TPTP Problem File: ITP212_4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP212_4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Assertions 00821_023885
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0024_Assertions_00821_023885 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 7654 (2430 unt;1385 typ; 0 def)
% Number of atoms : 13047 (5245 equ)
% Maximal formula atoms : 28 ( 2 avg)
% Number of connectives : 12484 (1710 ~; 181 |; 933 &)
% (1139 <=>;8521 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Maximal term depth : 37 ( 2 avg)
% Number of FOOLs : 464 ( 298 fml; 166 var)
% Number of X terms : 326 ( 0 []; 312 ite; 14 let)
% Number of types : 13 ( 12 usr)
% Number of type conns : 1419 ( 965 >; 454 *; 0 +; 0 <<)
% Number of predicates : 287 ( 284 usr; 2 prp; 0-7 aty)
% Number of functors : 1097 (1097 usr; 65 con; 0-7 aty)
% Number of variables : 23195 (20916 !; 326 ?;23195 :)
% (1953 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TX1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 15:05:32.099
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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_Code__Evaluation_Oterm,type,
code_term: $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_Old__Datatype_Onode,type,
old_node: ( $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_Sum__Type_Osum,type,
sum_sum: ( $tType * $tType ) > $tType ).
tff(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
tff(ty_t_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
tff(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
tff(ty_t_List_Olist,type,
list: $tType > $tType ).
tff(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
tff(ty_t_Set_Oset,type,
set: $tType > $tType ).
tff(ty_t_Rat_Orat,type,
rat: $tType ).
tff(ty_t_Num_Onum,type,
num: $tType ).
tff(ty_t_Nat_Onat,type,
nat: $tType ).
tff(ty_t_Int_Oint,type,
int: $tType ).
tff(ty_t_itself,type,
itself: $tType > $tType ).
tff(ty_t_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tf_b,type,
b: $tType ).
tff(ty_tf_a,type,
a: $tType ).
% Explicit typings (1357)
tff(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
tff(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[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_Osup,type,
sup:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_cl_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_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
tff(sy_c_ATP_058Lamp__a____,type,
aTP_Lamp_a:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aa____,type,
aTP_Lamp_aa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ab____,type,
aTP_Lamp_ab:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aj____,type,
aTP_Lamp_aj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ak____,type,
aTP_Lamp_ak:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__al____,type,
aTP_Lamp_al:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__am____,type,
aTP_Lamp_am:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__at____,type,
aTP_Lamp_at:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__au____,type,
aTP_Lamp_au:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__av____,type,
aTP_Lamp_av:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aw____,type,
aTP_Lamp_aw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ax____,type,
aTP_Lamp_ax:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ay____,type,
aTP_Lamp_ay:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__az____,type,
aTP_Lamp_az:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ba____,type,
aTP_Lamp_ba:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bb____,type,
aTP_Lamp_bb:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bq____,type,
aTP_Lamp_bq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__br____,type,
aTP_Lamp_br:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bs____,type,
aTP_Lamp_bs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bt____,type,
aTP_Lamp_bt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bu____,type,
aTP_Lamp_bu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bv____,type,
aTP_Lamp_bv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bw____,type,
aTP_Lamp_bw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cb____,type,
aTP_Lamp_cb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cc____,type,
aTP_Lamp_cc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cd____,type,
aTP_Lamp_cd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ce____,type,
aTP_Lamp_ce:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cf____,type,
aTP_Lamp_cf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cg____,type,
aTP_Lamp_cg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ch____,type,
aTP_Lamp_ch:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ci____,type,
aTP_Lamp_ci:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cj____,type,
aTP_Lamp_cj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ck____,type,
aTP_Lamp_ck:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cl____,type,
aTP_Lamp_cl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cm____,type,
aTP_Lamp_cm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cn____,type,
aTP_Lamp_cn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__co____,type,
aTP_Lamp_co:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cu____,type,
aTP_Lamp_cu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cv____,type,
aTP_Lamp_cv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cw____,type,
aTP_Lamp_cw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cx____,type,
aTP_Lamp_cx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cy____,type,
aTP_Lamp_cy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cz____,type,
aTP_Lamp_cz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__da____,type,
aTP_Lamp_da:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__db____,type,
aTP_Lamp_db:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ds____,type,
aTP_Lamp_ds:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dt____,type,
aTP_Lamp_dt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__du____,type,
aTP_Lamp_du:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dv____,type,
aTP_Lamp_dv:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ei____,type,
aTP_Lamp_ei:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ej____,type,
aTP_Lamp_ej:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__em____,type,
aTP_Lamp_em:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__en____,type,
aTP_Lamp_en:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__eo____,type,
aTP_Lamp_eo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__eq____,type,
aTP_Lamp_eq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__er____,type,
aTP_Lamp_er:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__es____,type,
aTP_Lamp_es:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__et____,type,
aTP_Lamp_et:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eu____,type,
aTP_Lamp_eu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ew____,type,
aTP_Lamp_ew:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ex____,type,
aTP_Lamp_ex:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ez____,type,
aTP_Lamp_ez:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fa____,type,
aTP_Lamp_fa:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fn____,type,
aTP_Lamp_fn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fo____,type,
aTP_Lamp_fo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fp____,type,
aTP_Lamp_fp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fq____,type,
aTP_Lamp_fq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fr____,type,
aTP_Lamp_fr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fs____,type,
aTP_Lamp_fs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ft____,type,
aTP_Lamp_ft:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fu____,type,
aTP_Lamp_fu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fv____,type,
aTP_Lamp_fv:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fz____,type,
aTP_Lamp_fz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ga____,type,
aTP_Lamp_ga:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gi____,type,
aTP_Lamp_gi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gj____,type,
aTP_Lamp_gj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gk____,type,
aTP_Lamp_gk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gl____,type,
aTP_Lamp_gl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gm____,type,
aTP_Lamp_gm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gn____,type,
aTP_Lamp_gn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__go____,type,
aTP_Lamp_go:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gp____,type,
aTP_Lamp_gp:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gs____,type,
aTP_Lamp_gs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gt____,type,
aTP_Lamp_gt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gu____,type,
aTP_Lamp_gu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gv____,type,
aTP_Lamp_gv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gw____,type,
aTP_Lamp_gw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gx____,type,
aTP_Lamp_gx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gy____,type,
aTP_Lamp_gy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hl____,type,
aTP_Lamp_hl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hm____,type,
aTP_Lamp_hm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ho____,type,
aTP_Lamp_ho:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hp____,type,
aTP_Lamp_hp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hq____,type,
aTP_Lamp_hq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hr____,type,
aTP_Lamp_hr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hs____,type,
aTP_Lamp_hs:
!>[A: $tType,B: $tType] : 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hy____,type,
aTP_Lamp_hy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hz____,type,
aTP_Lamp_hz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ia____,type,
aTP_Lamp_ia:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ic____,type,
aTP_Lamp_ic:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__id____,type,
aTP_Lamp_id:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ie____,type,
aTP_Lamp_ie:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__if____,type,
aTP_Lamp_if:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ig____,type,
aTP_Lamp_ig:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__il____,type,
aTP_Lamp_il:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__im____,type,
aTP_Lamp_im:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__in____,type,
aTP_Lamp_in:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__io____,type,
aTP_Lamp_io:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ip____,type,
aTP_Lamp_ip:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iq____,type,
aTP_Lamp_iq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ir____,type,
aTP_Lamp_ir:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__is____,type,
aTP_Lamp_is:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__it____,type,
aTP_Lamp_it:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iu____,type,
aTP_Lamp_iu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iw____,type,
aTP_Lamp_iw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ix____,type,
aTP_Lamp_ix:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iy____,type,
aTP_Lamp_iy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__iz____,type,
aTP_Lamp_iz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jb____,type,
aTP_Lamp_jb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jc____,type,
aTP_Lamp_jc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jd____,type,
aTP_Lamp_jd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__je____,type,
aTP_Lamp_je:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jf____,type,
aTP_Lamp_jf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jg____,type,
aTP_Lamp_jg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jh____,type,
aTP_Lamp_jh:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jm____,type,
