5 )
=> heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H2,H5,X,one_one(nat)) ) ).
% effect_returnI
tff(fact_4620_effect__returnE,axiom,
! [A: $tType,X: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N: nat] :
( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H2,H5,R2,N)
=> ~ ( ( R2 = X )
=> ( ( H5 = H2 )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_returnE
tff(fact_4621_execute__return,axiom,
! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X)) = comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nz(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% execute_return
tff(fact_4622_execute__of__list,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_of_list(A,Xs)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nw(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Xs)),array_alloc(A,Xs,H2))) ) ).
% execute_of_list
tff(fact_4623_execute__new,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A,H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_new(A,N,X)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X),H2))) ) ).
% execute_new
tff(fact_4624_effect__of__listI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H5: heap_ext(product_unit),Xs: list(A),H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,Xs,H2) )
=> heap_Time_effect(array(A),array_of_list(A,Xs),H2,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% effect_of_listI
tff(fact_4625_effect__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H5: heap_ext(product_unit),N: nat,X: A,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X),H2) )
=> heap_Time_effect(array(A),array_new(A,N,X),H2,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_newI
tff(fact_4626_execute__lookup,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,ref_lookup(A,R2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),ref_get(A,H2,R2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ).
% execute_lookup
tff(fact_4627_hoare__tripleE,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,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),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
=> ~ ! [R3: A,H4: heap_ext(product_unit)] :
( ? [T5: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),T5)))
=> ( 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,Q,R3)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),hoare_new_addrs(H2,As2,H4))))
=> ( relH(collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_oa(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H2),As2)),H2,H4)
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H2)),lim(product_unit,H4))) ) ) ) ) ) ).
% hoare_tripleE
tff(fact_4628_hoare__tripleI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( ! [H3: heap_ext(product_unit),As: 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)),H3),As)))
=> ? [R6: A,H6: heap_ext(product_unit),T6: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H3) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R6),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H6),T6))) )
& 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,Q,R6)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),hoare_new_addrs(H3,As,H6))))
& relH(collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_oa(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H3),As)),H3,H6)
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H3)),lim(product_unit,H6))) ) )
=> hoare_hoare_triple(A,P,C2,Q) ) ).
% hoare_tripleI
tff(fact_4629_hoare__triple__effect,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H2: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,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),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
=> ? [H4: heap_ext(product_unit),R3: A,T5: nat] :
( heap_Time_effect(A,C2,H2,H4,R3,T5)
& 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,Q,R3)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),hoare_new_addrs(H2,As2,H4)))) ) ) ) ).
% hoare_triple_effect
tff(fact_4630_effect__lookupE,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),X: A,N: nat] :
( heap_Time_effect(A,ref_lookup(A,R2),H2,H5,X,N)
=> ~ ( ( H5 = H2 )
=> ( ( X = ref_get(A,H2,R2) )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_lookupE
tff(fact_4631_effect__lookupI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H5: heap_ext(product_unit),H2: heap_ext(product_unit),X: A,R2: ref(A),N: nat] :
( ( H5 = H2 )
=> ( ( X = ref_get(A,H2,R2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,ref_lookup(A,R2),H2,H5,X,N) ) ) ) ) ).
% effect_lookupI
tff(fact_4632_hoare__triple__def,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,C2,Q)
<=> ! [H: heap_ext(product_unit),As7: 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),As7)))
=> ? [R7: A,H7: heap_ext(product_unit)] :
( ? [T2: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R7),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H7),T2)))
& 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,Q,R7)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),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),hoare_new_addrs(H,As7,H7))))
& relH(collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_oa(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H),As7)),H,H7)
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H)),lim(product_unit,H7))) ) ) ) ).
% hoare_triple_def
tff(fact_4633_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_4634_map__project__def,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),A4: set(A)] : map_project(A,B,F2,A4) = collect(B,aa(set(A),fun(B,bool),aTP_Lamp_ob(fun(A,option(B)),fun(set(A),fun(B,bool)),F2),A4)) ).
% map_project_def
tff(fact_4635_effect__makeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F2: fun(nat,A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A),N5: nat] :
( heap_Time_effect(array(A),array_make(A,N,F2),H2,H5,R2,N5)
=> ~ ( ( R2 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2)) )
=> ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)),H2)) )
=> ( ( array_get(A,H5,R2) = aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),N)) )
=> ( array_present(A,H5,R2)
=> ( ~ array_present(A,H2,R2)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N5 ) ) ) ) ) ) ) ) ).
% effect_makeE
tff(fact_4636_effect__of__listE,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A),N5: nat] :
( heap_Time_effect(array(A),array_of_list(A,Xs),H2,H5,R2,N5)
=> ~ ( ( R2 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H2)) )
=> ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H2)) )
=> ( ( array_get(A,H5,R2) = Xs )
=> ( array_present(A,H5,R2)
=> ( ~ array_present(A,H2,R2)
=> ( N5 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ) ) ) ) ) ) ).
% effect_of_listE
tff(fact_4637_effect__newE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A),N5: nat] :
( heap_Time_effect(array(A),array_new(A,N,X),H2,H5,R2,N5)
=> ~ ( ( R2 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X),H2)) )
=> ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X),H2)) )
=> ( ( array_get(A,H5,R2) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) )
=> ( array_present(A,H5,R2)
=> ( ~ array_present(A,H2,R2)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N5 ) ) ) ) ) ) ) ) ).
% effect_newE
tff(fact_4638_execute__tap,axiom,
! [A: $tType,F2: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_tap(A,F2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,F2,H2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ).
% execute_tap
tff(fact_4639_execute__nth_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,H2: heap_ext(product_unit),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_nth(A,A3,I2)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H2,A3)),I2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ) ).
% execute_nth(1)
tff(fact_4640_execute__assert_I1_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit)] :
( pp(aa(A,bool,P,X))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,X)),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ).
% execute_assert(1)
tff(fact_4641_effect__lengthE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: nat,N: nat] :
( heap_Time_effect(nat,array_len(A,A3),H2,H5,R2,N)
=> ~ ( ( R2 = array_length(A,H5,A3) )
=> ( ( H5 = H2 )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_lengthE
tff(fact_4642_effect__lengthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H5: heap_ext(product_unit),H2: heap_ext(product_unit),R2: nat,A3: array(A),N: nat] :
( ( H5 = H2 )
=> ( ( R2 = array_length(A,H2,A3) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(nat,array_len(A,A3),H2,H5,R2,N) ) ) ) ) ).
% effect_lengthI
tff(fact_4643_effect__assertI,axiom,
! [A: $tType,P: fun(A,bool),X: A,H5: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
( pp(aa(A,bool,P,X))
=> ( ( H5 = H2 )
=> ( ( R2 = X )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,heap_Time_assert(A,P,X),H2,H5,R2,N) ) ) ) ) ).
% effect_assertI
tff(fact_4644_effect__assertE,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N: nat] :
( heap_Time_effect(A,heap_Time_assert(A,P,X),H2,H5,R2,N)
=> ~ ( pp(aa(A,bool,P,X))
=> ( ( R2 = X )
=> ( ( H5 = H2 )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_assertE
tff(fact_4645_execute__len,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(nat,array_len(A,A3)),H2) = aa(product_prod(nat,product_prod(heap_ext(product_unit),nat)),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),some(product_prod(nat,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat))),product_Pair(nat,product_prod(heap_ext(product_unit),nat)),array_length(A,H2,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ).
% execute_len
tff(fact_4646_effect__tapI,axiom,
! [A: $tType,H5: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,F2: fun(heap_ext(product_unit),A)] :
( ( H5 = H2 )
=> ( ( R2 = aa(heap_ext(product_unit),A,F2,H2) )
=> heap_Time_effect(A,heap_Time_tap(A,F2),H2,H5,R2,one_one(nat)) ) ) ).
% effect_tapI
tff(fact_4647_effect__tapE,axiom,
! [A: $tType,F2: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N: nat] :
( heap_Time_effect(A,heap_Time_tap(A,F2),H2,H5,R2,N)
=> ~ ( ( H5 = H2 )
=> ( ( R2 = aa(heap_ext(product_unit),A,F2,H2) )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_tapE
tff(fact_4648_effect__freezeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H5: heap_ext(product_unit),H2: heap_ext(product_unit),R2: list(A),A3: array(A),N: nat] :
( ( H5 = H2 )
=> ( ( R2 = array_get(A,H2,A3) )
=> ( ( N = array_length(A,H2,A3) )
=> heap_Time_effect(list(A),array_freeze(A,A3),H2,H5,R2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ) ) ).
% effect_freezeI
tff(fact_4649_effect__freezeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: list(A),N: nat] :
( heap_Time_effect(list(A),array_freeze(A,A3),H2,H5,R2,N)
=> ~ ( ( H5 = H2 )
=> ( ( R2 = array_get(A,H2,A3) )
=> ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),array_length(A,H2,A3)),one_one(nat)) ) ) ) ) ) ).
% effect_freezeE
tff(fact_4650_execute__freeze,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(list(A),array_freeze(A,A3)),H2) = aa(product_prod(list(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,H2,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,H2,A3))))) ) ).
% execute_freeze
tff(fact_4651_freeze__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_oc(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ).
% freeze_def
tff(fact_4652_effect__nthE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I2: nat,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N: nat] :
( heap_Time_effect(A,array_nth(A,A3,I2),H2,H5,R2,N)
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),I2) )
=> ( ( H5 = H2 )
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_nthE
tff(fact_4653_effect__nthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,H2: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),R2: A,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( ( H5 = H2 )
=> ( ( R2 = aa(nat,A,nth(A,array_get(A,H2,A3)),I2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,array_nth(A,A3,I2),H2,H5,R2,N) ) ) ) ) ) ).
% effect_nthI
tff(fact_4654_Array__Time_Onth__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I2: nat] : array_nth(A,A3,I2) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_od(array(A),fun(nat,fun(heap_ext(product_unit),bool)),A3),I2),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oe(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),A3),I2)) ) ).
% Array_Time.nth_def
tff(fact_4655_execute__upd_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,H2: heap_ext(product_unit),A3: array(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_upd(A,I2,X,A3)),H2) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I2,X,H2)),one_one(nat)))) ) ) ) ).
% execute_upd(1)
tff(fact_4656_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_4657_upd__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,X: A,A3: array(A)] : array_upd(A,I2,X,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_of(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I2),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_og(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I2),X),A3)) ) ).
% upd_def
tff(fact_4658_effect__updE,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,V2: A,A3: array(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A),N: nat] :
( heap_Time_effect(array(A),array_upd(A,I2,V2,A3),H2,H5,R2,N)
=> ~ ( ( R2 = A3 )
=> ( ( H5 = array_update(A,A3,I2,V2,H2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_updE
tff(fact_4659_effect__updI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I2: nat,H2: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),V2: A,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H2,A3)))
=> ( ( H5 = array_update(A,A3,I2,V2,H2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(array(A),array_upd(A,I2,V2,A3),H2,H5,A3,N) ) ) ) ) ).
% effect_updI
tff(fact_4660_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_4661_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_oh(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_4662_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_oi(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).
% sorted_find_Min
tff(fact_4663_tap__def,axiom,
! [A: $tType,F2: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F2) = heap_Time_Heap2(A,aTP_Lamp_oj(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F2)) ).
% tap_def
tff(fact_4664_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_4665_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_4666_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_4667_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_4668_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_4669_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_4670_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_ok(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_4671_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_4672_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_4673_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_4674_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: 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),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter(B),P),Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_ol(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F2),P),Xs) ).
% map_filter_map_filter
tff(fact_4675_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) )
<=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
& ( X = aa(nat,A,nth(A,Xs),I4) )
& ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% find_Some_iff2
tff(fact_4676_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) )
<=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
& ( X = aa(nat,A,nth(A,Xs),I4) )
& ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% find_Some_iff
tff(fact_4677_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_on(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_4678_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_oo(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Inf_fin.eq_fold'
tff(fact_4679_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_oq(option(A),fun(A,option(A)),Y),X) ) ).
% inf_option_def
tff(fact_4680_map__filter__def,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F2,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),F2)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter(A),aTP_Lamp_or(fun(A,option(B)),fun(A,bool),F2)),Xs)) ).
% map_filter_def
tff(fact_4681_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_ot(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_4682_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_ot(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_4683_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] : array_alloc(A,Xs,H2) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),array2(A,lim(product_unit,H2))),array_set(A,array2(A,lim(product_unit,H2)),Xs,lim_update(product_unit,aTP_Lamp_ou(heap_ext(product_unit),fun(nat,nat),H2),H2))) ) ).
% Array_Time.alloc_def
tff(fact_4684_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_4685_effect__refE,axiom,
! [A: $tType] :
( heap(A)
=> ! [V2: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: ref(A),N: nat] :
( heap_Time_effect(ref(A),ref_ref(A,V2),H2,H5,R2,N)
=> ~ ( ( ref_get(A,H5,R2) = V2 )
=> ( ref_present(A,H5,R2)
=> ( ~ ref_present(A,H2,R2)
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_refE
tff(fact_4686_execute__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [V2: A,H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(ref(A),ref_ref(A,V2)),H2) = aa(product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ov(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,V2,H2))) ) ).
% execute_ref
tff(fact_4687_ref__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [V2: A] : ref_ref(A,V2) = heap_Time_heap(ref(A),aTP_Lamp_ow(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),V2)) ) ).
% ref_def
tff(fact_4688_effect__refI,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H5: heap_ext(product_unit),V2: A,H2: heap_ext(product_unit),N: nat] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R2),H5) = ref_alloc(A,V2,H2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(ref(A),ref_ref(A,V2),H2,H5,R2,N) ) ) ) ).
% effect_refI
tff(fact_4689_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: A,H2: heap_ext(product_unit)] : ref_alloc(A,X,H2) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),ref2(A,lim(product_unit,H2))),ref_set(A,ref2(A,lim(product_unit,H2)),X,lim_update(product_unit,aTP_Lamp_ou(heap_ext(product_unit),fun(nat,nat),H2),H2))) ) ).
% Ref_Time.alloc_def
tff(fact_4690_effect__updateE,axiom,
! [A: $tType] :
( heap(A)
=> ! [R5: ref(A),V2: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: product_unit,N: nat] :
( heap_Time_effect(product_unit,ref_update(A,R5,V2),H2,H5,R2,N)
=> ~ ( ( H5 = ref_set(A,R5,V2,H2) )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_updateE
tff(fact_4691_effect__updateI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H5: heap_ext(product_unit),R2: ref(A),V2: A,H2: heap_ext(product_unit),N: nat,X: product_unit] :
( ( H5 = ref_set(A,R2,V2,H2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(product_unit,ref_update(A,R2,V2),H2,H5,X,N) ) ) ) ).