aTP_Lamp_jm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jn____,type,
aTP_Lamp_jn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jo____,type,
aTP_Lamp_jo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jq____,type,
aTP_Lamp_jq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jr____,type,
aTP_Lamp_jr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__js____,type,
aTP_Lamp_js:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jt____,type,
aTP_Lamp_jt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ju____,type,
aTP_Lamp_ju:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jv____,type,
aTP_Lamp_jv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jw____,type,
aTP_Lamp_jw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jx____,type,
aTP_Lamp_jx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jy____,type,
aTP_Lamp_jy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ka____,type,
aTP_Lamp_ka:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kb____,type,
aTP_Lamp_kb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kc____,type,
aTP_Lamp_kc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kd____,type,
aTP_Lamp_kd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ke____,type,
aTP_Lamp_ke:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kg____,type,
aTP_Lamp_kg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kh____,type,
aTP_Lamp_kh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ki____,type,
aTP_Lamp_ki:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kn____,type,
aTP_Lamp_kn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ko____,type,
aTP_Lamp_ko:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kp____,type,
aTP_Lamp_kp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kq____,type,
aTP_Lamp_kq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kr____,type,
aTP_Lamp_kr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kt____,type,
aTP_Lamp_kt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ku____,type,
aTP_Lamp_ku:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kv____,type,
aTP_Lamp_kv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kw____,type,
aTP_Lamp_kw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lj____,type,
aTP_Lamp_lj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lk____,type,
aTP_Lamp_lk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ll____,type,
aTP_Lamp_ll:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lm____,type,
aTP_Lamp_lm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lo____,type,
aTP_Lamp_lo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lp____,type,
aTP_Lamp_lp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lq____,type,
aTP_Lamp_lq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lr____,type,
aTP_Lamp_lr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mi____,type,
aTP_Lamp_mi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mj____,type,
aTP_Lamp_mj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mk____,type,
aTP_Lamp_mk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mm____,type,
aTP_Lamp_mm:
!>[A: $tType,B: $tType] : 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mv____,type,
aTP_Lamp_mv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mw____,type,
aTP_Lamp_mw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mx____,type,
aTP_Lamp_mx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__my____,type,
aTP_Lamp_my:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mz____,type,
aTP_Lamp_mz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__na____,type,
aTP_Lamp_na:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nb____,type,
aTP_Lamp_nb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nc____,type,
aTP_Lamp_nc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nd____,type,
aTP_Lamp_nd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ne____,type,
aTP_Lamp_ne:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nf____,type,
aTP_Lamp_nf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ng____,type,
aTP_Lamp_ng:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nh____,type,
aTP_Lamp_nh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ni____,type,
aTP_Lamp_ni:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nj____,type,
aTP_Lamp_nj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nk____,type,
aTP_Lamp_nk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nl____,type,
aTP_Lamp_nl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nm____,type,
aTP_Lamp_nm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nn____,type,
aTP_Lamp_nn:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nt____,type,
aTP_Lamp_nt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oe____,type,
aTP_Lamp_oe:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oh____,type,
aTP_Lamp_oh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oi____,type,
aTP_Lamp_oi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oj____,type,
aTP_Lamp_oj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ol____,type,
aTP_Lamp_ol:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__om____,type,
aTP_Lamp_om:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__on____,type,
aTP_Lamp_on:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oo____,type,
aTP_Lamp_oo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__or____,type,
aTP_Lamp_or:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__os____,type,
aTP_Lamp_os:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ot____,type,
aTP_Lamp_ot:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ou____,type,
aTP_Lamp_ou:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ov____,type,
aTP_Lamp_ov:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pt____,type,
aTP_Lamp_pt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pv____,type,
aTP_Lamp_pv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pw____,type,
aTP_Lamp_pw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__px____,type,
aTP_Lamp_px:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__py____,type,
aTP_Lamp_py:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qa____,type,
aTP_Lamp_qa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qb____,type,
aTP_Lamp_qb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qc____,type,
aTP_Lamp_qc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qd____,type,
aTP_Lamp_qd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qe____,type,
aTP_Lamp_qe:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qf____,type,
aTP_Lamp_qf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qg____,type,
aTP_Lamp_qg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qh____,type,
aTP_Lamp_qh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qi____,type,
aTP_Lamp_qi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ql____,type,
aTP_Lamp_ql:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qm____,type,
aTP_Lamp_qm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qn____,type,
aTP_Lamp_qn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qo____,type,
aTP_Lamp_qo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qp____,type,
aTP_Lamp_qp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qq____,type,
aTP_Lamp_qq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qr____,type,
aTP_Lamp_qr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qu____,type,
aTP_Lamp_qu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qv____,type,
aTP_Lamp_qv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qw____,type,
aTP_Lamp_qw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qx____,type,
aTP_Lamp_qx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qy____,type,
aTP_Lamp_qy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qz____,type,
aTP_Lamp_qz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ra____,type,
aTP_Lamp_ra:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rb____,type,
aTP_Lamp_rb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rc____,type,
aTP_Lamp_rc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rd____,type,
aTP_Lamp_rd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__re____,type,
aTP_Lamp_re:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rf____,type,
aTP_Lamp_rf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rg____,type,
aTP_Lamp_rg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rh____,type,
aTP_Lamp_rh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ri____,type,
aTP_Lamp_ri:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rj____,type,
aTP_Lamp_rj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rk____,type,
aTP_Lamp_rk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rl____,type,
aTP_Lamp_rl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rm____,type,
aTP_Lamp_rm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rn____,type,
aTP_Lamp_rn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ro____,type,
aTP_Lamp_ro:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rp____,type,
aTP_Lamp_rp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rq____,type,
aTP_Lamp_rq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rr____,type,
aTP_Lamp_rr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rs____,type,
aTP_Lamp_rs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rt____,type,
aTP_Lamp_rt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ru____,type,
aTP_Lamp_ru:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rv____,type,
aTP_Lamp_rv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rw____,type,
aTP_Lamp_rw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rx____,type,
aTP_Lamp_rx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ry____,type,
aTP_Lamp_ry:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rz____,type,
aTP_Lamp_rz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sa____,type,
aTP_Lamp_sa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sc____,type,
aTP_Lamp_sc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sd____,type,
aTP_Lamp_sd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__se____,type,
aTP_Lamp_se:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sg____,type,
aTP_Lamp_sg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sh____,type,
aTP_Lamp_sh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__si____,type,
aTP_Lamp_si:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sj____,type,
aTP_Lamp_sj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sk____,type,
aTP_Lamp_sk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sl____,type,
aTP_Lamp_sl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sm____,type,
aTP_Lamp_sm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sn____,type,
aTP_Lamp_sn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__so____,type,
aTP_Lamp_so:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sp____,type,
aTP_Lamp_sp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sq____,type,
aTP_Lamp_sq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sr____,type,
aTP_Lamp_sr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ss____,type,
aTP_Lamp_ss:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__st____,type,
aTP_Lamp_st:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__su____,type,
aTP_Lamp_su:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_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_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn) ).
tff(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Oentails,type,
entails: ( assn * assn ) > $o ).
tff(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( fun(A,assn) > assn ) ).
tff(sy_c_Assertions_Oin__range,type,
in_range: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Ois__pure__assn,type,
is_pure_assn: assn > $o ).
tff(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
tff(sy_c_Assertions_Oproper,type,
proper: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o) ).
tff(sy_c_Assertions_Opure__assn,type,
pure_assn: $o > assn ).
tff(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( $o > fun(product_prod(A,set(B)),$o) ) ).
tff(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : fun(product_prod($o,product_prod(A,set(B))),fun(product_prod($o,product_prod(A,set(B))),$o)) ).