% effect_updateI
tff(fact_4692_update__wp__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: 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,R2,Y)),wand_assn(sngr_assn(A,R2,X),aa(product_unit,assn,Q,product_Unity))),ref_update(A,R2,X),Q) ) ).
% update_wp_rule
tff(fact_4693_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),V2: A] : ref_update(A,R2,V2) = heap_Time_heap(product_unit,aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ox(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R2),V2)) ) ).
% Ref_Time.update_def
tff(fact_4694_execute__update,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),V2: A,H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(product_unit,ref_update(A,R2,V2)),H2) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R2,V2,H2)),one_one(nat)))) ) ).
% execute_update
tff(fact_4695_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_4696_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I2: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( 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),Ys)),aa(nat,A,nth(A,Xs),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I2)) ) ) ) ) ).
% map_of_zip_nth
tff(fact_4697_assert_H__rule,axiom,
! [P: assn,Phi: bool] :
( ! [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(Phi) )
=> hoare_hoare_triple(product_unit,P,refine_Imp_assert(Phi),aTP_Lamp_oy(assn,fun(product_unit,assn),P)) ) ).
% assert'_rule
tff(fact_4698_map__of__zip__map,axiom,
! [A: $tType,B: $tType,Xs: list(A),F2: 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),F2),Xs))),X3) = aa(B,option(B),some(B),aa(A,B,F2,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),F2),Xs))),X3) = none(B) ) ) ) ).
% map_of_zip_map
tff(fact_4699_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
( ! [X4: A] : aa(A,option(B),map_of(A,B,Xys),X4) = none(B)
<=> ( Xys = nil(product_prod(A,B)) ) ) ).
% map_of_eq_empty_iff
tff(fact_4700_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
( ( aTP_Lamp_oz(A,option(B)) = map_of(A,B,Xys) )
<=> ( Xys = nil(product_prod(A,B)) ) ) ).
% empty_eq_map_of_iff
tff(fact_4701_map__of__zip__is__None,axiom,
! [B: $tType,A: $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) )
=> ( ( 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),Ys)),X) = none(B) )
<=> ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ) ).
% map_of_zip_is_None
tff(fact_4702_natural__zero__minus__one,axiom,
minus_minus(code_natural,zero_zero(code_natural),one_one(code_natural)) = zero_zero(code_natural) ).
% natural_zero_minus_one
tff(fact_4703_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).
% map_of_Cons_code(1)
tff(fact_4704_map__of_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,X3: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X3) = none(B) ).
% map_of.simps(1)
tff(fact_4705_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
=> ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
=> ( pp(aa(list(B),bool,distinct(B),Xs))
=> ( ( map_of(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)) = map_of(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),Zs)) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
tff(fact_4706_map__of__zip__is__Some,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)))
<=> ? [Y3: B] : 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),Ys)),X) = aa(B,option(B),some(B),Y3) ) ) ).
% map_of_zip_is_Some
tff(fact_4707_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( log(X,Xa) = Y )
=> ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,divide_divide(code_natural,Xa,X))) ) ) ) ) ).
% log.elims
tff(fact_4708_log_Osimps,axiom,
! [B2: code_natural,I2: code_natural] :
( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
=> ( log(B2,I2) = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
=> ( log(B2,I2) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,divide_divide(code_natural,I2,B2))) ) ) ) ).
% log.simps
tff(fact_4709_next_Osimps,axiom,
! [V2: code_natural,W: code_natural] : aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),V2),W)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one2)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(one2))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one2))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(one2)))))))))))))))) ).
% next.simps
tff(fact_4710_Random_Orange__def,axiom,
! [K: code_natural] : range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))),K),aTP_Lamp_pb(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_pc(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).
% Random.range_def
tff(fact_4711_inc__shift__def,axiom,
! [V2: code_natural,K: code_natural] :
( ( ( V2 = K )
=> ( inc_shift(V2,K) = one_one(code_natural) ) )
& ( ( V2 != K )
=> ( inc_shift(V2,K) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural)) ) ) ) ).
% inc_shift_def
tff(fact_4712_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( log(X,Xa) = Y )
=> ( accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa))
=> ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,divide_divide(code_natural,Xa,X))) ) ) )
=> ~ accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa)) ) ) ) ).
% log.pelims
tff(fact_4713_iterate_Osimps,axiom,
! [A: $tType,B: $tType,K: code_natural,F2: fun(B,fun(A,product_prod(B,A))),X: B] :
( ( ( K = zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),X) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X) ) )
& ( ( K != zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F2),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F2,X),iterate(B,A,minus_minus(code_natural,K,one_one(code_natural)),F2)) ) ) ) ).
% iterate.simps
tff(fact_4714_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( ( ( X = zero_zero(code_natural) )
=> ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
& ( ( X != zero_zero(code_natural) )
=> ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,minus_minus(code_natural,X,one_one(code_natural)),Xa)) ) ) ) ) ).
% iterate.elims
tff(fact_4715_select,axiom,
! [A: $tType,Xs: list(A),S3: 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),S3)),aa(list(A),set(A),set2(A),Xs))) ) ).
% select
tff(fact_4716_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb)))
=> ~ ( ( ( ( X = zero_zero(code_natural) )
=> ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
& ( ( X != zero_zero(code_natural) )
=> ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,minus_minus(code_natural,X,one_one(code_natural)),Xa)) ) ) )
=> ~ accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb))) ) ) ) ).
% iterate.pelims
tff(fact_4717_ran__map__of__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) )
=> ( pp(aa(list(A),bool,distinct(A),Xs))
=> ( ran(A,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),Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).
% ran_map_of_zip
tff(fact_4718_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_4719_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z2: B] :
( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
=> ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
=> ( ~ pp(member2(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( ( fun_upd(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),Ys)),X,aa(B,option(B),some(B),Y)) = fun_upd(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),Zs)),X,aa(B,option(B),some(B),Z2)) )
=> ( 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),Ys)) = 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),Zs)) ) ) ) ) ) ).
% map_of_zip_upd
tff(fact_4720_assert_H__bind__rule,axiom,
! [A: $tType,P: assn,Phi: bool,C2: heap_Time_Heap(A),Q: fun(A,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(Phi) )
=> ( ( pp(Phi)
=> hoare_hoare_triple(A,P,C2,Q) )
=> hoare_hoare_triple(A,P,heap_Time_bind(product_unit,A,refine_Imp_assert(Phi),aTP_Lamp_pd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),C2)),Q) ) ) ).
% assert'_bind_rule
tff(fact_4721_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list(A),F2: fun(A,B),V2: 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,F2,Y,V2)),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) ) ) ).
% map_fun_upd
tff(fact_4722_return__bind,axiom,
! [A: $tType,B: $tType,X: B,F2: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),F2) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_pe(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F2)) ).
% return_bind
tff(fact_4723_bind__return,axiom,
! [A: $tType,F2: heap_Time_Heap(A)] : heap_Time_bind(A,A,F2,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_pd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F2)) ).
% bind_return
tff(fact_4724_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( heap(B)
=> ! [R2: ref(B),F2: heap_Time_Heap(A)] : heap_Time_bind(B,A,ref_lookup(B,R2),aTP_Lamp_pf(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),F2)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_pd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F2)) ) ).
% lookup_chain
tff(fact_4725_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F2: heap_Time_Heap(B),R: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
( ! [R3: A] : hoare_hoare_triple(B,P,F2,aa(fun(B,assn),fun(B,assn),aTP_Lamp_lr(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,F2,G),Q) ) ) ).
% bind_rule'
tff(fact_4726_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_4727_map__upds__fold__map__upd,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M2,Ks,Vs) = aa(list(product_prod(A,B)),fun(A,option(B)),aa(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B))),aa(fun(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),fun(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B)))),foldl(fun(A,option(B)),product_prod(A,B)),aTP_Lamp_ph(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))))),M2),aa(list(B),list(product_prod(A,B)),aa(list(A),fun(list(B),list(product_prod(A,B))),zip(A,B),Ks),Vs)) ).
% map_upds_fold_map_upd
tff(fact_4728_pick__same,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pick(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),code_natural_of_nat(L)) = aa(nat,A,nth(A,Xs),L) ) ) ).
% pick_same
tff(fact_4729_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_4730_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_4731_one__natural__def,axiom,
one_one(code_natural) = code_natural_of_nat(one_one(nat)) ).
% one_natural_def
tff(fact_4732_map__upds__def,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),Xs: list(A),Ys: list(B)] : map_upds(A,B,M2,Xs,Ys) = map_add(A,B,M2,map_of(A,B,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)))) ).
% map_upds_def
tff(fact_4733_dom__map__of__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) )
=> ( dom(A,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),Ys))) = aa(list(A),set(A),set2(A),Xs) ) ) ).
% dom_map_of_zip
tff(fact_4734_Suc__natural__minus__one,axiom,
! [N: code_natural] : minus_minus(code_natural,code_Suc(N),one_one(code_natural)) = N ).
% Suc_natural_minus_one
tff(fact_4735_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_4736_ran__add,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F2)),dom(A,B,G)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,F2,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F2)),ran(A,B,G)) ) ) ).
% ran_add
tff(fact_4737_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A4: fun(A,option(B)),B3: fun(A,option(B)),C5: 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,C5)) = map_add(A,B,B3,map_add(A,B,A4,C5)) ) ) ).
% map_add_left_comm
tff(fact_4738_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_4739_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)),N5: 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),N5))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M6)),dom(A,B,N5)) = 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,N5))) ) ) ) ) ).
% map_add_distinct_le
tff(fact_4740_dom__if,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),F2: 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_pi(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F2),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,F2)),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_ef(fun(A,bool),fun(A,bool),P)))) ).
% dom_if
tff(fact_4741_dom__override__on,axiom,
! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),A4: set(A)] : dom(A,B,override_on(A,option(B),F2,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,F2),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_pj(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4)))),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_pk(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A4))) ).
% dom_override_on
tff(fact_4742_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_4743_Code__Numeral_Osub__code_I9_J,axiom,
! [M2: num,N: num] : code_sub(bit0(M2),bit1(N)) = minus_minus(code_integer,code_dup(code_sub(M2,N)),one_one(code_integer)) ).
% Code_Numeral.sub_code(9)
tff(fact_4744_Code__Numeral_Osub__code_I8_J,axiom,
! [M2: num,N: num] : code_sub(bit1(M2),bit0(N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_dup(code_sub(M2,N))),one_one(code_integer)) ).
% Code_Numeral.sub_code(8)
tff(fact_4745_AboveS__disjoint,axiom,
! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),order_AboveS(A,R2,A4)) = bot_bot(set(A)) ).
% AboveS_disjoint
tff(fact_4746_antisym__Restr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
( antisym(A,R2)
=> 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))),R2),product_Sigma(A,A,A4,aTP_Lamp_jt(set(A),fun(A,set(A)),A4)))) ) ).
% antisym_Restr
tff(fact_4747_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_4748_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_4749_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_4750_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_4751_Int_Osub__code_I9_J,axiom,
! [M2: num,N: num] : sub(bit0(M2),bit1(N)) = minus_minus(int,dup(sub(M2,N)),one_one(int)) ).
% Int.sub_code(9)
tff(fact_4752_Int_Osub__code_I8_J,axiom,
! [M2: num,N: num] : sub(bit1(M2),bit0(N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(M2,N))),one_one(int)) ).
% Int.sub_code(8)
tff(fact_4753_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_4754_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_4755_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,P7: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P7))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P7))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_pl(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z2),K),G),H2)),set_ord_atMost(nat,P7)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_pm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,minus_minus(nat,P7,aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% comm_monoid_set.zero_middle
tff(fact_4756_Fract__def,axiom,
fract = aa(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,rat)),map_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),id(int),map_fun(int,int,product_prod(int,int),rat,id(int),abs_Rat)),aTP_Lamp_iu(int,fun(int,product_prod(int,int)))) ).
% Fract_def
tff(fact_4757_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_4758_comm__monoid__set_Ounion__diff2,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(B),B3))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B3)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),minus_minus(set(B),A4,B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),minus_minus(set(B),B3,A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3))) ) ) ) ) ).
% comm_monoid_set.union_diff2
tff(fact_4759_comm__monoid__set_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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,F2,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),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B3)) ) ) ) ) ) ).
% comm_monoid_set.union_inter_neutral
tff(fact_4760_comm__monoid__set_Ounion__inter,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(B),B3))
=> ( aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),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,F2,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),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B3)) ) ) ) ) ).
% comm_monoid_set.union_inter
tff(fact_4761_comm__monoid__set_Ointer__restrict,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),G: fun(B,A),B3: set(B)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(set(B),fun(B,A),aa(fun(B,A),fun(set(B),fun(B,A)),aTP_Lamp_pn(A,fun(fun(B,A),fun(set(B),fun(B,A))),Z2),G),B3)),A4) ) ) ) ).
% comm_monoid_set.inter_restrict
tff(fact_4762_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_am(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),G)),A4) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),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,F2,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_4763_comm__monoid__set_OInt__Diff,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),G: fun(B,A),B3: set(B)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A4) = aa(A,A,aa(A,fun(A,A),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),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,F2,Z2),G),minus_minus(set(B),A4,B3))) ) ) ) ).
% comm_monoid_set.Int_Diff
tff(fact_4764_comm__monoid__set_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,T: set(B),S: set(B),H2: fun(B,A),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),T))
=> ( pp(aa(set(B),bool,finite_finite(B),S))
=> ( ! [I3: B] :
( pp(member2(B,I3,minus_minus(set(B),T,S)))
=> ( aa(B,A,H2,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,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),S) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),H2),T) ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong
tff(fact_4765_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,A4: set(B),B3: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),A4))
=> ( pp(aa(set(B),bool,finite_finite(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,F2,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),F2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),G),B3)) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
tff(fact_4766_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,C5: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> pp(aa(set(B),bool,finite_finite(B),X2)) )
=> ( ! [X2: set(B)] :
( pp(member2(set(B),X2,C5))
=> ! [Xa4: set(B)] :
( pp(member2(set(B),Xa4,C5))
=> ( ( 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,F2,Z2),G),complete_Sup_Sup(set(B),C5)) = 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),F2,Z2),groups_comm_monoid_F(A,B,F2,Z2)),G),C5) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
tff(fact_4767_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G: fun(nat,A),P7: nat] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( 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),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P7))) = aa(A,A,aa(A,fun(A,A),F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F2,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P7)))) ) ) ) ).