tff(sy_c_Assertions_OrelH,type,
relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).
tff(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).
tff(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array(A) * list(A) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).
tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref(A) * A ) > assn ) ).
tff(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref(A) * A ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) ).
tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_Assertions_Owand__assn,type,
wand_assn: ( assn * assn ) > assn ).
tff(sy_c_Assertions_Owand__raw,type,
wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) ) > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).
tff(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set(product_prod($o,$o)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(set(A),set(A))) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( set(fun(B,set(A))) > fun(B,set(A)) ) ).
tff(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( fun(A,$o) > fun(A,fun(A,$o)) ) ).
tff(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,fun(C,$o)) * product_prod(A,C) ) > product_prod(A,B) ) ).
tff(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,$o)) * fun(A,fun(C,$o)) * B * C ) > A ) ).
tff(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).
tff(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(set(A),fun(set(B),$o)) ) ).
tff(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,$o)) * fun(A,fun(B,$o)) * product_prod(C,B) ) > product_prod(A,B) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(A) * fun(product_prod(A,B),C) * A ) > fun(B,C) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(A,A)),set(A)) ) ).
tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_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_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).
tff(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,$o)) * fun(C,fun(D,$o)) ) > fun(product_prod(A,C),fun(product_prod(B,D),$o)) ) ).
tff(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(A)) ).
tff(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(B)) ).
tff(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).
tff(sy_c_Binomial_Obinomial,type,
binomial: ( nat * nat ) > nat ).
tff(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > fun(nat,A) ) ).
tff(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).
tff(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).
tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: ( num * num ) > num ).
tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > fun(nat,$o) ) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).
tff(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: ( nat * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
bit_un6697907153464112080or_num: ( num * num ) > num ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
bit_un4773296044027857193um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__axioms,type,
boolea6902313364301356556axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * fun(A,fun(A,A)) ) > $o ) ).
tff(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
tff(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).
tff(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_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_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,$o) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,A) ) > A ) ).
tff(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( fun(A,A) > set(A) ) ).
tff(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,fun(B,$o)) * fun(A,B) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit8047198070973881523_above:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit8119078960628432327_below:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: fun(nat,fun(nat,$o)) ).
tff(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod(int,int) > int ).
tff(sy_c_Divides_Odivmod__nat,type,
divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).
tff(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( product_prod(A,A) > $o ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).
tff(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > set(A) ) ).
tff(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).
tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Ocofinite,type,
cofinite:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( fun(fun(A,$o),$o) > filter(A) ) ).
tff(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).
tff(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).
tff(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( set(A) > filter(set(A)) ) ).
tff(sy_c_Filter_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( set(A) > filter(A) ) ).
tff(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).
tff(sy_c_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_Ofinite,type,
finite_finite:
!>[A: $tType] : fun(set(A),$o) ).
tff(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) * B ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).
tff(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).
tff(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).
tff(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).
tff(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * A * B ) > $o ) ).
tff(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).
tff(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).
tff(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_Obezw,type,
bezw: ( nat * nat ) > product_prod(int,int) ).
tff(sy_c_GCD_Obezw__rel,type,
bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).
tff(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).
tff(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).
tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * set(B) ) > A ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).
tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : fun(list(A),A) ).
tff(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).
tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : fun(list(A),A) ).
tff(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,$o) > A ) ).
tff(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( fun(A,$o) > $o ) ).
tff(sy_c_HOL_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_Oarray_Oset__array,type,
set_array:
!>[A: $tType] : ( array(A) > set(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_Oref_Oset__ref,type,
set_ref:
!>[A: $tType] : ( ref(A) > set(A) ) ).
tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : ( set(heap_Time_Heap(A)) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( fun(A,$o) * A ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).
tff(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).
tff(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( fun(heap_ext(product_unit),$o) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_OtimeFrame,type,
heap_Time_timeFrame:
!>[A: $tType] : ( ( nat * option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ).
tff(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o)) ).
tff(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : ( A > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > heap_Time_Heap(product_unit) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set(A) * nat ) > A ) ).
tff(sy_c_Int_OAbs__Integ,type,
abs_Integ: fun(product_prod(nat,nat),int) ).
tff(sy_c_Int_ORep__Integ,type,
rep_Integ: fun(int,product_prod(nat,nat)) ).
tff(sy_c_Int_Ointrel,type,
intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).
tff(sy_c_Int_Onat,type,
nat2: int > nat ).
tff(sy_c_Int_Opcr__int,type,
pcr_int: fun(product_prod(nat,nat),fun(int,$o)) ).
tff(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( ( A * int ) > A ) ).
tff(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : set(A) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : fun(int,A) ).
tff(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices_Osemilattice,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: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) * B ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
lattic8678736583308907530nf_fin:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
lattic4630905495605216202up_fin:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).
tff(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list(A) * A ) > nat ) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).
tff(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).
tff(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).
tff(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).
tff(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).
tff(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * set(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( set(A) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > fun(list(A),list(A)) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) * list(A) ) > B ) ).
tff(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).
tff(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * list(A) ) > list(Aa) ) ).
tff(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) * list(A) ) > C ) ).
tff(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( list(A) > set(A) ) ).
tff(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
tff(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).
tff(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).
tff(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).
tff(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( set(A) > set(list(A)) ) ).
tff(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).
tff(sy_c_List_Omap__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: $tType] : fun(product_prod(fun(A,B),product_prod(list(A),list(B))),fun(product_prod(fun(A,B),product_prod(list(A),list(B))),$o)) ).
tff(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).
tff(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).
tff(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( list(A) > fun(nat,A) ) ).
tff(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).
tff(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( fun(A,$o) > fun(list(A),product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( ( nat * A ) > list(A) ) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).
tff(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).
tff(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).
tff(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oupt,type,
upt: ( nat * nat ) > list(nat) ).
tff(sy_c_List_Oupto,type,
upto: ( int * int ) > list(int) ).
tff(sy_c_List_Oupto__rel,type,
upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).
tff(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).
tff(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).
tff(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).
tff(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).
tff(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( fun(A,$o) > option(A) ) ).
tff(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod($o,A),product_prod($o,B))) ) ).
tff(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( set(A) > option(set(A)) ) ).
tff(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).
tff(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,$o)) * list(A) * list(B) ) > $o ) ).
tff(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o)) ).
tff(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).
tff(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( fun(B,option(A)) * set(B) * A ) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),$o)) ).
tff(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).
tff(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).
tff(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),$o)) ).
tff(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).
tff(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),$o)) ).
tff(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( fun(A,$o) * product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o)) ).
tff(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).
tff(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * fun(A,B) ) > $o ) ).
tff(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( ( A * option(A) ) > A ) ).
tff(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Misc_Ozipf,type,
zipf:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * list(A) * list(B) ) > list(C) ) ).
tff(sy_c_Misc_Ozipf__rel,type,
zipf_rel:
!>[A: $tType,B: $tType,C: $tType] : fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),$o)) ).
tff(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( ( A * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m7189776963980413722m_mset:
!>[A: $tType] : ( multiset(A) > A ) ).
tff(sy_c_Multiset_Ocomm__monoid__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(multiset(A),A) ) ).
tff(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : fun(multiset(A),A) ).
tff(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * multiset(A) ) > B ) ).
tff(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * multiset(A) ) > multiset(B) ) ).
tff(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).
tff(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).
tff(sy_c_Multiset_Oms__strict,type,
ms_strict: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).
tff(sy_c_Multiset_Oms__weak,type,
ms_weak: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).
tff(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( list(A) > multiset(A) ) ).
tff(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( set(B) > multiset(B) ) ).
tff(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).
tff(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).
tff(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : fun(fun(A,nat),multiset(A)) ).
tff(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : fun(multiset(A),fun(A,nat)) ).
tff(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( fun(C,fun(B,$o)) > fun(fun(C,nat),fun(multiset(B),$o)) ) ).
tff(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset(product_prod(nat,nat)) * multiset(product_prod(nat,nat)) ) > $o ).
tff(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( ( nat * A ) > multiset(A) ) ).
tff(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : fun(multiset(A),set(A)) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list(A) * list(A) ) > option($o) ) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).
tff(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).
tff(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).
tff(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).
tff(sy_c_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: $tType] : fun(A,nat) ).
tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: fun(nat,fun(nat,$o)) ).
tff(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list(nat) > nat ).
tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).
tff(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > product_prod(nat,nat) ).
tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).
tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).
tff(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: product_prod(nat,nat) > nat ).
tff(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set(nat) ).
tff(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set(nat) > nat ).
tff(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > sum_sum(nat,nat) ).
tff(sy_c_Num_OBitM,type,
bitM: num > num ).
tff(sy_c_Num_Oinc,type,
inc: num > num ).
tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_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: fun(num,num) ).
tff(sy_c_Num_Onum_OBit1,type,
bit1: fun(num,num) ).
tff(sy_c_Num_Onum_OOne,type,
one2: num ).
tff(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).
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_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( sum_sum(A,nat) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))) ).
tff(sy_c_Old__Datatype_OScons,type,
old_Scons:
!>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_Odprod,type,
old_dprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Odsum,type,
old_dsum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).
tff(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) > old_node(A,B) ) ).
tff(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( ( nat * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_Ouprod,type,
old_uprod:
!>[A: $tType,B: $tType] : ( ( set(set(old_node(A,B))) * set(set(old_node(A,B))) ) > set(set(old_node(A,B))) ) ).
tff(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : option(A) ).
tff(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : fun(A,option(A)) ).
tff(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).
tff(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).
tff(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( option(A) > set(A) ) ).
tff(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( option(A) > A ) ).
tff(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( set(option(A)) > set(A) ) ).
tff(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).
tff(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( fun(A,$o) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,$o)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : fun(A,fun(A,$o)) ).
tff(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,B),$o) ) ).
tff(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( fun(A,$o) > A ) ).
tff(sy_c_Orderings_Oorder__class_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_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A ) > $o ) ).
tff(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).
tff(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > fun(set(A),A) ) ).
tff(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > fun(A,fun(A,$o)) ) ).
tff(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( 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: $tType] : ( fun(A,fun(B,$o)) > fun(fun(C,A),fun(fun(C,B),$o)) ) ).
tff(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(set(A),A) ) > $o ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : fun(A,fun(nat,A)) ).
tff(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( fun(product_unit,seq(A)) > pred(A) ) ).
tff(sy_c_Predicate_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).
tff(sy_c_Predicate_Oif__pred,type,
if_pred: $o > pred(product_unit) ).
tff(sy_c_Predicate_Ois__empty,type,
is_empty:
!>[A: $tType] : ( pred(A) > $o ) ).
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: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o)) ).
tff(sy_c_Predicate_Onot__pred,type,
not_pred: pred(product_unit) > pred(product_unit) ).
tff(sy_c_Predicate_Opred_OPred,type,
pred2:
!>[A: $tType] : ( fun(A,$o) > pred(A) ) ).
tff(sy_c_Predicate_Opred_Oeval,type,
eval:
!>[A: $tType] : ( pred(A) > fun(A,$o) ) ).
tff(sy_c_Predicate_Opred__of__seq,type,
pred_of_seq:
!>[A: $tType] : ( seq(A) > pred(A) ) ).
tff(sy_c_Predicate_Opred__of__set,type,
pred_of_set:
!>[A: $tType] : ( set(A) > pred(A) ) ).
tff(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : seq(A) ).
tff(sy_c_Predicate_Oseq_OInsert,type,
insert:
!>[A: $tType] : ( ( A * pred(A) ) > seq(A) ) ).
tff(sy_c_Predicate_Oseq_Ocase__seq,type,
case_seq:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(pred(A),B)) * fun(pred(A),fun(seq(A),B)) * seq(A) ) > B ) ).
tff(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : ( pred(A) > set(A) ) ).
tff(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : ( seq(A) > set(A) ) ).
tff(sy_c_Predicate_Osingle,type,
single:
!>[A: $tType] : fun(A,pred(A)) ).
tff(sy_c_Predicate_Osingleton,type,
singleton:
!>[A: $tType] : ( ( fun(product_unit,A) * pred(A) ) > A ) ).
tff(sy_c_Predicate__Compile_Ocontains__pred,type,
predic7144156976422707464s_pred:
!>[A: $tType] : ( ( set(A) * A ) > pred(product_unit) ) ).
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) * product_prod(A,B) ) > product_prod(C,B) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(product_prod(A,B),C) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).
tff(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).
tff(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).
tff(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).
tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: fun($o,product_unit) ).
tff(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: fun(product_unit,$o) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
tff(sy_c_Random_Oinc__shift,type,
inc_shift: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).
tff(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o)) ).
tff(sy_c_Random_Olog,type,
log: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Olog__rel,type,
log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),$o)) ).
tff(sy_c_Random_Ominus__shift,type,
minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Onext,type,
next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).
tff(sy_c_Random_Orange,type,
range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Osplit__seed,type,
split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).
tff(sy_c_Random__Pred_ORandom,type,
random_Random:
!>[A: $tType] : ( fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))) * product_prod(code_natural,code_natural) ) > product_prod(pred(B),product_prod(code_natural,code_natural)) ) ).
tff(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ).
tff(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(product_unit),product_prod(code_natural,code_natural)) ).
tff(sy_c_Random__Pred_Osingle,type,
random_single:
!>[A: $tType] : ( A > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(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: $tType] : ( rat > A ) ).
tff(sy_c_Rat_Onormalize,type,
normalize: product_prod(int,int) > product_prod(int,int) ).
tff(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
tff(sy_c_Rat_Opcr__rat,type,
pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).
tff(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod(int,int) ).
tff(sy_c_Rat_Oratrel,type,
ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).
tff(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( ( A * heap_ext(product_unit) ) > product_prod(ref(A),heap_ext(product_unit)) ) ).
tff(sy_c_Ref__Time_Ochange,type,
ref_change:
!>[A: $tType] : ( ( fun(A,A) * ref(A) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).
tff(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > $o ) ).
tff(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).
tff(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).
tff(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).
tff(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(A,$o) ) ).
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: $tType] : ( fun(A,fun(B,$o)) > fun(B,$o) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).
tff(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(B,fun(A,$o)) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).
tff(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,$o)) * fun(B,fun(C,$o)) ) > fun(A,fun(C,$o)) ) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : fun($o,A) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : fun(fun(A,$o),set(A)) ).
tff(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Set_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).
tff(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).
tff(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).
tff(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).
tff(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > fun(set(A),set(A)) ) ).
tff(sy_c_Set_Ois__empty,type,
is_empty2:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).
tff(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( ( A * set(A) ) > set(A) ) ).
tff(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(B) ) > set(A) ) ).
tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).
tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).
tff(sy_c_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr3752724095348155675AtMost:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr287244882034783167ssThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > set(A) ) ).
tff(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).
tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) ) > fun(sum_sum(A,B),C) ) ).
tff(sy_c_Syntax__Match_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_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,$o)) ) ).
tff(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,$o) ) ).
tff(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : set(product_prod(set(A),set(A))) ).
tff(sy_c_Wellfounded_Oless__than,type,
less_than: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).
tff(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $o ) ).
tff(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).
tff(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wfrec_Owfrec__rel,type,
wfrec_rel:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) * A * B ) > $o ) ).
tff(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).
tff(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).
tff(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).
tff(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).
tff(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
pred_s596693808085603175closed:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > set(set(A)) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( fun(A,$o) > A ) ).
tff(sy_c_fNot,type,
fNot: fun($o,$o) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,$o)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * set(A) ) > $o ) ).
tff(sy_v_A,type,
a2: assn ).
tff(sy_v_B,type,
b2: assn ).
tff(sy_v_C,type,
c: assn ).