% comm_monoid_set.ub_add_nat
tff(fact_4768_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_4769_comm__monoid__set_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType,F2: fun(A,fun(A,A)),Z2: A,I: set(B),A4: fun(B,set(C)),G: fun(C,A)] :
( groups778175481326437816id_set(A,F2,Z2)
=> ( pp(aa(set(B),bool,finite_finite(B),I))
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> pp(aa(set(C),bool,finite_finite(C),aa(B,set(C),A4,X2))) )
=> ( ! [X2: B] :
( pp(member2(B,X2,I))
=> ! [Xa4: B] :
( pp(member2(B,Xa4,I))
=> ( ( 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,F2,Z2),G),complete_Sup_Sup(set(C),image2(B,set(C),A4,I))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F2,Z2),aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_po(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F2),Z2),A4),G)),I) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
tff(fact_4770_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_4771_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_4772_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_4773_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_4774_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_4775_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_4776_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_4777_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_4778_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_4779_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_4780_lenlex__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R2)),aTP_Lamp_pp(list(A),product_prod(nat,list(A)))) ).
% lenlex_def
tff(fact_4781_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
<=> ! [X4: A] :
( pp(member2(A,X4,field2(A,R2)))
=> 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))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_mq(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% Card_order_iff_Restr_underS
tff(fact_4782_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_4783_measures__def,axiom,
! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_pr(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).
% measures_def
tff(fact_4784_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_fn(nat,fun(nat,bool),N)),aTP_Lamp_ps(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_pt(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq
tff(fact_4785_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_pu(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_pt(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq2
tff(fact_4786_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_pv(list(option(A)),fun(A,option(list(A))),Xs),X) ).
% those.simps(2)
tff(fact_4787_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_4788_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_4789_sym__Int__converse,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),converse(A,A,R2))) ).
% sym_Int_converse
tff(fact_4790_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)))] :
( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),A1),A22)))
=> ( ! [F8: fun(code_natural,A),N3: code_natural,M5: code_natural] :
( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F8),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N3),M5)))
=> ( ! [X3: product_unit] :
( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),M5),N3))
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F8),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N3),one_one(code_natural))),M5)) )
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F8),N3),M5)) ) )
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,A0),A1),A22)) ) ) ).
% Predicate.iterate_upto.pinduct
tff(fact_4791_one__integer__code,axiom,
one_one(code_integer) = code_Pos(one2) ).
% one_integer_code
tff(fact_4792_listrel__sym,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( sym(A,R2)
=> sym(list(A),listrel(A,A,R2)) ) ).
% listrel_sym
tff(fact_4793_sym__Int,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( sym(A,R2)
=> ( sym(A,S3)
=> sym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S3)) ) ) ).
% sym_Int
tff(fact_4794_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
( ( iterate_upto(A,X,Xa,Xb) = Y )
=> ( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb)))
=> ~ ( ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_pw(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb)) )
=> ~ accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb))) ) ) ) ).
% Predicate.iterate_upto.pelims
tff(fact_4795_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F2: fun(code_natural,A),N: code_natural,M2: code_natural] :
( accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N),M2)))
=> ( iterate_upto(A,F2,N,M2) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_pw(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F2),N),M2)) ) ) ).
% Predicate.iterate_upto.psimps
tff(fact_4796_Inf__fin__def,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ( lattic7752659483105999362nf_fin(A) = lattic1715443433743089157tice_F(A,inf_inf(A)) ) ) ).
% Inf_fin_def
tff(fact_4797_Predicate_Oiterate__upto_Oelims,axiom,
! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
( ( iterate_upto(A,X,Xa,Xb) = Y )
=> ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_pw(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb)) ) ) ).
% Predicate.iterate_upto.elims
tff(fact_4798_Predicate_Oiterate__upto_Osimps,axiom,
! [A: $tType,F2: fun(code_natural,A),N: code_natural,M2: code_natural] : iterate_upto(A,F2,N,M2) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_pw(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F2),N),M2)) ).
% Predicate.iterate_upto.simps
tff(fact_4799_semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F2: fun(A,fun(A,A)),X: A,Xs: list(A)] :
( lattic149705377957585745ce_set(A,F2)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),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),F2),Xs),X) ) ) ).
% semilattice_set.set_eq_fold
tff(fact_4800_one__natural_Orep__eq,axiom,
code_nat_of_natural(one_one(code_natural)) = one_one(nat) ).
% one_natural.rep_eq
tff(fact_4801_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_4802_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> lattic149705377957585745ce_set(A,inf_inf(A)) ) ).
% Inf_fin.semilattice_set_axioms
tff(fact_4803_filter__rev__def,axiom,
! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).
% filter_rev_def
tff(fact_4804_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_4805_nat__of__num__code_I1_J,axiom,
nat_of_num(one2) = one_one(nat) ).
% nat_of_num_code(1)
tff(fact_4806_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_4807_one__int__code,axiom,
one_one(int) = pos(one2) ).
% one_int_code
tff(fact_4808_disjnt__def,axiom,
! [A: $tType,A4: set(A),B3: set(A)] :
( 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_4809_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),C5: fun(A,set(B)),B3: set(A)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,A4,C5),product_Sigma(A,B,B3,C5))
<=> ( ! [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),C5,X4) = bot_bot(set(B)) ) )
| disjnt(A,A4,B3) ) ) ).
% disjnt_Sigma_iff
tff(fact_4810_one__natural_Otransfer,axiom,
code_pcr_natural(one_one(nat),one_one(code_natural)) ).
% one_natural.transfer
tff(fact_4811_list_Opred__rel,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))
<=> 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),bNF_eq_onp(A,P)),X),X)) ) ).
% list.pred_rel
tff(fact_4812_list_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),bNF_Grp(A,B,A4,F2)) = bNF_Grp(list(A),list(B),collect(list(A),aTP_Lamp_bu(set(A),fun(list(A),bool),A4)),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2)) ).
% list.rel_Grp
tff(fact_4813_list_Orel__eq__onp,axiom,
! [A: $tType,P: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),bNF_eq_onp(A,P)) = bNF_eq_onp(list(A),aa(fun(A,bool),fun(list(A),bool),list_all(A),P)) ).
% list.rel_eq_onp
tff(fact_4814_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R) = relcompp(list(A),list(product_prod(A,B)),list(B),conversep(list(product_prod(A,B)),list(A),bNF_Grp(list(product_prod(A,B)),list(A),collect(list(product_prod(A,B)),aTP_Lamp_kd(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R)),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)))),bNF_Grp(list(product_prod(A,B)),list(B),collect(list(product_prod(A,B)),aTP_Lamp_kd(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R)),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)))) ).
% list.rel_compp_Grp
tff(fact_4815_one__integer_Otransfer,axiom,
code_pcr_integer(one_one(int),one_one(code_integer)) ).
% one_integer.transfer
tff(fact_4816_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_4817_list_Orel__conversep,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(list(B),fun(list(A),bool)),list_all2(B,A),conversep(A,B,R)) = conversep(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ).
% list.rel_conversep
tff(fact_4818_list_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R: fun(A,fun(B,bool)),S: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),relcompp(A,B,C,R,S)) = relcompp(list(A),list(B),list(C),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(fun(B,fun(C,bool)),fun(list(B),fun(list(C),bool)),list_all2(B,C),S)) ).
% list.rel_compp
tff(fact_4819_list_Orel__flip,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A3: list(B),B2: list(A)] :
( pp(aa(list(A),bool,aa(list(B),fun(list(A),bool),aa(fun(B,fun(A,bool)),fun(list(B),fun(list(A),bool)),list_all2(B,A),conversep(A,B,R)),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),R),B2),A3)) ) ).
% list.rel_flip
tff(fact_4820_converse__meet,axiom,
! [A: $tType,B: $tType,R2: fun(B,fun(A,bool)),S3: fun(B,fun(A,bool))] : conversep(B,A,aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),inf_inf(fun(B,fun(A,bool))),R2),S3)) = 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))),conversep(B,A,R2)),conversep(B,A,S3)) ).
% converse_meet
tff(fact_4821_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B3: fun(A,fun(B,bool)),P22: fun(A,bool),A4: fun(C,fun(A,bool)),P1: fun(C,bool),P23: fun(C,bool)] :
( ( domainp(A,B,B3) = P22 )
=> ( ( domainp(C,A,A4) = P1 )
=> ( pp(aa(fun(A,bool),bool,aa(fun(C,bool),fun(fun(A,bool),bool),bNF_rel_fun(C,A,bool,bool,A4,fequal(bool)),P23),P22))
=> ( domainp(C,B,relcompp(C,A,B,A4,B3)) = aa(fun(C,bool),fun(C,bool),aa(fun(C,bool),fun(fun(C,bool),fun(C,bool)),inf_inf(fun(C,bool)),P1),P23) ) ) ) ) ).
% pcr_Domainp_par
tff(fact_4822_composed__equiv__rel__eq__onp,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool)),P: fun(A,bool),P2: fun(B,bool),P8: fun(A,bool)] :
( left_unique(A,B,R)
=> ( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),bNF_rel_fun(A,B,bool,bool,R,fequal(bool)),P),P2))
=> ( ( domainp(A,B,R) = P8 )
=> ( relcompp(A,B,A,R,relcompp(B,B,A,bNF_eq_onp(B,P2),conversep(A,B,R))) = bNF_eq_onp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),P8),P)) ) ) ) ) ).
% composed_equiv_rel_eq_onp
tff(fact_4823_ListMem__iff,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( listMem(A,X,Xs)
<=> pp(member2(A,X,aa(list(A),set(A),set2(A),Xs))) ) ).
% ListMem_iff
tff(fact_4824_list_Oleft__unique__rel,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
( left_unique(A,B,R)
=> left_unique(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ) ).
% list.left_unique_rel
tff(fact_4825_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_4826_elem,axiom,
! [A: $tType,X: A,Xs: list(A)] : listMem(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ).
% elem
tff(fact_4827_ListMem_Ocases,axiom,
! [A: $tType,A1: A,A22: list(A)] :
( listMem(A,A1,A22)
=> ( ! [Xs2: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A1),Xs2)
=> ~ ! [Xs2: list(A)] :
( ? [Y2: A] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)
=> ~ listMem(A,A1,Xs2) ) ) ) ).
% ListMem.cases
tff(fact_4828_ListMem_Osimps,axiom,
! [A: $tType,A1: A,A22: list(A)] :
( listMem(A,A1,A22)
<=> ( ? [X4: A,Xs4: list(A)] :
( ( A1 = X4 )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) ) )
| ? [X4: A,Xs4: list(A),Y3: A] :
( ( A1 = X4 )
& ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs4) )
& listMem(A,X4,Xs4) ) ) ) ).
% ListMem.simps
tff(fact_4829_insert,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A] :
( listMem(A,X,Xs)
=> listMem(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) ) ).
% insert
tff(fact_4830_multiset_Odomain__par,axiom,
! [B: $tType,C: $tType,T: fun(C,fun(B,bool)),DT: fun(C,bool),DS: fun(nat,bool),P23: fun(fun(C,nat),bool)] :
( ( domainp(C,B,T) = DT )
=> ( ( domainp(nat,nat,fequal(nat)) = DS )
=> ( left_unique(C,B,T)
=> ( pp(aa(fun(fun(B,nat),bool),bool,aa(fun(fun(C,nat),bool),fun(fun(fun(B,nat),bool),bool),bNF_rel_fun(fun(C,nat),fun(B,nat),bool,bool,bNF_rel_fun(C,B,nat,nat,T,fequal(nat)),fequal(bool)),P23),aTP_Lamp_py(fun(B,nat),bool)))
=> ( domainp(fun(C,nat),multiset(B),pcr_multiset(C,B,T)) = aa(fun(fun(C,nat),bool),fun(fun(C,nat),bool),aa(fun(fun(C,nat),bool),fun(fun(fun(C,nat),bool),fun(fun(C,nat),bool)),inf_inf(fun(fun(C,nat),bool)),basic_pred_fun(C,nat,DT,DS)),P23) ) ) ) ) ) ).
% multiset.domain_par
tff(fact_4831_rat_Odomain__par,axiom,
! [DR1: fun(int,bool),DR2: fun(int,bool),P23: fun(product_prod(int,int),bool)] :
( ( domainp(int,int,fequal(int)) = DR1 )
=> ( ( domainp(int,int,fequal(int)) = DR2 )
=> ( pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal(bool)),P23),aTP_Lamp_pz(product_prod(int,int),bool)))
=> ( domainp(product_prod(int,int),rat,pcr_rat) = aa(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),inf_inf(fun(product_prod(int,int),bool)),basic_pred_prod(int,int,DR1,DR2)),P23) ) ) ) ) ).
% rat.domain_par
tff(fact_4832_wfP__SUP,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,fun(B,bool)))] :
( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,bool)),R2,I3))
=> ( ! [I3: A,J2: A] :
( ( aa(A,fun(B,fun(B,bool)),R2,I3) != aa(A,fun(B,fun(B,bool)),R2,J2) )
=> ( 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)),R2,I3))),rangep(B,B,aa(A,fun(B,fun(B,bool)),R2,J2))) = bot_bot(fun(B,bool)) ) )
=> wfP(B,complete_Sup_Sup(fun(B,fun(B,bool)),image2(A,fun(B,fun(B,bool)),R2,top_top(set(A))))) ) ) ).
% wfP_SUP
tff(fact_4833_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_4834_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F10: $tType,E5: $tType,C5: fun(A,fun(B,bool)),A4: fun(C,fun(E5,bool)),B3: fun(D,fun(F10,bool))] :
( bi_total(A,B,C5)
=> pp(aa(fun(fun(E5,fun(F10,bool)),fun(fun(B,E5),fun(fun(B,F10),bool))),bool,aa(fun(fun(C,fun(D,bool)),fun(fun(A,C),fun(fun(A,D),bool))),fun(fun(fun(E5,fun(F10,bool)),fun(fun(B,E5),fun(fun(B,F10),bool))),bool),bNF_rel_fun(fun(C,fun(D,bool)),fun(E5,fun(F10,bool)),fun(fun(A,C),fun(fun(A,D),bool)),fun(fun(B,E5),fun(fun(B,F10),bool)),bNF_rel_fun(C,E5,fun(D,bool),fun(F10,bool),A4,bNF_rel_fun(D,F10,bool,bool,B3,fequal(bool))),bNF_rel_fun(fun(A,C),fun(B,E5),fun(fun(A,D),bool),fun(fun(B,F10),bool),bNF_rel_fun(A,B,C,E5,C5,A4),bNF_rel_fun(fun(A,D),fun(B,F10),bool,bool,bNF_rel_fun(A,B,D,F10,C5,B3),fequal(bool)))),partial_fun_ord(C,D,A)),partial_fun_ord(E5,F10,B))) ) ).