% Relevant facts (5460)
tff(fact_0_ent__iffI,axiom,
! [Aa2: assn,Ba: assn] :
( entails(Aa2,Ba)
=> ( entails(Ba,Aa2)
=> ( Aa2 = Ba ) ) ) ).
% ent_iffI
tff(fact_1_ent__refl,axiom,
! [P: assn] : entails(P,P) ).
% ent_refl
tff(fact_2_ent__disjE,axiom,
! [Aa2: assn,Ca: assn,Ba: assn] :
( entails(Aa2,Ca)
=> ( entails(Ba,Ca)
=> entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),Ba),Ca) ) ) ).
% ent_disjE
tff(fact_3_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( entails(P,Q)
=> ( entails(Q,R)
=> entails(P,R) ) ) ).
% ent_trans
tff(fact_4_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R)
=> entails(P,R) ) ).
% ent_disjI1
tff(fact_5_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),R)
=> entails(Q,R) ) ).
% ent_disjI2
tff(fact_6_ent__star__mono,axiom,
! [P: assn,P2: assn,Q: assn,Q2: assn] :
( entails(P,P2)
=> ( entails(Q,Q2)
=> 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_7_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_8_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_9_ent__disjI1__direct,axiom,
! [Aa2: assn,Ba: assn] : entails(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),Ba)) ).
% ent_disjI1_direct
tff(fact_10_ent__disjI2__direct,axiom,
! [Ba: assn,Aa2: assn] : entails(Ba,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),Aa2),Ba)) ).
% ent_disjI2_direct
tff(fact_11_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_12_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_13_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_14_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_15_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_16_sup__apply,axiom,
! [A: $tType,B: $tType] :
( semilattice_sup(A)
=> ! [F: fun(B,A),G: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F),G),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,X)),aa(B,A,G,X)) ) ).
% sup_apply
tff(fact_17_ent__mp,axiom,
! [P: assn,Q: assn] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),wand_assn(P,Q)),Q) ).
% ent_mp
tff(fact_18_ent__wandI,axiom,
! [Q: assn,P: assn,R: assn] :
( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P),R)
=> entails(P,wand_assn(Q,R)) ) ).
% ent_wandI
tff(fact_19_ent__pure__pre__iff,axiom,
! [P: assn,B2: $o,Q: assn] :
( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),Q)
<=> ( (B2)
=> entails(P,Q) ) ) ).
% ent_pure_pre_iff
tff(fact_20_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_21_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_22_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_23_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( pure_assn((P)) = pure_assn((Q)) )
<=> ( (P)
<=> (Q) ) ) ).
% pure_assn_eq_conv
tff(fact_24_merge__pure__star,axiom,
! [A3: $o,B2: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((A3))),pure_assn((B2))) = pure_assn(( (A3)
& (B2) )) ).
% merge_pure_star
tff(fact_25_merge__pure__or,axiom,
! [A3: $o,B2: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn((A3))),pure_assn((B2))) = pure_assn(( (A3)
| (B2) )) ).
% merge_pure_or
tff(fact_26_is__pure__assn__pure,axiom,
! [P: $o] : is_pure_assn(pure_assn((P))) ).
% is_pure_assn_pure
tff(fact_27_is__pure__assnE,axiom,
! [A3: assn] :
( is_pure_assn(A3)
=> ~ ! [P3: $o] : A3 != pure_assn((P3)) ) ).
% is_pure_assnE
tff(fact_28_is__pure__assn__def,axiom,
! [A3: assn] :
( is_pure_assn(A3)
<=> ? [P4: $o] : A3 = pure_assn((P4)) ) ).
% is_pure_assn_def
tff(fact_29_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(A,B),G: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X2) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X2)),aa(A,B,G,X2)) ) ).
% sup_fun_def
tff(fact_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B2: $o] :
( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn((B2))))
<=> ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H)
=> (B2) )
& entails(P,Q) ) ) ).
% ent_pure_post_iff
tff(fact_39_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Ba: A,K: A,B2: A,A3: A] :
( ( Ba = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Ba) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_40_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: A,K: A,A3: A,B2: A] :
( ( Aa2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_41_ent__pure__pre__iff__sng,axiom,
! [B2: $o,Q: assn] :
( entails(pure_assn((B2)),Q)
<=> ( (B2)
=> entails(one_one(assn),Q) ) ) ).
% ent_pure_pre_iff_sng
tff(fact_42_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_43_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_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: fun(A,$o)] :
( member(A,A3,aa(fun(A,$o),set(A),collect(A),P))
<=> aa(A,$o,P,A3) ) ).
% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
! [A: $tType,Aa2: set(A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,$o),Aa2)) = Aa2 ).
% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
( ! [X3: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,Q,X3) )
=> ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).
% Collect_cong
tff(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
% ext
tff(fact_49_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_50_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_51_mod__pure__star__dist,axiom,
! [P: assn,B2: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2)))),H2)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
& (B2) ) ) ).
% mod_pure_star_dist
tff(fact_52_pure__true,axiom,
pure_assn($true) = one_one(assn) ).
% pure_true
tff(fact_53_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( pure_assn((P)) = one_one(assn) )
<=> (P) ) ).
% pure_assn_eq_emp_iff
tff(fact_54_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X) = aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Y) )
<=> ( X = Y ) ) ).
% Rep_assn_inject
tff(fact_55_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_56_mult__1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).
% mult_1
tff(fact_57_mod__or__dist,axiom,
! [P: assn,Q: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),H2)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H2) ) ) ).
% mod_or_dist
tff(fact_58_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: $o] :
( entails(P,pure_assn((B2)))
<=> ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H)
=> (B2) )
& entails(P,one_one(assn)) ) ) ).
% ent_pure_post_iff_sng
tff(fact_59_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
% one_reorient
tff(fact_60_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_61_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).
% mult.comm_neutral
tff(fact_62_ent__fwd,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Q: assn] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
=> ( entails(P,Q)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H2) ) ) ).
% ent_fwd
tff(fact_63_entailsD,axiom,
! [P: assn,Q: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( entails(P,Q)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H2) ) ) ).
% entailsD
tff(fact_64_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H3)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H3) )
=> entails(P,Q) ) ).
% entailsI
tff(fact_65_entails__def,axiom,
! [P: assn,Q: assn] :
( entails(P,Q)
<=> ! [H: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H) ) ) ).
% entails_def
tff(fact_66_mod__starD,axiom,
! [Aa2: assn,Ba: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),Ba)),H2)
=> ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Aa2),H1)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Ba),H22) ) ) ).
% mod_starD
tff(fact_67_mod__starE,axiom,
! [A3: assn,B2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),H2)
=> ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,A3),X_1)
=> ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,B2),H_2) ) ) ).
% mod_starE
tff(fact_68_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_69_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn(one_one(assn)) ).
% is_pure_assn_basic_simps(2)
tff(fact_70_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_71_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_72_ent__false__iff,axiom,
! [P: assn] :
( entails(P,bot_bot(assn))
<=> ! [H: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H) ) ).
% ent_false_iff
tff(fact_73_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> comm_monoid(A,times_times(A),one_one(A)) ) ).
% mult.comm_monoid_axioms
tff(fact_74_mod__star__trueE,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H2)
=> ~ ! [H4: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H4) ) ).
% mod_star_trueE
tff(fact_75_mod__star__trueI,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H2) ) ).
% mod_star_trueI
tff(fact_76_Rep__assn__inverse,axiom,
! [X: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X)) = X ).
% Rep_assn_inverse
tff(fact_77_wand__assn__def,axiom,
! [P: assn,Q: assn] : wand_assn(P,Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,wand_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).
% wand_assn_def
tff(fact_78_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_79_syntax__nomatch__def,axiom,
! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax2379306206330768139omatch(A,B,Pat,Obj) ).