% fun_ord_parametric
tff(fact_4835_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_qa(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),A4)),vimage(D,B))) ) ) ).
% vimage_right_total_transfer
tff(fact_4836_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_qb(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),A4)),complete_Inf_Inf(set(B)))) ) ) ).
% right_total_Inter_transfer
tff(fact_4837_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_qc(fun(A,fun(B,bool)),fun(set(A),set(A)),A4)),uminus_uminus(set(B)))) ) ) ).
% right_total_Compl_transfer
tff(fact_4838_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_4839_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_4840_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_4841_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_4842_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_4843_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_4844_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_4845_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_4846_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_4847_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_4848_fun__lub__parametric,axiom,
! [A: $tType,C: $tType,E5: $tType,F10: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool)),C5: fun(E5,fun(F10,bool))] :
( bi_total(A,B,A4)
=> ( bi_unique(A,B,A4)
=> pp(aa(fun(fun(set(B),D),fun(set(fun(F10,B)),fun(F10,D))),bool,aa(fun(fun(set(A),C),fun(set(fun(E5,A)),fun(E5,C))),fun(fun(fun(set(B),D),fun(set(fun(F10,B)),fun(F10,D))),bool),bNF_rel_fun(fun(set(A),C),fun(set(B),D),fun(set(fun(E5,A)),fun(E5,C)),fun(set(fun(F10,B)),fun(F10,D)),bNF_rel_fun(set(A),set(B),C,D,bNF_rel_set(A,B,A4),B3),bNF_rel_fun(set(fun(E5,A)),set(fun(F10,B)),fun(E5,C),fun(F10,D),bNF_rel_set(fun(E5,A),fun(F10,B),bNF_rel_fun(E5,F10,A,B,C5,A4)),bNF_rel_fun(E5,F10,C,D,C5,B3))),partial_fun_lub(A,C,E5)),partial_fun_lub(B,D,F10))) ) ) ).
% fun_lub_parametric
tff(fact_4849_syntax__nomatch__def,axiom,
! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax2379306206330768139omatch(A,B,Pat,Obj) ).
% syntax_nomatch_def
tff(fact_4850_comp__fun__idem__inf,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> finite_comp_fun_idem(A,A,inf_inf(A)) ) ).
% comp_fun_idem_inf
tff(fact_4851_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F2: fun(multiset(A),B),A4: multiset(A),B3: multiset(A)] :
( mono(multiset(A),B,subseteq_mset(A),F2)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F2,inter_mset(A,A4,B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(multiset(A),B,F2,A4)),aa(multiset(A),B,F2,B3)))) ) ) ).
% subset_mset.mono_inf
tff(fact_4852_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_4853_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),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,complete_Sup_Sup(multiset(A),A4),complete_Sup_Sup(multiset(A),B3)))) ) ) ) ).
% subset_mset.cSup_inter_less_eq
tff(fact_4854_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_4855_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_4856_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_4857_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_4858_ATP_Olambda__1,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_iq(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).
% ATP.lambda_1
tff(fact_4859_ATP_Olambda__2,axiom,
! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_hm(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(set(A),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).
% ATP.lambda_2
tff(fact_4860_ATP_Olambda__3,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_pp(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_3
tff(fact_4861_ATP_Olambda__4,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_cw(nat,nat),Uu) = minus_minus(nat,Uu,aa(nat,nat,suc,zero_zero(nat))) ).
% ATP.lambda_4
tff(fact_4862_ATP_Olambda__5,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_bw(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).
% ATP.lambda_5
tff(fact_4863_ATP_Olambda__6,axiom,
! [Uu: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aTP_Lamp_pz(product_prod(int,int),bool),Uu))
<=> pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Uu),Uu)) ) ).
% ATP.lambda_6
tff(fact_4864_ATP_Olambda__7,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_gi(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_7
tff(fact_4865_ATP_Olambda__8,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_ms(A,bool),Uu))
<=> pp(dvd_dvd(A,Uu,one_one(A))) ) ) ).
% ATP.lambda_8
tff(fact_4866_ATP_Olambda__9,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,list(A),aTP_Lamp_mm(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).
% ATP.lambda_9
tff(fact_4867_ATP_Olambda__10,axiom,
! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_dq(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_10
tff(fact_4868_ATP_Olambda__11,axiom,
! [B: $tType,Uu: list(B)] :
( pp(aa(list(B),bool,aTP_Lamp_kg(list(B),bool),Uu))
<=> ( Uu = nil(B) ) ) ).
% ATP.lambda_11
tff(fact_4869_ATP_Olambda__12,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_kf(list(A),bool),Uu))
<=> ( Uu = nil(A) ) ) ).
% ATP.lambda_12
tff(fact_4870_ATP_Olambda__13,axiom,
! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_pb(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_pa(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).
% ATP.lambda_13
tff(fact_4871_ATP_Olambda__14,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ey(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_14
tff(fact_4872_ATP_Olambda__15,axiom,
! [B: $tType,Uu: list(B)] :
( pp(aa(list(B),bool,aTP_Lamp_ez(list(B),bool),Uu))
<=> ( Uu != nil(B) ) ) ).
% ATP.lambda_15
tff(fact_4873_ATP_Olambda__16,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_ei(list(A),bool),Uu))
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_16
tff(fact_4874_ATP_Olambda__17,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_ph(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = product_case_prod(A,B,fun(A,option(B)),aTP_Lamp_pg(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).
% ATP.lambda_17
tff(fact_4875_ATP_Olambda__18,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_go(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_gn(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_18
tff(fact_4876_ATP_Olambda__19,axiom,
! [B: $tType,Uu: fun(B,nat)] :
( pp(aa(fun(B,nat),bool,aTP_Lamp_py(fun(B,nat),bool),Uu))
<=> pp(aa(set(B),bool,finite_finite(B),collect(B,aTP_Lamp_px(fun(B,nat),fun(B,bool),Uu)))) ) ).
% ATP.lambda_19
tff(fact_4877_ATP_Olambda__20,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ex(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_20
tff(fact_4878_ATP_Olambda__21,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_cd(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_21
tff(fact_4879_ATP_Olambda__22,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gg(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),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_22
tff(fact_4880_ATP_Olambda__23,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_im(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_23
tff(fact_4881_ATP_Olambda__24,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_je(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu))),one_one(int))) ).
% ATP.lambda_24
tff(fact_4882_ATP_Olambda__25,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_jd(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu))),one2)) ).
% ATP.lambda_25
tff(fact_4883_ATP_Olambda__26,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_jg(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).
% ATP.lambda_26
tff(fact_4884_ATP_Olambda__27,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ik(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),abs_abs(int,Uu))) ).
% ATP.lambda_27
tff(fact_4885_ATP_Olambda__28,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_iu(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Uu),Uua)) ).
% ATP.lambda_28
tff(fact_4886_ATP_Olambda__29,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ds(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),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_29
tff(fact_4887_ATP_Olambda__30,axiom,
! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_kn(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_km(A,fun(list(A),bool))),Uua),Uu)) ) ).
% ATP.lambda_30
tff(fact_4888_ATP_Olambda__31,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_oq(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_op(A,fun(A,option(A)),Uua),Uu) ) ).
% ATP.lambda_31
tff(fact_4889_ATP_Olambda__32,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_do(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_dn(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_32
tff(fact_4890_ATP_Olambda__33,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gb(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_33
tff(fact_4891_ATP_Olambda__34,axiom,
! [Uu: rat,Uua: int] :
( pp(aa(int,bool,aTP_Lamp_lm(rat,fun(int,bool),Uu),Uua))
<=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
& pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).
% ATP.lambda_34
tff(fact_4892_ATP_Olambda__35,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ge(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_35
tff(fact_4893_ATP_Olambda__36,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fe(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_36
tff(fact_4894_ATP_Olambda__37,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gf(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_37
tff(fact_4895_ATP_Olambda__38,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fp(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_38
tff(fact_4896_ATP_Olambda__39,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_oc(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,Uua,Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,Uua,Uu)))) ) ).
% ATP.lambda_39
tff(fact_4897_ATP_Olambda__40,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
( pp(aa(set(set(A)),bool,aTP_Lamp_hn(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
<=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
& ( Uua != bot_bot(set(set(A))) ) ) ) ).
% ATP.lambda_40
tff(fact_4898_ATP_Olambda__41,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ew(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).
% ATP.lambda_41
tff(fact_4899_ATP_Olambda__42,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_lo(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_42
tff(fact_4900_ATP_Olambda__43,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_di(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_43
tff(fact_4901_ATP_Olambda__44,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dw(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_44
tff(fact_4902_ATP_Olambda__45,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fm(fun(nat,A),fun(nat,A),Uu),Uua) = minus_minus(A,aa(nat,A,Uu,Uua),aa(nat,A,Uu,minus_minus(nat,Uua,one_one(nat)))) ) ).
% ATP.lambda_45
tff(fact_4903_ATP_Olambda__46,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fj(fun(nat,A),fun(nat,A),Uu),Uua) = minus_minus(A,aa(nat,A,Uu,Uua),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).
% ATP.lambda_46
tff(fact_4904_ATP_Olambda__47,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: A] : aa(A,assn,aTP_Lamp_nk(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_47
tff(fact_4905_ATP_Olambda__48,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_aq(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_48
tff(fact_4906_ATP_Olambda__49,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_cb(fun(B,A),fun(B,bool),Uu),Uua))
<=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).
% ATP.lambda_49
tff(fact_4907_ATP_Olambda__50,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nx(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nw(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ).
% ATP.lambda_50
tff(fact_4908_ATP_Olambda__51,axiom,
! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_pr(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_pq(A,fun(fun(A,nat),nat),Uua)),Uu) ).
% ATP.lambda_51
tff(fact_4909_ATP_Olambda__52,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_ch(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_cg(list(A),fun(A,list(A))),Uua)),Uu) ).
% ATP.lambda_52
tff(fact_4910_ATP_Olambda__53,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_ct(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_53
tff(fact_4911_ATP_Olambda__54,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_aj(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_54
tff(fact_4912_ATP_Olambda__55,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_kd(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_55
tff(fact_4913_ATP_Olambda__56,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_qb(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_56
tff(fact_4914_ATP_Olambda__57,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_qc(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_57
tff(fact_4915_ATP_Olambda__58,axiom,
! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_pv(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_58
tff(fact_4916_ATP_Olambda__59,axiom,
! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_ck(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_59
tff(fact_4917_ATP_Olambda__60,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ja(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_60
tff(fact_4918_ATP_Olambda__61,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_bu(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_61
tff(fact_4919_ATP_Olambda__62,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fw(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_62
tff(fact_4920_ATP_Olambda__63,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_bz(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = 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_63
tff(fact_4921_ATP_Olambda__64,axiom,
! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_ha(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_64
tff(fact_4922_ATP_Olambda__65,axiom,
! [Uu: set(nat),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_cz(set(nat),fun(nat,bool),Uu),Uua))
<=> pp(member2(nat,aa(nat,nat,suc,Uua),Uu)) ) ).
% ATP.lambda_65
tff(fact_4923_ATP_Olambda__66,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_dc(nat,fun(list(A),bool),Uu),Uua))
<=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).
% ATP.lambda_66
tff(fact_4924_ATP_Olambda__67,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aTP_Lamp_ou(heap_ext(product_unit),fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),lim(product_unit,Uu)),one_one(nat)) ).
% ATP.lambda_67
tff(fact_4925_ATP_Olambda__68,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_gk(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_68
tff(fact_4926_ATP_Olambda__69,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_gl(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_69
tff(fact_4927_ATP_Olambda__70,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_70
tff(fact_4928_ATP_Olambda__71,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gc(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).
% ATP.lambda_71
tff(fact_4929_ATP_Olambda__72,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ov(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_Pair(ref(A),product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ) ).
% ATP.lambda_72
tff(fact_4930_ATP_Olambda__73,axiom,
! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nz(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ).
% ATP.lambda_73
tff(fact_4931_ATP_Olambda__74,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ci(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_74
tff(fact_4932_ATP_Olambda__75,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_dr(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_75
tff(fact_4933_ATP_Olambda__76,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dj(A,fun(nat,A),Uu),Uua) = minus_minus(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_76
tff(fact_4934_ATP_Olambda__77,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_km(A,fun(list(A),bool)),Uu),Uua))
<=> pp(member2(A,Uu,aa(list(A),set(A),set2(A),Uua))) ) ).
% ATP.lambda_77
tff(fact_4935_ATP_Olambda__78,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_et(list(A),fun(A,bool),Uu),Uua))
<=> pp(member2(A,Uua,aa(list(A),set(A),set2(A),Uu))) ) ).
% ATP.lambda_78
tff(fact_4936_ATP_Olambda__79,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_db(nat,fun(list(A),bool),Uu),Uua))
<=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_79
tff(fact_4937_ATP_Olambda__80,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_kq(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).
% ATP.lambda_80
tff(fact_4938_ATP_Olambda__81,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_kp(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua = last(A,Uu) ) ) ).
% ATP.lambda_81
tff(fact_4939_ATP_Olambda__82,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_co(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_82
tff(fact_4940_ATP_Olambda__83,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mv(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_83
tff(fact_4941_ATP_Olambda__84,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mt(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_4942_ATP_Olambda__85,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_fn(nat,fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).
% ATP.lambda_85
tff(fact_4943_ATP_Olambda__86,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_cp(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_86
tff(fact_4944_ATP_Olambda__87,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mw(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_87
tff(fact_4945_ATP_Olambda__88,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mu(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_88
tff(fact_4946_ATP_Olambda__89,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_dd(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_89
tff(fact_4947_ATP_Olambda__90,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bg(A,fun(A,A),Uu),Uua) = minus_minus(A,Uua,Uu) ) ).
% ATP.lambda_90
tff(fact_4948_ATP_Olambda__91,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_bd(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_91
tff(fact_4949_ATP_Olambda__92,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bi(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).
% ATP.lambda_92
tff(fact_4950_ATP_Olambda__93,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_cj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).
% ATP.lambda_93
tff(fact_4951_ATP_Olambda__94,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: ref(A)] : aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_as(A,fun(ref(A),assn)),Uu),Uua) = sngr_assn(A,Uua,Uu) ) ).
% ATP.lambda_94
tff(fact_4952_ATP_Olambda__95,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_at(list(A),fun(array(A),assn)),Uu),Uua) = snga_assn(A,Uua,Uu) ) ).
% ATP.lambda_95
tff(fact_4953_ATP_Olambda__96,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_gq(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_96
tff(fact_4954_ATP_Olambda__97,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_jk(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_97
tff(fact_4955_ATP_Olambda__98,axiom,
! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_cg(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_98
tff(fact_4956_ATP_Olambda__99,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_99
tff(fact_4957_ATP_Olambda__100,axiom,
! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_mc(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_100
tff(fact_4958_ATP_Olambda__101,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_ft(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).
% ATP.lambda_101
tff(fact_4959_ATP_Olambda__102,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_kl(A,fun(A,bool),Uu),Uua))
<=> ( Uua = Uu ) ) ).
% ATP.lambda_102
tff(fact_4960_ATP_Olambda__103,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_gj(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_103
tff(fact_4961_ATP_Olambda__104,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_gh(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_104
tff(fact_4962_ATP_Olambda__105,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ow(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ov(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,Uu,Uua)) ) ).
% ATP.lambda_105
tff(fact_4963_ATP_Olambda__106,axiom,
! [B: $tType,Uu: fun(B,nat),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_px(fun(B,nat),fun(B,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(B,nat,Uu,Uua))) ) ).
% ATP.lambda_106
tff(fact_4964_ATP_Olambda__107,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_pu(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).
% ATP.lambda_107
tff(fact_4965_ATP_Olambda__108,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_gr(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_108
tff(fact_4966_ATP_Olambda__109,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_gs(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_109
tff(fact_4967_ATP_Olambda__110,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_110
tff(fact_4968_ATP_Olambda__111,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ev(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_111
tff(fact_4969_ATP_Olambda__112,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).
% ATP.lambda_112
tff(fact_4970_ATP_Olambda__113,axiom,
! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_pq(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).
% ATP.lambda_113
tff(fact_4971_ATP_Olambda__114,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_oo(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_114
tff(fact_4972_ATP_Olambda__115,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(bool)),aTP_Lamp_ot(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_os(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua)) ).
% ATP.lambda_115
tff(fact_4973_ATP_Olambda__116,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_on(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_om(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).
% ATP.lambda_116
tff(fact_4974_ATP_Olambda__117,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_kw(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_kv(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).
% ATP.lambda_117
tff(fact_4975_ATP_Olambda__118,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_id(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_ic(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua)) ) ).
% ATP.lambda_118
tff(fact_4976_ATP_Olambda__119,axiom,
! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oj(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Uu,Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat)))) ).
% ATP.lambda_119
tff(fact_4977_ATP_Olambda__120,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_or(fun(A,option(B)),fun(A,bool),Uu),Uua))
<=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).
% ATP.lambda_120
tff(fact_4978_ATP_Olambda__121,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_in(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_im(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).
% ATP.lambda_121
tff(fact_4979_ATP_Olambda__122,axiom,
! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_pa(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_122
tff(fact_4980_ATP_Olambda__123,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dp(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_123
tff(fact_4981_ATP_Olambda__124,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dm(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_124
tff(fact_4982_ATP_Olambda__125,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_es(list(A),fun(A,bool),Uu),Uua))
<=> ~ pp(member2(A,Uua,aa(list(A),set(A),set2(A),Uu))) ) ).
% ATP.lambda_125
tff(fact_4983_ATP_Olambda__126,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_hy(list(A),fun(A,bool),Uu),Uua))
<=> ( Uua != last(A,Uu) ) ) ).
% ATP.lambda_126
tff(fact_4984_ATP_Olambda__127,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_op(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_127
tff(fact_4985_ATP_Olambda__128,axiom,
! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_pc(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),modulo_modulo(code_natural,Uua,Uu)) ).
% ATP.lambda_128
tff(fact_4986_ATP_Olambda__129,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_il(A,fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uua)) ) ) ).
% ATP.lambda_129
tff(fact_4987_ATP_Olambda__130,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ej(set(A),fun(A,bool),Uu),Uua))
<=> ~ pp(member2(A,Uua,Uu)) ) ).
% ATP.lambda_130
tff(fact_4988_ATP_Olambda__131,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_eg(A,fun(A,bool)),Uu),Uua))
<=> ( Uu != Uua ) ) ).
% ATP.lambda_131
tff(fact_4989_ATP_Olambda__132,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_er(A,fun(A,bool),Uu),Uua))
<=> ( Uua != Uu ) ) ).
% ATP.lambda_132
tff(fact_4990_ATP_Olambda__133,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_jz(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)),insert3(A),Uu),bot_bot(set(A)))) ).
% ATP.lambda_133
tff(fact_4991_ATP_Olambda__134,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_ea(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_134
tff(fact_4992_ATP_Olambda__135,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_cu(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_135
tff(fact_4993_ATP_Olambda__136,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_gv(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_136
tff(fact_4994_ATP_Olambda__137,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_he(fun(A,set(B)),fun(A,nat),Uu),Uua) = finite_card(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_137
tff(fact_4995_ATP_Olambda__138,axiom,
! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_cf(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_138
tff(fact_4996_ATP_Olambda__139,axiom,
! [B: $tType,Uu: fun(B,bool),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_an(fun(B,bool),fun(B,bool),Uu),Uua))
<=> ~ pp(aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_139
tff(fact_4997_ATP_Olambda__140,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ef(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_140
tff(fact_4998_ATP_Olambda__141,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ps(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aTP_Lamp_fn(nat,fun(nat,bool),Uu)) ).
% ATP.lambda_141
tff(fact_4999_ATP_Olambda__142,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_is(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_142
tff(fact_5000_ATP_Olambda__143,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_iz(list(A),fun(A,bool),Uu),Uua))
<=> ? [I4: nat] :
( ( Uua = aa(nat,A,nth(A,Uu),I4) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).
% ATP.lambda_143
tff(fact_5001_ATP_Olambda__144,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_ma(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_144
tff(fact_5002_ATP_Olambda__145,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_mn(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).
% ATP.lambda_145
tff(fact_5003_ATP_Olambda__146,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_ok(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).
% ATP.lambda_146
tff(fact_5004_ATP_Olambda__147,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_147
tff(fact_5005_ATP_Olambda__148,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_js(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).
% ATP.lambda_148
tff(fact_5006_ATP_Olambda__149,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_ld(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert3(A),Uu) ).
% ATP.lambda_149
tff(fact_5007_ATP_Olambda__150,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_ju(set(A),fun(list(A),set(list(A))),Uu),Uua) = aa(set(A),set(list(A)),lists(A),Uu) ).
% ATP.lambda_150
tff(fact_5008_ATP_Olambda__151,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_io(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),minus_minus(code_integer,Uub,aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_151
tff(fact_5009_ATP_Olambda__152,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hi(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),minus_minus(nat,Uub,aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_152
tff(fact_5010_ATP_Olambda__153,axiom,
! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hj(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),minus_minus(int,Uub,aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_153
tff(fact_5011_ATP_Olambda__154,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_hl(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_154
tff(fact_5012_ATP_Olambda__155,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_jv(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_155
tff(fact_5013_ATP_Olambda__156,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_en(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_156
tff(fact_5014_ATP_Olambda__157,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_ca(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_157
tff(fact_5015_ATP_Olambda__158,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_al(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_158
tff(fact_5016_ATP_Olambda__159,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_159
tff(fact_5017_ATP_Olambda__160,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_mz(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_160
tff(fact_5018_ATP_Olambda__161,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_nd(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_161
tff(fact_5019_ATP_Olambda__162,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_my(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_162
tff(fact_5020_ATP_Olambda__163,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_nc(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_163
tff(fact_5021_ATP_Olambda__164,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua),one_one(code_integer))),minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu),Uub))) ).
% ATP.lambda_164
tff(fact_5022_ATP_Olambda__165,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua),one_one(code_integer))),minus_minus(code_integer,abs_abs(code_integer,Uu),Uub))) ).
% ATP.lambda_165
tff(fact_5023_ATP_Olambda__166,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ji(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua),one_one(code_integer))),minus_minus(code_integer,Uu,Uub))) ).
% ATP.lambda_166
tff(fact_5024_ATP_Olambda__167,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_eq(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_167
tff(fact_5025_ATP_Olambda__168,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_bn(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_168
tff(fact_5026_ATP_Olambda__169,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_kc(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_169
tff(fact_5027_ATP_Olambda__170,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_ol(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_170
tff(fact_5028_ATP_Olambda__171,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_hq(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,mergesort_by_rel(A,Uu,Uua),mergesort_by_rel(A,Uu,Uub)) ).
% ATP.lambda_171
tff(fact_5029_ATP_Olambda__172,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_pg(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).
% ATP.lambda_172
tff(fact_5030_ATP_Olambda__173,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_kx(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_173
tff(fact_5031_ATP_Olambda__174,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_ku(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_174
tff(fact_5032_ATP_Olambda__175,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_175
tff(fact_5033_ATP_Olambda__176,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_jf(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).
% ATP.lambda_176
tff(fact_5034_ATP_Olambda__177,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oe(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uub,Uu)),Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),one_one(nat))) ) ).
% ATP.lambda_177
tff(fact_5035_ATP_Olambda__178,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_dz(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_178
tff(fact_5036_ATP_Olambda__179,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_lb(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = minus_minus(nat,Uu,one_one(nat)) )
& ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).
% ATP.lambda_179
tff(fact_5037_ATP_Olambda__180,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_ia(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_180
tff(fact_5038_ATP_Olambda__181,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_me(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_181
tff(fact_5039_ATP_Olambda__182,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_iy(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,Xs6: list(A),Ys6: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys6)) )
& 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_182
tff(fact_5040_ATP_Olambda__183,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_lc(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
& ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).
% ATP.lambda_183
tff(fact_5041_ATP_Olambda__184,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_ad(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_184
tff(fact_5042_ATP_Olambda__185,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ab(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_185
tff(fact_5043_ATP_Olambda__186,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_jq(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_186
tff(fact_5044_ATP_Olambda__187,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_cv(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_187
tff(fact_5045_ATP_Olambda__188,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_da(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_188
tff(fact_5046_ATP_Olambda__189,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_cr(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_189
tff(fact_5047_ATP_Olambda__190,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_ks(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
& ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).
% ATP.lambda_190
tff(fact_5048_ATP_Olambda__191,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_hb(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_191
tff(fact_5049_ATP_Olambda__192,axiom,
! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_ce(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_192
tff(fact_5050_ATP_Olambda__193,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_bv(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_193
tff(fact_5051_ATP_Olambda__194,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_em(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_194
tff(fact_5052_ATP_Olambda__195,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_oi(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_195
tff(fact_5053_ATP_Olambda__196,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_du(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_196
tff(fact_5054_ATP_Olambda__197,axiom,
! [Uu: heap_ext(product_unit),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_oa(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uu)))
& ~ pp(member2(nat,Uub,Uua)) ) ) ).
% ATP.lambda_197
tff(fact_5055_ATP_Olambda__198,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_cc(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_198
tff(fact_5056_ATP_Olambda__199,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_pt(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_199
tff(fact_5057_ATP_Olambda__200,axiom,
! [Uu: set(nat),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_nh(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_ng(set(nat),fun(nat,fun(nat,bool)),Uu),Uub))),Uua)) ) ) ).
% ATP.lambda_200
tff(fact_5058_ATP_Olambda__201,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_bx(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_201
tff(fact_5059_ATP_Olambda__202,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_by(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_202
tff(fact_5060_ATP_Olambda__203,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ng(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_203
tff(fact_5061_ATP_Olambda__204,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ba(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member2(A,Uub,Uu))
& pp(member2(A,Uub,Uua)) ) ) ).
% ATP.lambda_204
tff(fact_5062_ATP_Olambda__205,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_bm(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_205
tff(fact_5063_ATP_Olambda__206,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_ni(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_206
tff(fact_5064_ATP_Olambda__207,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_be(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_207
tff(fact_5065_ATP_Olambda__208,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_jl(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_208
tff(fact_5066_ATP_Olambda__209,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_nn(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_209
tff(fact_5067_ATP_Olambda__210,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_np(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_210
tff(fact_5068_ATP_Olambda__211,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_nq(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_211
tff(fact_5069_ATP_Olambda__212,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_qa(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_212
tff(fact_5070_ATP_Olambda__213,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_df(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_213
tff(fact_5071_ATP_Olambda__214,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_de(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_214
tff(fact_5072_ATP_Olambda__215,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_bc(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_215
tff(fact_5073_ATP_Olambda__216,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dn(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_216
tff(fact_5074_ATP_Olambda__217,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_bo(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_217
tff(fact_5075_ATP_Olambda__218,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_ar(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_218
tff(fact_5076_ATP_Olambda__219,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_ff(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_219
tff(fact_5077_ATP_Olambda__220,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_ly(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_220
tff(fact_5078_ATP_Olambda__221,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_hu(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_221
tff(fact_5079_ATP_Olambda__222,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_nf(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_222
tff(fact_5080_ATP_Olambda__223,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_ht(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_223
tff(fact_5081_ATP_Olambda__224,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_ne(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_224
tff(fact_5082_ATP_Olambda__225,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_jy(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_225
tff(fact_5083_ATP_Olambda__226,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_lf(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_226
tff(fact_5084_ATP_Olambda__227,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_ag(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_227
tff(fact_5085_ATP_Olambda__228,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_af(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_228
tff(fact_5086_ATP_Olambda__229,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_bb(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_229
tff(fact_5087_ATP_Olambda__230,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_dt(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_230
tff(fact_5088_ATP_Olambda__231,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ek(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).
% ATP.lambda_231
tff(fact_5089_ATP_Olambda__232,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_lj(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_232
tff(fact_5090_ATP_Olambda__233,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_ip(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).
% ATP.lambda_233
tff(fact_5091_ATP_Olambda__234,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_dx(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_234
tff(fact_5092_ATP_Olambda__235,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_lx(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_235
tff(fact_5093_ATP_Olambda__236,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_fa(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).