% syntax_nomatch_def
tff(fact_80_mult_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> syntax_ac_operator(A,times_times(A)) ) ).
% mult.ac_operator_axioms
tff(fact_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).
% boolean_algebra.disj_one_right
tff(fact_93_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).
% boolean_algebra.disj_one_left
tff(fact_94_star__false__right,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).
% star_false_right
tff(fact_95_star__false__left,axiom,
! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).
% star_false_left
tff(fact_96_pure__false,axiom,
pure_assn($false) = bot_bot(assn) ).
% pure_false
tff(fact_97_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( pure_assn((P)) = bot_bot(assn) )
<=> ~ (P) ) ).
% pure_assn_eq_false_iff
tff(fact_98_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_99_assn__basic__inequalities_I5_J,axiom,
top_top(assn) != bot_bot(assn) ).
% assn_basic_inequalities(5)
tff(fact_100_assn__basic__inequalities_I3_J,axiom,
bot_bot(assn) != one_one(assn) ).
% assn_basic_inequalities(3)
tff(fact_101_assn__basic__inequalities_I1_J,axiom,
top_top(assn) != one_one(assn) ).
% assn_basic_inequalities(1)
tff(fact_102_ac__operator_Osafe__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A,Y: A,A3: A,B2: A] :
( syntax_ac_operator(A,F)
=> ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),F,X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ) ).
% ac_operator.safe_commute
tff(fact_103_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( comm_monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% comm_monoid.comm_neutral
tff(fact_104_monoid_Oright__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% monoid.right_neutral
tff(fact_105_monoid_Oleft__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% monoid.left_neutral
tff(fact_106_comm__monoid__list_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups1828464146339083142d_list(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% comm_monoid_list.axioms(1)
tff(fact_107_comm__monoid__list__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups4802862169904069756st_set(A,F,Z2)
=> groups1828464146339083142d_list(A,F,Z2) ) ).
% comm_monoid_list_set.axioms(1)
tff(fact_108_ac__operator_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A4: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A4),B3)),C3) = aa(A,A,aa(A,fun(A,A),F,A4),aa(A,A,aa(A,fun(A,A),F,B3),C3))
=> ( ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),F,A4),B3) = aa(A,A,aa(A,fun(A,A),F,B3),A4)
=> syntax_ac_operator(A,F) ) ) ).
% ac_operator.intro
tff(fact_109_ac__operator_Ocommute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).
% ac_operator.commute
tff(fact_110_ac__operator_Oleft__assoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).
% ac_operator.left_assoc
tff(fact_111_ac__operator_Oright__assoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).
% ac_operator.right_assoc
tff(fact_112_ac__operator_Oleft__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) = aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C2)) ) ) ).
% ac_operator.left_commute
tff(fact_113_ac__operator_Oright__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( syntax_ac_operator(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),C2)),B2) ) ) ).
% ac_operator.right_commute
tff(fact_114_ac__operator__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( syntax_ac_operator(A,F)
<=> ( ! [A5: A,B4: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B4)),C4) = aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B4),C4))
& ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F,A5),B4) = aa(A,A,aa(A,fun(A,A),F,B4),A5) ) ) ).
% ac_operator_def
tff(fact_115_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_116_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.monoid_axioms
tff(fact_117_syntax__fo__nomatch__def,axiom,
! [A: $tType,B: $tType,Pat: A,Obj: B] : syntax7388354845996824322omatch(A,B,Pat,Obj) ).
% syntax_fo_nomatch_def
tff(fact_118_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).
% boolean_algebra.disj_zero_right
tff(fact_119_mod__false,axiom,
! [H2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,bot_bot(assn)),H2) ).
% mod_false
tff(fact_120_ent__true,axiom,
! [P: assn] : entails(P,top_top(assn)) ).
% ent_true
tff(fact_121_ent__false,axiom,
! [P: assn] : entails(bot_bot(assn),P) ).
% ent_false
tff(fact_122_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn(bot_bot(assn)) ).
% is_pure_assn_basic_simps(1)
tff(fact_123_Abs__assn__eqI_I2_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H3)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H3) )
=> ( Pr = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) ) ) ).
% Abs_assn_eqI(2)
tff(fact_124_Abs__assn__eqI_I1_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Pr: assn] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,H3)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Pr),H3) )
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,P) = Pr ) ) ).
% Abs_assn_eqI(1)
tff(fact_125_sngr__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P5: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P5,X)),sngr_assn(A,P5,Y)) = bot_bot(assn) ) ).
% sngr_same_false
tff(fact_126_snga__same__false,axiom,
! [A: $tType] :
( heap(A)
=> ! [P5: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P5,X)),snga_assn(A,P5,Y)) = bot_bot(assn) ) ).
% snga_same_false
tff(fact_127_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).
% top_apply
tff(fact_128_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).
% bot_apply
tff(fact_129_times__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,times_assn_raw(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q))) ).
% times_assn_def
tff(fact_130_one__assn__def,axiom,
one_one(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,one_assn_raw) ).
% one_assn_def
tff(fact_131_pure__assn__def,axiom,
! [B2: $o] : pure_assn((B2)) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,pure_assn_raw(heap_ext(product_unit),nat,(B2))) ).
% pure_assn_def
tff(fact_132_comm__monoid__list_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> ( groups_monoid_list(A,F,Z2)
=> groups1828464146339083142d_list(A,F,Z2) ) ) ).
% comm_monoid_list.intro
tff(fact_133_comm__monoid__list__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups1828464146339083142d_list(A,F,Z2)
<=> ( comm_monoid(A,F,Z2)
& groups_monoid_list(A,F,Z2) ) ) ).
% comm_monoid_list_def
tff(fact_134_mod__true,axiom,
! [H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,top_top(assn)),H2)
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2) ) ).
% mod_true
tff(fact_135_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> monoid_axioms(A,F,Z2) ) ).
% monoid.axioms(2)
tff(fact_136_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_137_monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F,Z2)
<=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
& ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5 ) ) ).
% monoid_axioms_def
tff(fact_138_monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A4) = A4
=> ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),Z2) = A4
=> monoid_axioms(A,F,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_139_models__in__range,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2) ) ).
% models_in_range
tff(fact_140_monoid__list__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups_monoid_list(A,F,Z2)
<=> monoid(A,F,Z2) ) ).
% monoid_list_def
tff(fact_141_monoid__list_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups_monoid_list(A,F,Z2)
=> monoid(A,F,Z2) ) ).
% monoid_list.axioms
tff(fact_142_monoid__list_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> groups_monoid_list(A,F,Z2) ) ).
% monoid_list.intro
tff(fact_143_comm__monoid__list_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups1828464146339083142d_list(A,F,Z2)
=> groups_monoid_list(A,F,Z2) ) ).
% comm_monoid_list.axioms(2)
tff(fact_144_top__assn__def,axiom,
top_top(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,in_range) ).
% top_assn_def
tff(fact_145_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X2: A] : aa(A,B,bot_bot(fun(A,B)),X2) = bot_bot(B) ) ).
% bot_fun_def
tff(fact_146_sngr__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : sngr_assn(A,R2,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,sngr_assn_raw(A,R2,X)) ) ).
% sngr_assn_def
tff(fact_147_snga__assn__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),A3: list(A)] : snga_assn(A,R2,A3) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,snga_assn_raw(A,R2,A3)) ) ).
% snga_assn_def
tff(fact_148_monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semigroup(A,F)
=> ( monoid_axioms(A,F,Z2)
=> monoid(A,F,Z2) ) ) ).
% monoid.intro
tff(fact_149_monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
<=> ( semigroup(A,F)
& monoid_axioms(A,F,Z2) ) ) ).