% ATP.lambda_236
tff(fact_5094_ATP_Olambda__237,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_ed(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_237
tff(fact_5095_ATP_Olambda__238,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_lq(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_238
tff(fact_5096_ATP_Olambda__239,axiom,
! [E5: $tType,Uu: fun(E5,assn),Uua: assn,Uub: E5] : aa(E5,assn,aa(assn,fun(E5,assn),aTP_Lamp_lt(fun(E5,assn),fun(assn,fun(E5,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(E5,assn,Uu,Uub)),Uua) ).
% ATP.lambda_239
tff(fact_5097_ATP_Olambda__240,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_lw(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_240
tff(fact_5098_ATP_Olambda__241,axiom,
! [G3: $tType,H8: $tType,Uu: fun(G3,set(H8)),Uua: set(H8),Uub: G3] : aa(G3,set(H8),aa(set(H8),fun(G3,set(H8)),aTP_Lamp_bs(fun(G3,set(H8)),fun(set(H8),fun(G3,set(H8))),Uu),Uua),Uub) = aa(set(H8),set(H8),aa(set(H8),fun(set(H8),set(H8)),inf_inf(set(H8)),aa(G3,set(H8),Uu,Uub)),Uua) ).
% ATP.lambda_241
tff(fact_5099_ATP_Olambda__242,axiom,
! [C: $tType,Uu: fun(C,assn),Uua: assn,Uub: C] : aa(C,assn,aa(assn,fun(C,assn),aTP_Lamp_lv(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_242
tff(fact_5100_ATP_Olambda__243,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_bh(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_243
tff(fact_5101_ATP_Olambda__244,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_hv(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_244
tff(fact_5102_ATP_Olambda__245,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_lz(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_245
tff(fact_5103_ATP_Olambda__246,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_ho(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_246
tff(fact_5104_ATP_Olambda__247,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_ke(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_247
tff(fact_5105_ATP_Olambda__248,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_ii(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_248
tff(fact_5106_ATP_Olambda__249,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nu(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),Uua),upt(zero_zero(nat),Uu)),Uub)) ) ).
% ATP.lambda_249
tff(fact_5107_ATP_Olambda__250,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ny(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_nt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),Uu),Uua),Uub)) ) ).
% ATP.lambda_250
tff(fact_5108_ATP_Olambda__251,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lh(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ~ pp(aa(A,bool,Uu,Uub))
| ( Uua = Uub ) ) ) ).
% ATP.lambda_251
tff(fact_5109_ATP_Olambda__252,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_dy(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_252
tff(fact_5110_ATP_Olambda__253,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_kr(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
<=> ( ? [Y3: A,Ys4: list(A)] :
( ( Uua = nil(A) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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,Xs4),Ys4)) ) ) ) ) ).
% ATP.lambda_253
tff(fact_5111_ATP_Olambda__254,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_gz(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_254
tff(fact_5112_ATP_Olambda__255,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_ns(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)),insert3(A),Uua),bot_bot(set(A)))) ).
% ATP.lambda_255
tff(fact_5113_ATP_Olambda__256,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_ig(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_if(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub)) ) ).
% ATP.lambda_256
tff(fact_5114_ATP_Olambda__257,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_kz(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_257
tff(fact_5115_ATP_Olambda__258,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gd(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_258
tff(fact_5116_ATP_Olambda__259,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fz(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_259
tff(fact_5117_ATP_Olambda__260,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)))) ) ).
% ATP.lambda_260
tff(fact_5118_ATP_Olambda__261,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nw(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Uu)))) ) ).
% ATP.lambda_261
tff(fact_5119_ATP_Olambda__262,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_nm(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_262
tff(fact_5120_ATP_Olambda__263,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_fo(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_263
tff(fact_5121_ATP_Olambda__264,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_pj(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_264
tff(fact_5122_ATP_Olambda__265,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,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu)))) ) ).
% ATP.lambda_265
tff(fact_5123_ATP_Olambda__266,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_of(nat,fun(array(A),fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),array_length(A,Uub,Uua))) ) ) ).
% ATP.lambda_266
tff(fact_5124_ATP_Olambda__267,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_od(array(A),fun(nat,fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),array_length(A,Uub,Uu))) ) ) ).
% ATP.lambda_267
tff(fact_5125_ATP_Olambda__268,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_eo(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_268
tff(fact_5126_ATP_Olambda__269,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_no(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_269
tff(fact_5127_ATP_Olambda__270,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_hx(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_270
tff(fact_5128_ATP_Olambda__271,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_gn(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_271
tff(fact_5129_ATP_Olambda__272,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_mb(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_272
tff(fact_5130_ATP_Olambda__273,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_mr(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_273
tff(fact_5131_ATP_Olambda__274,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_cy(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_274
tff(fact_5132_ATP_Olambda__275,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_fd(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_275
tff(fact_5133_ATP_Olambda__276,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_ib(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_276
tff(fact_5134_ATP_Olambda__277,axiom,
! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_lr(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_277
tff(fact_5135_ATP_Olambda__278,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_ee(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_278
tff(fact_5136_ATP_Olambda__279,axiom,
! [F10: $tType,Uu: assn,Uua: fun(F10,assn),Uub: F10] : aa(F10,assn,aa(fun(F10,assn),fun(F10,assn),aTP_Lamp_ls(assn,fun(fun(F10,assn),fun(F10,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Uu),aa(F10,assn,Uua,Uub)) ).
% ATP.lambda_279
tff(fact_5137_ATP_Olambda__280,axiom,
! [D: $tType,Uu: assn,Uua: fun(D,assn),Uub: D] : aa(D,assn,aa(fun(D,assn),fun(D,assn),aTP_Lamp_lu(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_280
tff(fact_5138_ATP_Olambda__281,axiom,
! [I6: $tType,J5: $tType,Uu: set(I6),Uua: fun(J5,set(I6)),Uub: J5] : aa(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_bt(set(I6),fun(fun(J5,set(I6)),fun(J5,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),Uu),aa(J5,set(I6),Uua,Uub)) ).
% ATP.lambda_281
tff(fact_5139_ATP_Olambda__282,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_hp(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_282
tff(fact_5140_ATP_Olambda__283,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_br(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_283
tff(fact_5141_ATP_Olambda__284,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_bj(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_284
tff(fact_5142_ATP_Olambda__285,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_hw(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_285
tff(fact_5143_ATP_Olambda__286,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_fb(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_286
tff(fact_5144_ATP_Olambda__287,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_gw(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_287
tff(fact_5145_ATP_Olambda__288,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_eh(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).
% ATP.lambda_288
tff(fact_5146_ATP_Olambda__289,axiom,
! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_nr(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_289
tff(fact_5147_ATP_Olambda__290,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dh(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),minus_minus(nat,Uua,Uub))) ) ).
% ATP.lambda_290
tff(fact_5148_ATP_Olambda__291,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_ep(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_291
tff(fact_5149_ATP_Olambda__292,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_mp(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = image2(B,A,Uua,field2(B,Uu)) ).
% ATP.lambda_292
tff(fact_5150_ATP_Olambda__293,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_jx(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_293
tff(fact_5151_ATP_Olambda__294,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_mq(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).
% ATP.lambda_294
tff(fact_5152_ATP_Olambda__295,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_ml(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).
% ATP.lambda_295
tff(fact_5153_ATP_Olambda__296,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_gu(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_296
tff(fact_5154_ATP_Olambda__297,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ox(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,Uu,Uua,Uub)),one_one(nat))) ) ).
% ATP.lambda_297
tff(fact_5155_ATP_Olambda__298,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_kj(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_298
tff(fact_5156_ATP_Olambda__299,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_ki(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_299
tff(fact_5157_ATP_Olambda__300,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_mk(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).
% ATP.lambda_300
tff(fact_5158_ATP_Olambda__301,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_fg(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_301
tff(fact_5159_ATP_Olambda__302,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_kv(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).
% ATP.lambda_302
tff(fact_5160_ATP_Olambda__303,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_pe(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu) ).
% ATP.lambda_303
tff(fact_5161_ATP_Olambda__304,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_fc(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ~ pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_304
tff(fact_5162_ATP_Olambda__305,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_le(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_305
tff(fact_5163_ATP_Olambda__306,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_ir(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [I4: nat] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).
% ATP.lambda_306
tff(fact_5164_ATP_Olambda__307,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_jb(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
<=> ? [I4: nat] :
( ( Uub = aa(nat,A,nth(A,Uu),I4) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)))
& pp(member2(nat,I4,Uua)) ) ) ).
% ATP.lambda_307
tff(fact_5165_ATP_Olambda__308,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [Uu: A,Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_iv(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [B4: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B4) )
& pp(member2(A,B4,Uua)) ) ) ) ).
% ATP.lambda_308
tff(fact_5166_ATP_Olambda__309,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mi(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [A5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A5) )
& pp(member2(A,A5,Uu)) ) ) ) ).
% ATP.lambda_309
tff(fact_5167_ATP_Olambda__310,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_ob(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_310
tff(fact_5168_ATP_Olambda__311,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_ix(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [A5: A,V3: 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),A5),V3)) )
| ? [U3: list(A),Aa3: A,B4: 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),B4),Uu))
& ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U3),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),U3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),W4)) ) ) ) ) ).
% ATP.lambda_311
tff(fact_5169_ATP_Olambda__312,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_mj(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [A5: A,B4: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B4) )
& pp(member2(A,A5,Uu))
& pp(member2(A,B4,Uua)) ) ) ) ).
% ATP.lambda_312
tff(fact_5170_ATP_Olambda__313,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_it(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [X4: A,Xs4: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& pp(member2(A,X4,Uu))
& pp(member2(list(A),Xs4,Uua)) ) ) ).
% ATP.lambda_313
tff(fact_5171_ATP_Olambda__314,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_iw(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [Us2: list(A),Z4: A,Z7: A,Vs2: list(A)] :
( ( Uua = 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),Z4),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),Z4),Z7),Uu))
& ( Uub = 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),Z7),Vs2)) ) ) ) ).
% ATP.lambda_314
tff(fact_5172_ATP_Olambda__315,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,minus_minus(nat,Uuc,aa(nat,nat,suc,zero_zero(nat)))))) ) ).
% ATP.lambda_315
tff(fact_5173_ATP_Olambda__316,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_dk(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_316
tff(fact_5174_ATP_Olambda__317,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_dl(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_317
tff(fact_5175_ATP_Olambda__318,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fv(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_318
tff(fact_5176_ATP_Olambda__319,axiom,
! [A: $tType,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_pm(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_319
tff(fact_5177_ATP_Olambda__320,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_cl(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_320
tff(fact_5178_ATP_Olambda__321,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_cx(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_321
tff(fact_5179_ATP_Olambda__322,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_pn(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_322
tff(fact_5180_ATP_Olambda__323,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_dg(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_323
tff(fact_5181_ATP_Olambda__324,axiom,
! [A: $tType,Uu: fun(code_natural,A),Uua: code_natural,Uub: code_natural,Uuc: product_unit] : aa(product_unit,seq(A),aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_pw(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Uu),Uua),Uub),Uuc) = if(seq(A),aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Uub),Uua),empty(A),insert2(A,aa(code_natural,A,Uu,Uua),iterate_upto(A,Uu,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),one_one(code_natural)),Uub))) ).
% ATP.lambda_324
tff(fact_5182_ATP_Olambda__325,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_ln(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_325
tff(fact_5183_ATP_Olambda__326,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_cn(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_326
tff(fact_5184_ATP_Olambda__327,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_eu(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_327
tff(fact_5185_ATP_Olambda__328,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_am(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_328
tff(fact_5186_ATP_Olambda__329,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_pi(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_329
tff(fact_5187_ATP_Olambda__330,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_jh(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_330
tff(fact_5188_ATP_Olambda__331,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_mg(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_mf(fun(A,fun(A,bool)),fun(A,fun(bool,fun(A,fun(list(A),bool)))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_331
tff(fact_5189_ATP_Olambda__332,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_la(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_332
tff(fact_5190_ATP_Olambda__333,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_ax(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_333
tff(fact_5191_ATP_Olambda__334,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_gy(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_334
tff(fact_5192_ATP_Olambda__335,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_kk(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_335
tff(fact_5193_ATP_Olambda__336,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_kh(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_336
tff(fact_5194_ATP_Olambda__337,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_kt(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_337
tff(fact_5195_ATP_Olambda__338,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_ec(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_eb(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_338
tff(fact_5196_ATP_Olambda__339,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_if(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_ie(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua)) ) ).
% ATP.lambda_339
tff(fact_5197_ATP_Olambda__340,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_ao(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_340
tff(fact_5198_ATP_Olambda__341,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_fr(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,minus_minus(nat,Uub,Uuc))) ) ).
% ATP.lambda_341
tff(fact_5199_ATP_Olambda__342,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_fq(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),minus_minus(nat,Uub,Uuc))) ) ).
% ATP.lambda_342
tff(fact_5200_ATP_Olambda__343,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_jc(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,Xs6: list(A),Ys6: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
& ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys6)) )
& 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_343
tff(fact_5201_ATP_Olambda__344,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_fl(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_344
tff(fact_5202_ATP_Olambda__345,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_fh(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_345
tff(fact_5203_ATP_Olambda__346,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_ll(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_346
tff(fact_5204_ATP_Olambda__347,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_fk(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_347
tff(fact_5205_ATP_Olambda__348,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_ic(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_348
tff(fact_5206_ATP_Olambda__349,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_hg(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_349
tff(fact_5207_ATP_Olambda__350,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_hh(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_350
tff(fact_5208_ATP_Olambda__351,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_bf(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_351
tff(fact_5209_ATP_Olambda__352,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_nj(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_352
tff(fact_5210_ATP_Olambda__353,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_fs(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_353
tff(fact_5211_ATP_Olambda__354,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_cs(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_354
tff(fact_5212_ATP_Olambda__355,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_na(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_355
tff(fact_5213_ATP_Olambda__356,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_eb(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_356
tff(fact_5214_ATP_Olambda__357,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_bp(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_357
tff(fact_5215_ATP_Olambda__358,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_bk(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_358
tff(fact_5216_ATP_Olambda__359,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_gx(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_359
tff(fact_5217_ATP_Olambda__360,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_el(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_360
tff(fact_5218_ATP_Olambda__361,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_ie(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_id(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_361
tff(fact_5219_ATP_Olambda__362,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_fi(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_fh(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_362
tff(fact_5220_ATP_Olambda__363,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_ko(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,Ys4: list(A)] :
( ( Uub = nil(A) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) ) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y3)) )
| ? [X4: A,Y3: A,Xs4: list(A),Ys4: list(A)] :
( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys4) )
& ~ 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,Xs4),Ys4)) ) ) ) ).
% ATP.lambda_363
tff(fact_5221_ATP_Olambda__364,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_ap(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_364
tff(fact_5222_ATP_Olambda__365,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_md(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_365
tff(fact_5223_ATP_Olambda__366,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_og(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),one_one(nat))) ) ).
% ATP.lambda_366
tff(fact_5224_ATP_Olambda__367,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_lg(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = 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)),insert3(A),Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_367
tff(fact_5225_ATP_Olambda__368,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_os(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_368
tff(fact_5226_ATP_Olambda__369,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_hf(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_369
tff(fact_5227_ATP_Olambda__370,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_gm(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_370
tff(fact_5228_ATP_Olambda__371,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_hc(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_371
tff(fact_5229_ATP_Olambda__372,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_hd(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_372
tff(fact_5230_ATP_Olambda__373,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_nb(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_na(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_373
tff(fact_5231_ATP_Olambda__374,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_bq(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = complete_Sup_Sup(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_bp(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc),Uub)) ).
% ATP.lambda_374
tff(fact_5232_ATP_Olambda__375,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_bl(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = complete_Sup_Sup(A,image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_bk(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_375
tff(fact_5233_ATP_Olambda__376,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_oh(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_376
tff(fact_5234_ATP_Olambda__377,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_om(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_377
tff(fact_5235_ATP_Olambda__378,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_po(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_378
tff(fact_5236_ATP_Olambda__379,axiom,
! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_pl(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),Uua),aa(nat,A,Uub,Uud),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uud),Uua),Uu,aa(nat,A,Uuc,minus_minus(nat,Uud,aa(nat,nat,suc,zero_zero(nat)))))) ).
% ATP.lambda_379
tff(fact_5237_ATP_Olambda__380,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_mo(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_380
tff(fact_5238_ATP_Olambda__381,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_mf(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_381
tff(fact_5239_ATP_Olambda__382,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_ga(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_382
tff(fact_5240_ATP_Olambda__383,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_fx(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_383
tff(fact_5241_ATP_Olambda__384,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_fy(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_384
tff(fact_5242_ATP_Olambda__385,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_nl(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uub,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_385
tff(fact_5243_ATP_Olambda__386,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_kb(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ? [A5: A,B4: C] :
( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A5)),aa(C,B,Uua,B4)) )
& pp(member2(A,A5,Uub))
& pp(member2(C,B4,Uuc)) ) ) ) ).
% ATP.lambda_386
tff(fact_5244_ATP_Olambda__387,axiom,
! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_pd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_387
tff(fact_5245_ATP_Olambda__388,axiom,
! [B: $tType,A: $tType] :
( heap(B)
=> ! [Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_pf(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ) ).
% ATP.lambda_388
tff(fact_5246_ATP_Olambda__389,axiom,
! [Uu: assn,Uua: product_unit] : aa(product_unit,assn,aTP_Lamp_oy(assn,fun(product_unit,assn),Uu),Uua) = Uu ).
% ATP.lambda_389
tff(fact_5247_ATP_Olambda__390,axiom,
! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_lp(assn,fun(A,assn),Uu),Uua) = Uu ).
% ATP.lambda_390
tff(fact_5248_ATP_Olambda__391,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_lk(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_391
tff(fact_5249_ATP_Olambda__392,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_ka(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_392
tff(fact_5250_ATP_Olambda__393,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_jw(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_393
tff(fact_5251_ATP_Olambda__394,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_jr(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).
% ATP.lambda_394
tff(fact_5252_ATP_Olambda__395,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_jt(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_395
tff(fact_5253_ATP_Olambda__396,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_hz(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).
% ATP.lambda_396
tff(fact_5254_ATP_Olambda__397,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ih(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_397
tff(fact_5255_ATP_Olambda__398,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_ky(B,fun(C,B),Uu),Uua) = Uu ) ).
% ATP.lambda_398
tff(fact_5256_ATP_Olambda__399,axiom,
! [A: $tType,B: $tType] :
( semiring_1(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_mx(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_399
tff(fact_5257_ATP_Olambda__400,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_ij(A,fun(list(A),A)),Uu),Uua) = Uu ) ).
% ATP.lambda_400
tff(fact_5258_ATP_Olambda__401,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_dv(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_401
tff(fact_5259_ATP_Olambda__402,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_jn(A,fun(list(A),A)),Uu),Uua) = Uu ).
% ATP.lambda_402
tff(fact_5260_ATP_Olambda__403,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jo(A,fun(nat,A),Uu),Uua) = Uu ).
% ATP.lambda_403
tff(fact_5261_ATP_Olambda__404,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_jm(A,fun(B,A),Uu),Uua) = Uu ).
% ATP.lambda_404
tff(fact_5262_ATP_Olambda__405,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_gt(A,fun(list(A),list(A))),Uu),Uua) = Uua ).
% ATP.lambda_405
tff(fact_5263_ATP_Olambda__406,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_hs(A,fun(list(A),bool)),Uu),Uua))
<=> $false ) ).
% ATP.lambda_406
tff(fact_5264_ATP_Olambda__407,axiom,
! [A: $tType,Uu: A,Uua: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_hr(A,fun(list(A),bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_407
tff(fact_5265_ATP_Olambda__408,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_au(A,fun(A,bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_408
tff(fact_5266_ATP_Olambda__409,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_ac(nat,nat),Uu) = Uu ).
% ATP.lambda_409
tff(fact_5267_ATP_Olambda__410,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_hk(int,int),Uu) = Uu ).
% ATP.lambda_410
tff(fact_5268_ATP_Olambda__411,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_az(A,A),Uu) = Uu ) ).
% ATP.lambda_411
tff(fact_5269_ATP_Olambda__412,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_aw(A,A),Uu) = Uu ) ).
% ATP.lambda_412
tff(fact_5270_ATP_Olambda__413,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu) = Uu ).
% ATP.lambda_413
tff(fact_5271_ATP_Olambda__414,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_av(A,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_414
tff(fact_5272_ATP_Olambda__415,axiom,
! [Uu: product_unit] : aa(product_unit,assn,aTP_Lamp_nv(product_unit,assn),Uu) = one_one(assn) ).
% ATP.lambda_415
tff(fact_5273_ATP_Olambda__416,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_ai(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_416
tff(fact_5274_ATP_Olambda__417,axiom,
! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_gp(A,nat),Uu) = one_one(nat) ).
% ATP.lambda_417
tff(fact_5275_ATP_Olambda__418,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_oz(A,option(B)),Uu) = none(B) ).
% ATP.lambda_418
tff(fact_5276_ATP_Olambda__419,axiom,
! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_ah(product_prod(heap_ext(product_unit),set(nat)),bool),Uu))
<=> $false ) ).
% ATP.lambda_419
tff(fact_5277_ATP_Olambda__420,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_li(A,bool),Uu))
<=> $true ) ).
% ATP.lambda_420
% Type constructors (603)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_1,axiom,
bounded_lattice(assn) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_2,axiom,
! [A10: $tType] : bounded_lattice(pred(A10)) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice_3,axiom,
! [A10: $tType] :
( bounded_lattice_top(A10)
=> bounded_lattice(option(A10)) ) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_4,axiom,
bounded_lattice(bool) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_5,axiom,
! [A10: $tType] : bounded_lattice(set(A10)) ).
tff(tcon_fun___Lattices_Obounded__lattice_6,axiom,
! [A10: $tType,A11: $tType] :
( bounded_lattice(A11)
=> bounded_lattice(fun(A10,A11)) ) ).
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A10: $tType,A11: $tType] :
( comple6319245703460814977attice(A11)
=> condit1219197933456340205attice(fun(A10,A11)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A10: $tType,A11: $tType] :
( comple592849572758109894attice(A11)
=> comple592849572758109894attice(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A10: $tType,A11: $tType] :
( bounded_lattice(A11)
=> bounde4967611905675639751up_bot(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A10: $tType,A11: $tType] :
( bounded_lattice(A11)
=> bounde4346867609351753570nf_top(fun(A10,A11)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A10: $tType,A11: $tType] :
( comple6319245703460814977attice(A11)
=> comple6319245703460814977attice(fun(A10,A11)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A10: $tType,A11: $tType] :
( boolea8198339166811842893lgebra(A11)
=> boolea8198339166811842893lgebra(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A10: $tType,A11: $tType] :
( bounded_lattice(A11)
=> bounded_lattice_top(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A10: $tType,A11: $tType] :
( bounded_lattice(A11)
=> bounded_lattice_bot(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A10: $tType,A11: $tType] :
( semilattice_sup(A11)
=> semilattice_sup(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A10: $tType,A11: $tType] :
( semilattice_inf(A11)
=> semilattice_inf(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A10: $tType,A11: $tType] :
( distrib_lattice(A11)
=> distrib_lattice(fun(A10,A11)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A11: $tType] :
( preorder(A11)
=> preorder(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Olattice,axiom,
! [A10: $tType,A11: $tType] :
( lattice(A11)
=> lattice(fun(A10,A11)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A11: $tType] :
( order(A11)
=> order(fun(A10,A11)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A11: $tType] :
( ord(A11)
=> ord(fun(A10,A11)) ) ).
tff(tcon_fun___Groups_Ouminus,axiom,
! [A10: $tType,A11: $tType] :
( uminus(A11)
=> uminus(fun(A10,A11)) ) ).
tff(tcon_fun___Lattices_Oinf,axiom,
! [A10: $tType,A11: $tType] :
( semilattice_inf(A11)
=> inf(fun(A10,A11)) ) ).
tff(tcon_fun___Groups_Ominus,axiom,
! [A10: $tType,A11: $tType] :
( minus(A11)
=> minus(fun(A10,A11)) ) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_7,axiom,
condit1219197933456340205attice(int) ).
tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations(int) ).
tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel(int) ).
tff(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative(int) ).
tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel(int) ).
tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le(int) ).
tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict(int) ).
tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict(int) ).
tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize(int) ).
tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add(int) ).
tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel(int) ).
tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits(int) ).
tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring(int) ).
tff(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring(int) ).
tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_8,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_9,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Lattices_Odistrib__lattice_10,axiom,
distrib_lattice(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(int) ).
tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(int) ).
tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide(int) ).
tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral(int) ).
tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn(int) ).
tff(tcon_Int_Oint___Orderings_Opreorder_11,axiom,
preorder(int) ).
tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
tff(tcon_Int_Oint___Lattices_Olattice_12,axiom,
lattice(int) ).
tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd(int) ).
tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
tff(tcon_Int_Oint___Orderings_Oorder_13,axiom,
order(int) ).
tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral(int) ).
tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0(int) ).
tff(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(tcon_Int_Oint___Orderings_Oord_14,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_15,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Lattices_Oinf_16,axiom,
inf(int) ).
tff(tcon_Int_Oint___Groups_Otimes,axiom,
times(int) ).
tff(tcon_Int_Oint___Groups_Ominus_17,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__lattice_18,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_19,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_20,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_21,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_22,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_23,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_24,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_25,axiom,
normal6328177297339901930cative(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_26,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_27,axiom,
semiri6575147826004484403cancel(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_28,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_29,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_30,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_31,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_32,axiom,
linord2810124833399127020strict(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_33,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_34,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_35,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_37,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
linord8928482502909563296strict(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_41,axiom,
semiri6843258321239162965malize(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
ordere2520102378445227354miring(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_48,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_49,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring_50,axiom,
linordered_semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_51,axiom,
ordered_semiring_0(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_52,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_53,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_54,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_55,axiom,
distrib_lattice(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_56,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_57,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_58,axiom,
comm_monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_59,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring_60,axiom,
ordered_semiring(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_61,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_62,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_63,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_64,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_65,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_73,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_74,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_75,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_76,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_77,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_78,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_79,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_80,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_81,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_82,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_83,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Rings_Omult__zero_84,axiom,
mult_zero(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_85,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_86,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_87,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Lattices_Oinf_88,axiom,
inf(nat) ).
tff(tcon_Nat_Onat___Groups_Otimes_89,axiom,
times(nat) ).
tff(tcon_Nat_Onat___Groups_Ominus_90,axiom,
minus(nat) ).
tff(tcon_Nat_Onat___Power_Opower_91,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_92,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_93,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_94,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_95,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_96,axiom,
heap(nat) ).
tff(tcon_Num_Onum___Orderings_Opreorder_97,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_98,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_99,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_100,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Otimes_101,axiom,
times(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_102,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_103,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_104,axiom,
semiri6575147826004484403cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_105,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_106,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_107,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_108,axiom,
linord2810124833399127020strict(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_109,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_110,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_111,axiom,
linord715952674999750819strict(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___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___Lattices_Olattice_162,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_163,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Rings_Omult__zero_164,axiom,
mult_zero(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_165,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_166,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_167,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_168,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_169,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_170,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_171,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_172,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Lattices_Oinf_173,axiom,
inf(rat) ).
tff(tcon_Rat_Orat___Groups_Otimes_174,axiom,
times(rat) ).
tff(tcon_Rat_Orat___Groups_Ominus_175,axiom,
minus(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Rat_Orat___Power_Opower_176,axiom,
power(rat) ).
tff(tcon_Rat_Orat___Num_Onumeral_177,axiom,
numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Ozero_178,axiom,
zero(rat) ).
tff(tcon_Rat_Orat___Rings_Oring_179,axiom,
ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom_180,axiom,
idom(rat) ).
tff(tcon_Rat_Orat___Groups_Oone_181,axiom,
one(rat) ).
tff(tcon_Rat_Orat___Rings_Odvd_182,axiom,
dvd(rat) ).
tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_183,axiom,
! [A10: $tType] : condit1219197933456340205attice(set(A10)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_184,axiom,
! [A10: $tType] : comple592849572758109894attice(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_185,axiom,
! [A10: $tType] : bounde4967611905675639751up_bot(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_186,axiom,
! [A10: $tType] : bounde4346867609351753570nf_top(set(A10)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_187,axiom,
! [A10: $tType] : comple6319245703460814977attice(set(A10)) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_188,axiom,
! [A10: $tType] : boolea8198339166811842893lgebra(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_189,axiom,
! [A10: $tType] : bounded_lattice_top(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_190,axiom,
! [A10: $tType] : bounded_lattice_bot(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_191,axiom,
! [A10: $tType] : semilattice_sup(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_192,axiom,
! [A10: $tType] : semilattice_inf(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Odistrib__lattice_193,axiom,
! [A10: $tType] : distrib_lattice(set(A10)) ).