% monoid_def
tff(fact_150_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [P5: ref(A),X: A,H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,sngr_assn(A,P5,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_151_mod__h__bot__iff_I4_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [Q3: array(A),Y: list(A),H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,snga_assn(A,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_152_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid.axioms(2)
tff(fact_153_pure__assn__proper,axiom,
! [B2: $o] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,pure_assn_raw(heap_ext(product_unit),nat,(B2))) ).
% pure_assn_proper
tff(fact_154_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_155_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(7)
tff(fact_156_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,pure_assn((B2))),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> (B2) ) ).
% mod_h_bot_iff(1)
tff(fact_157_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(5)
tff(fact_158_in__range__dist__union,axiom,
! [H2: heap_ext(product_unit),As: set(nat),As2: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As2)))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)) ) ) ).
% in_range_dist_union
tff(fact_159_bool__assn__proper_I1_J,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,in_range) ).
% bool_assn_proper(1)
tff(fact_160_snga__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,snga_assn_raw(A,R2,X)) ) ).
% snga_assn_proper
tff(fact_161_sngr__assn__proper,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,sngr_assn_raw(A,R2,X)) ) ).
% sngr_assn_proper
tff(fact_162_wand__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,wand_raw(P,Q)) ).
% wand_proper
tff(fact_163_one__assn__proper,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,one_assn_raw) ).
% one_assn_proper
tff(fact_164_times__assn__proper,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,times_assn_raw(P,Q)) ) ) ).
% times_assn_proper
tff(fact_165_in__range__empty,axiom,
! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).
% in_range_empty
tff(fact_166_semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
<=> ! [A5: A,B4: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B4)),C4) = aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B4),C4)) ) ).
% semigroup_def
tff(fact_167_semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A4: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A4),B3)),C3) = aa(A,A,aa(A,fun(A,A),F,A4),aa(A,A,aa(A,fun(A,A),F,B3),C3))
=> semigroup(A,F) ) ).
% semigroup.intro
tff(fact_168_semigroup_Oassoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).
% semigroup.assoc
tff(fact_169_one__assn__raw_Ocases,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
~ ! [H3: heap_ext(product_unit),As3: set(nat)] : X != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) ).
% one_assn_raw.cases
tff(fact_170_properD1,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).
% properD1
tff(fact_171_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),Z2) = A4
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_172_comm__monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F,Z2)
<=> ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5 ) ).
% comm_monoid_axioms_def
tff(fact_173_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),bot_bot(set(nat)))) ) ).
% mod_h_bot_indep
tff(fact_174_Rep__assn,axiom,
! [X: assn] : member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X),aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).
% Rep_assn
tff(fact_175_Rep__assn__cases,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ~ ! [X3: assn] : Y != aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X3) ) ).
% Rep_assn_cases
tff(fact_176_Rep__assn__induct,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o),P: fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( ! [X3: assn] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,X3))
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,P,Y) ) ) ).
% Rep_assn_induct
tff(fact_177_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_178_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semigroup(A,sup_sup(A)) ) ).
% sup.semigroup_axioms
tff(fact_179_pure__assn__raw_Osimps,axiom,
! [B: $tType,A: $tType,B2: $o,H2: A,As: set(B)] :
( aa(product_prod(A,set(B)),$o,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),As))
<=> ( ( As = bot_bot(set(B)) )
& (B2) ) ) ).
% pure_assn_raw.simps
tff(fact_180_pure__assn__raw_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B)),Y: $o] :
( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
<=> (Y) )
=> ~ ! [H3: A,As3: 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),As3) )
=> ( (Y)
<=> ~ ( ( As3 = bot_bot(set(B)) )
& (X) ) ) ) ) ).
% pure_assn_raw.elims(1)
tff(fact_181_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ~ ! [H3: A,As3: 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),As3) )
=> ~ ( ( As3 = bot_bot(set(B)) )
& (X) ) ) ) ).
% pure_assn_raw.elims(2)
tff(fact_182_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ~ ! [H3: A,As3: 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),As3) )
=> ( ( As3 = bot_bot(set(B)) )
& (X) ) ) ) ).
% pure_assn_raw.elims(3)
tff(fact_183_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( ( X = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2) )
=> ~ member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ) ).
% Abs_assn_cases
tff(fact_184_Abs__assn__induct,axiom,
! [P: fun(assn,$o),X: assn] :
( ! [Y2: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y2,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> aa(assn,$o,P,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y2)) )
=> aa(assn,$o,P,X) ) ).
% Abs_assn_induct
tff(fact_185_Abs__assn__inject,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),X,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,X) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_assn_inject
tff(fact_186_one__assn__raw_Osimps,axiom,
! [H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ( As = bot_bot(set(nat)) ) ) ).
% one_assn_raw.simps
tff(fact_187_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ( As3 != bot_bot(set(nat)) ) ) ) ) ).
% one_assn_raw.elims(1)
tff(fact_188_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( As3 != bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(2)
tff(fact_189_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( As3 = bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(3)
tff(fact_190_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> semigroup(A,F) ) ).
% monoid.axioms(1)
tff(fact_191_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% semilattice_neutr.axioms(2)
tff(fact_192_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,top_top(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).
% mod_h_bot_iff(2)
tff(fact_193_mod__emp__simp,axiom,
! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,one_one(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).
% mod_emp_simp
tff(fact_194_Abs__assn__inverse,axiom,
! [Y: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( member(fun(product_prod(heap_ext(product_unit),set(nat)),$o),Y,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper))
=> ( aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,Y)) = Y ) ) ).
% Abs_assn_inverse
tff(fact_195_empty__iff,axiom,
! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).
% empty_iff
tff(fact_196_all__not__in__conv,axiom,
! [A: $tType,Aa2: set(A)] :
( ! [X4: A] : ~ member(A,X4,Aa2)
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% all_not_in_conv
tff(fact_197_Collect__empty__eq,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
<=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).
% Collect_empty_eq
tff(fact_198_empty__Collect__eq,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
<=> ! [X4: A] : ~ aa(A,$o,P,X4) ) ).
% empty_Collect_eq
tff(fact_199_Un__empty,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba) = bot_bot(set(A)) )
<=> ( ( Aa2 = bot_bot(set(A)) )
& ( Ba = bot_bot(set(A)) ) ) ) ).
% Un_empty
tff(fact_200_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_201_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
<=> ( ( A3 = A6 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
tff(fact_202_mod__star__conv,axiom,
! [Aa2: assn,Ba: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Aa2),Ba)),H2)
<=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
( ( H2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Aa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Ba),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As22)) ) ) ).
% mod_star_conv
tff(fact_203_star__assnI,axiom,
! [P: assn,H2: heap_ext(product_unit),As: set(nat),Q: assn,As2: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
=> ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As2) = bot_bot(set(nat)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As2))) ) ) ) ).
% star_assnI
tff(fact_204_proper__iff,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( relH(As,H2,H5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As)) ) ) ) ) ).
% proper_iff
tff(fact_205_proper__def,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
<=> ! [H: heap_ext(product_unit),H6: heap_ext(product_unit),As4: set(nat)] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)) )
& ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
& relH(As4,H,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As4)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As4)) ) ) ) ).
% proper_def
tff(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_inf__apply,axiom,
! [A: $tType,B: $tType] :
( semilattice_inf(A)
=> ! [F: fun(B,A),G: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),inf_inf(fun(B,A)),F),G),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,X)),aa(B,A,G,X)) ) ).
% inf_apply
tff(fact_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_relH__dist__union,axiom,
! [As: set(nat),As2: 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)),As),As2),H2,H5)
<=> ( relH(As,H2,H5)
& relH(As2,H2,H5) ) ) ).
% relH_dist_union
tff(fact_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(A,B),G: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X2) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X2)),aa(A,B,G,X2)) ) ).
% inf_fun_def
tff(fact_239_relH__trans,axiom,
! [As: set(nat),H12: heap_ext(product_unit),H23: heap_ext(product_unit),H32: heap_ext(product_unit)] :
( relH(As,H12,H23)
=> ( relH(As,H23,H32)
=> relH(As,H12,H32) ) ) ).