tff(tcon_Set_Oset___Orderings_Opreorder_194,axiom,
! [A10: $tType] : preorder(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Olattice_195,axiom,
! [A10: $tType] : lattice(set(A10)) ).
tff(tcon_Set_Oset___Orderings_Oorder_196,axiom,
! [A10: $tType] : order(set(A10)) ).
tff(tcon_Set_Oset___Orderings_Oord_197,axiom,
! [A10: $tType] : ord(set(A10)) ).
tff(tcon_Set_Oset___Groups_Ouminus_198,axiom,
! [A10: $tType] : uminus(set(A10)) ).
tff(tcon_Set_Oset___Lattices_Oinf_199,axiom,
! [A10: $tType] : inf(set(A10)) ).
tff(tcon_Set_Oset___Groups_Ominus_200,axiom,
! [A10: $tType] : minus(set(A10)) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_201,axiom,
condit1219197933456340205attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_202,axiom,
comple592849572758109894attice(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_203,axiom,
bounde4967611905675639751up_bot(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_204,axiom,
bounde4346867609351753570nf_top(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_205,axiom,
comple6319245703460814977attice(bool) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_206,axiom,
boolea8198339166811842893lgebra(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_207,axiom,
bounded_lattice_top(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_208,axiom,
bounded_lattice_bot(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_209,axiom,
semilattice_sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_210,axiom,
semilattice_inf(bool) ).
tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_211,axiom,
distrib_lattice(bool) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_212,axiom,
preorder(bool) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_213,axiom,
linorder(bool) ).
tff(tcon_HOL_Obool___Lattices_Olattice_214,axiom,
lattice(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder_215,axiom,
order(bool) ).
tff(tcon_HOL_Obool___Orderings_Oord_216,axiom,
ord(bool) ).
tff(tcon_HOL_Obool___Groups_Ouminus_217,axiom,
uminus(bool) ).
tff(tcon_HOL_Obool___Lattices_Oinf_218,axiom,
inf(bool) ).
tff(tcon_HOL_Obool___Groups_Ominus_219,axiom,
minus(bool) ).
tff(tcon_HOL_Obool___Heap_Oheap_220,axiom,
heap(bool) ).
tff(tcon_Heap_Oref___Heap_Oheap_221,axiom,
! [A10: $tType] : heap(ref(A10)) ).
tff(tcon_List_Olist___Heap_Oheap_222,axiom,
! [A10: $tType] :
( heap(A10)
=> heap(list(A10)) ) ).
tff(tcon_Heap_Oarray___Heap_Oheap_223,axiom,
! [A10: $tType] : heap(array(A10)) ).
tff(tcon_String_Ochar___Heap_Oheap_224,axiom,
heap(char) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_225,axiom,
! [A10: $tType] :
( comple6319245703460814977attice(A10)
=> condit1219197933456340205attice(option(A10)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_226,axiom,
! [A10: $tType] :
( comple592849572758109894attice(A10)
=> comple592849572758109894attice(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_227,axiom,
! [A10: $tType] :
( lattice(A10)
=> bounde4967611905675639751up_bot(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_228,axiom,
! [A10: $tType] :
( bounded_lattice_top(A10)
=> bounde4346867609351753570nf_top(option(A10)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A10: $tType] :
( comple5582772986160207858norder(A10)
=> comple5582772986160207858norder(option(A10)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_229,axiom,
! [A10: $tType] :
( comple6319245703460814977attice(A10)
=> comple6319245703460814977attice(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_230,axiom,
! [A10: $tType] :
( bounded_lattice_top(A10)
=> bounded_lattice_top(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_231,axiom,
! [A10: $tType] :
( lattice(A10)
=> bounded_lattice_bot(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_232,axiom,
! [A10: $tType] :
( semilattice_sup(A10)
=> semilattice_sup(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_233,axiom,
! [A10: $tType] :
( semilattice_inf(A10)
=> semilattice_inf(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_234,axiom,
! [A10: $tType] :
( distrib_lattice(A10)
=> distrib_lattice(option(A10)) ) ).
tff(tcon_Option_Ooption___Orderings_Owellorder_235,axiom,
! [A10: $tType] :
( wellorder(A10)
=> wellorder(option(A10)) ) ).
tff(tcon_Option_Ooption___Orderings_Opreorder_236,axiom,
! [A10: $tType] :
( preorder(A10)
=> preorder(option(A10)) ) ).
tff(tcon_Option_Ooption___Orderings_Olinorder_237,axiom,
! [A10: $tType] :
( linorder(A10)
=> linorder(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Olattice_238,axiom,
! [A10: $tType] :
( lattice(A10)
=> lattice(option(A10)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder_239,axiom,
! [A10: $tType] :
( order(A10)
=> order(option(A10)) ) ).
tff(tcon_Option_Ooption___Orderings_Oord_240,axiom,
! [A10: $tType] :
( preorder(A10)
=> ord(option(A10)) ) ).
tff(tcon_Option_Ooption___Lattices_Oinf_241,axiom,
! [A10: $tType] :
( inf(A10)
=> inf(option(A10)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_242,axiom,
! [A10: $tType] :
( heap(A10)
=> heap(option(A10)) ) ).
tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_243,axiom,
! [A10: $tType] : condit1219197933456340205attice(pred(A10)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_244,axiom,
! [A10: $tType] : comple592849572758109894attice(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_245,axiom,
! [A10: $tType] : bounde4967611905675639751up_bot(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_246,axiom,
! [A10: $tType] : bounde4346867609351753570nf_top(pred(A10)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_247,axiom,
! [A10: $tType] : comple6319245703460814977attice(pred(A10)) ).
tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_248,axiom,
! [A10: $tType] : boolea8198339166811842893lgebra(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_249,axiom,
! [A10: $tType] : bounded_lattice_top(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_250,axiom,
! [A10: $tType] : bounded_lattice_bot(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_251,axiom,
! [A10: $tType] : semilattice_sup(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_252,axiom,
! [A10: $tType] : semilattice_inf(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Odistrib__lattice_253,axiom,
! [A10: $tType] : distrib_lattice(pred(A10)) ).
tff(tcon_Predicate_Opred___Orderings_Opreorder_254,axiom,
! [A10: $tType] : preorder(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Olattice_255,axiom,
! [A10: $tType] : lattice(pred(A10)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder_256,axiom,
! [A10: $tType] : order(pred(A10)) ).
tff(tcon_Predicate_Opred___Orderings_Oord_257,axiom,
! [A10: $tType] : ord(pred(A10)) ).
tff(tcon_Predicate_Opred___Groups_Ouminus_258,axiom,
! [A10: $tType] : uminus(pred(A10)) ).
tff(tcon_Predicate_Opred___Lattices_Oinf_259,axiom,
! [A10: $tType] : inf(pred(A10)) ).
tff(tcon_Predicate_Opred___Groups_Ominus_260,axiom,
! [A10: $tType] : minus(pred(A10)) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_261,axiom,
bounde4967611905675639751up_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_262,axiom,
bounde4346867609351753570nf_top(assn) ).
tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_263,axiom,
boolea8198339166811842893lgebra(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_264,axiom,
bounded_lattice_top(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_265,axiom,
bounded_lattice_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_266,axiom,
semilattice_sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_267,axiom,
semilattice_inf(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_268,axiom,
distrib_lattice(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_269,axiom,
ab_semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_270,axiom,
comm_monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_271,axiom,
semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Opreorder_272,axiom,
preorder(assn) ).
tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_273,axiom,
monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Olattice_274,axiom,
lattice(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder_275,axiom,
order(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oord_276,axiom,
ord(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ouminus_277,axiom,
uminus(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Oinf_278,axiom,
inf(assn) ).
tff(tcon_Assertions_Oassn___Groups_Otimes_279,axiom,
times(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ominus_280,axiom,
minus(assn) ).
tff(tcon_Assertions_Oassn___Power_Opower_281,axiom,
power(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oone_282,axiom,
one(assn) ).
tff(tcon_Assertions_Oassn___Rings_Odvd_283,axiom,
dvd(assn) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_284,axiom,
heap(typerep) ).
tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_285,axiom,
! [A10: $tType] :
( preorder(A10)
=> ordere6658533253407199908up_add(multiset(A10)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_286,axiom,
! [A10: $tType] : cancel2418104881723323429up_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_287,axiom,
! [A10: $tType] : cancel1802427076303600483id_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_288,axiom,
! [A10: $tType] : cancel_semigroup_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_289,axiom,
! [A10: $tType] : comm_monoid_diff(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_290,axiom,
! [A10: $tType] : ab_semigroup_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_291,axiom,
! [A10: $tType] : comm_monoid_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_292,axiom,
! [A10: $tType] : semigroup_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Orderings_Opreorder_293,axiom,
! [A10: $tType] :
( preorder(A10)
=> preorder(multiset(A10)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_294,axiom,
! [A10: $tType] : monoid_add(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Orderings_Oorder_295,axiom,
! [A10: $tType] :
( preorder(A10)
=> order(multiset(A10)) ) ).
tff(tcon_Multiset_Omultiset___Orderings_Oord_296,axiom,
! [A10: $tType] :
( preorder(A10)
=> ord(multiset(A10)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ominus_297,axiom,
! [A10: $tType] : minus(multiset(A10)) ).
tff(tcon_Multiset_Omultiset___Groups_Ozero_298,axiom,
! [A10: $tType] : zero(multiset(A10)) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_299,axiom,
! [A10: $tType,A11: $tType] :
( ( heap(A10)
& heap(A11) )
=> heap(product_prod(A10,A11)) ) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_300,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_301,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_302,axiom,
bounde4967611905675639751up_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_303,axiom,
bounde4346867609351753570nf_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_304,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_305,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_306,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_307,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_308,axiom,
bounded_lattice_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_309,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_310,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_311,axiom,
distrib_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_312,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_313,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_314,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_315,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_316,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_317,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_318,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Oinf_319,axiom,
inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ominus_320,axiom,
minus(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_321,axiom,
heap(product_unit) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_322,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_323,axiom,
semiri1453513574482234551roduct(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_324,axiom,
euclid5411537665997757685th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_325,axiom,
euclid8789492081693882211th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_326,axiom,
ordere1937475149494474687imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_327,axiom,
euclid3128863361964157862miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_328,axiom,
euclid4440199948858584721cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_329,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_330,axiom,
euclid8851590272496341667cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_331,axiom,
semiri6575147826004484403cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_332,axiom,
strict9044650504122735259up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_333,axiom,
ordere580206878836729694up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_334,axiom,
ordere2412721322843649153imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_335,axiom,
bit_se359711467146920520ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_336,axiom,
linord2810124833399127020strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_337,axiom,
strict7427464778891057005id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_338,axiom,
ordere8940638589300402666id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_339,axiom,
euclid3725896446679973847miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_340,axiom,
linord715952674999750819strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_341,axiom,
bit_ri3973907225187159222ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_342,axiom,
linord181362715937106298miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_343,axiom,
linord8928482502909563296strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_344,axiom,
semiri3467727345109120633visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_345,axiom,
ordere6658533253407199908up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_346,axiom,
ordere166539214618696060dd_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_347,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_348,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_349,axiom,
cancel2418104881723323429up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_350,axiom,
ring_15535105094025558882visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_351,axiom,
cancel1802427076303600483id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_352,axiom,
linord4710134922213307826strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_353,axiom,
comm_s4317794764714335236cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_354,axiom,
bit_semiring_bits(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_355,axiom,
ordere2520102378445227354miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_356,axiom,
linord6961819062388156250ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_357,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_358,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_359,axiom,
linordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_360,axiom,
ordered_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_361,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_362,axiom,
ab_semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_363,axiom,
algebraic_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_364,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_365,axiom,
ab_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_366,axiom,
ordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_367,axiom,
ordered_ring_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_368,axiom,
semiring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_369,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_370,axiom,
semiring_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_371,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_372,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_373,axiom,
comm_semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_374,axiom,
comm_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_375,axiom,
semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_376,axiom,
semidom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_377,axiom,
semidom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_378,axiom,
semiring_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_379,axiom,
semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_380,axiom,
zero_less_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_381,axiom,
comm_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_382,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_383,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_384,axiom,
zero_neq_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_385,axiom,
ordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_386,axiom,
idom_abs_sgn(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_387,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_388,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_389,axiom,
monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_390,axiom,
comm_ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_391,axiom,
monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_392,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_393,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_394,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_395,axiom,
mult_zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_396,axiom,
comm_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_397,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_398,axiom,
neg_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_399,axiom,
ring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_400,axiom,
semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_401,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_402,axiom,
uminus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_403,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_404,axiom,
times(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_405,axiom,
minus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Power_Opower_406,axiom,
power(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_407,axiom,
numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_408,axiom,
zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring_409,axiom,
ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_410,axiom,
idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oone_411,axiom,
one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_412,axiom,
dvd(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_413,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_414,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_415,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_416,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_417,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_418,axiom,
semiri6575147826004484403cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_419,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_420,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_421,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_422,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_423,axiom,
linord2810124833399127020strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_424,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_425,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_426,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_427,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_428,axiom,
linord8928482502909563296strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_429,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_430,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_431,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_432,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_433,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_434,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_435,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_436,axiom,
ordere2520102378445227354miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_437,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_438,axiom,
linordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_439,axiom,
ordered_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_440,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_441,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_442,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_443,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_444,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_445,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_446,axiom,
ordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_447,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_448,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_449,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_450,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_451,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_452,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_453,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_454,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_455,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_456,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_457,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_458,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_459,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_460,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_461,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_462,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_463,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_464,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_465,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_466,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_467,axiom,
mult_zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_468,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_469,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_470,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Otimes_471,axiom,
times(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ominus_472,axiom,
minus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_473,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_474,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_475,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_476,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_477,axiom,
dvd(code_natural) ).
% Helper facts (15)
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 ) ) ).
% Conjectures (1)
tff(conj_0,conjecture,
fI_QUERY(p,q,f) ).
%------------------------------------------------------------------------------