% relH_trans
tff(fact_240_relH__sym,axiom,
! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As,H2,H5)
=> relH(As,H5,H2) ) ).
% relH_sym
tff(fact_241_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: A,K: A,A3: A,B2: A] :
( ( Aa2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Aa2),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_242_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Ba: A,K: A,B2: A,A3: A] :
( ( Ba = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Ba) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.inf2
tff(fact_243_times__assn__raw_Ocases,axiom,
! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [P3: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q4: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H3: heap_ext(product_unit),As3: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),P3),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),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),As3))) ).
% times_assn_raw.cases
tff(fact_244_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),X3: A,H3: heap_ext(product_unit),As3: 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))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))) ) ).
% sngr_assn_raw.cases
tff(fact_245_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),X3: list(A),H3: heap_ext(product_unit),As3: 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))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))) ) ).
% snga_assn_raw.cases
tff(fact_246_disjoint__iff__not__equal,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ! [Xa2: A] :
( member(A,Xa2,Ba)
=> ( X4 != Xa2 ) ) ) ) ).
% disjoint_iff_not_equal
tff(fact_247_Int__empty__right,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),bot_bot(set(A))) = bot_bot(set(A)) ).
% Int_empty_right
tff(fact_248_Int__empty__left,axiom,
! [A: $tType,Ba: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),Ba) = bot_bot(set(A)) ).
% Int_empty_left
tff(fact_249_disjoint__iff,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ~ member(A,X4,Ba) ) ) ).
% disjoint_iff
tff(fact_250_Int__emptyI,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ! [X3: A] :
( member(A,X3,Aa2)
=> ~ member(A,X3,Ba) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) ) ) ).
% Int_emptyI
tff(fact_251_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_252_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_253_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_254_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_255_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_256_distrib__imp1,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y2),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Z3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),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_257_distrib__imp2,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y2),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Z3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),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_258_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_259_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_260_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_261_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_262_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod($o,product_prod(A,set(B)))] :
~ ! [B3: $o,H3: A,As3: set(B)] : X != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(B3)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3)) ).
% pure_assn_raw.cases
tff(fact_263_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semigroup(A,inf_inf(A)) ) ).
% inf.semigroup_axioms
tff(fact_264_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_265_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.monoid_axioms
tff(fact_266_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).
% inf_top.semilattice_neutr_axioms
tff(fact_267_wand__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
& relH(As,H2,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))) ) ) ) ).
% wand_raw.simps
tff(fact_268_wand__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As3),As5) = bot_bot(set(nat)) )
& relH(As3,H3,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As5))) ) ) ) ) ) ).
% wand_raw.elims(1)
tff(fact_269_wand__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [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)),As3),As6) = bot_bot(set(nat)) )
& relH(As3,H3,H7)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As6)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As6))) ) ) ) ) ).
% wand_raw.elims(2)
tff(fact_270_wand__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [H4: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As3),As7) = bot_bot(set(nat)) )
& relH(As3,H3,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As7)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As7))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_271_mod__relH,axiom,
! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),P: assn] :
( relH(As,H2,H5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As)) ) ) ).
% mod_relH
tff(fact_272_relH__refl,axiom,
! [H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> relH(As,H2,H2) ) ).
% relH_refl
tff(fact_273_relH__in__rangeI_I1_J,axiom,
! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As,H2,H5)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ).
% relH_in_rangeI(1)
tff(fact_274_relH__in__rangeI_I2_J,axiom,
! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As,H2,H5)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As)) ) ).
% relH_in_rangeI(2)
tff(fact_275_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_276_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))] :
( ! [A4: A,B3: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C3),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E2),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3)))))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),$o,P,X) ) ).
% prod_induct7
tff(fact_277_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
( ! [A4: A,B3: B,C3: C,D2: D,E2: E,F3: F2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F3))))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),$o,P,X) ) ).
% prod_induct6
tff(fact_278_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
( ! [A4: A,B3: B,C3: C,D2: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))))
=> aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,X) ) ).
% prod_induct5
tff(fact_279_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
( ! [A4: A,B3: B,C3: C,D2: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))))
=> aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,X) ) ).
% prod_induct4
tff(fact_280_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),$o),X: product_prod(A,product_prod(B,C))] :
( ! [A4: A,B3: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)))
=> aa(product_prod(A,product_prod(B,C)),$o,P,X) ) ).
% prod_induct3
tff(fact_281_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))] :
~ ! [A4: A,B3: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C3),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E2),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3)))))) ).
% prod_cases7
tff(fact_282_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A4: A,B3: B,C3: C,D2: D,E2: E,F3: F2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F3))))) ).
% prod_cases6
tff(fact_283_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A4: A,B3: B,C3: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).
% prod_cases5
tff(fact_284_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A4: A,B3: B,C3: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).
% prod_cases4
tff(fact_285_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A4: A,B3: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).
% prod_cases3
tff(fact_286_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
=> ~ ( ( A3 = A6 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
tff(fact_287_prod__cases,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P5: product_prod(A,B)] :
( ! [A4: A,B3: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> aa(product_prod(A,B),$o,P,P5) ) ).
% prod_cases
tff(fact_288_surj__pair,axiom,
! [A: $tType,B: $tType,P5: product_prod(A,B)] :
? [X3: A,Y2: B] : P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) ).
% surj_pair
tff(fact_289_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A4: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) ).
% old.prod.exhaust
tff(fact_290_Un__empty__left,axiom,
! [A: $tType,Ba: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),Ba) = Ba ).
% Un_empty_left
tff(fact_291_Un__empty__right,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),bot_bot(set(A))) = Aa2 ).
% Un_empty_right
tff(fact_292_empty__not__UNIV,axiom,
! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).
% empty_not_UNIV
tff(fact_293_bot__set__def,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).
% bot_set_def
tff(fact_294_ex__in__conv,axiom,
! [A: $tType,Aa2: set(A)] :
( ? [X4: A] : member(A,X4,Aa2)
<=> ( Aa2 != bot_bot(set(A)) ) ) ).
% ex_in_conv
tff(fact_295_equals0I,axiom,
! [A: $tType,Aa2: set(A)] :
( ! [Y2: A] : ~ member(A,Y2,Aa2)
=> ( Aa2 = bot_bot(set(A)) ) ) ).
% equals0I
tff(fact_296_equals0D,axiom,
! [A: $tType,Aa2: set(A),A3: A] :
( ( Aa2 = bot_bot(set(A)) )
=> ~ member(A,A3,Aa2) ) ).
% equals0D
tff(fact_297_emptyE,axiom,
! [A: $tType,A3: A] : ~ member(A,A3,bot_bot(set(A))) ).
% emptyE
tff(fact_298_wand__assnI,axiom,
! [H2: heap_ext(product_unit),As: set(nat),Q: assn,R: assn] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> ( ! [H4: heap_ext(product_unit),As7: set(nat)] :
( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
=> ( relH(As,H2,H4)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As7))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,R),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7))) ) ) ) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,wand_assn(Q,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).
% wand_assnI
tff(fact_299_times__assn__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(P,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22)) ) ) ).
% times_assn_raw.simps
tff(fact_300_times__assn__raw_Oelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ~ ? [As1: set(nat),As22: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As22)) ) ) ) ) ).
% times_assn_raw.elims(1)
tff(fact_301_times__assn__raw_Oelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ~ ? [As12: set(nat),As23: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As12))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As23)) ) ) ) ).
% times_assn_raw.elims(2)
tff(fact_302_times__assn__raw_Oelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ? [As13: set(nat),As24: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As13))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As24)) ) ) ) ).
% times_assn_raw.elims(3)
tff(fact_303_properI,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( ! [As3: set(nat),H3: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)) )
=> ( ! [As3: set(nat),H3: heap_ext(product_unit),H4: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> ( relH(As3,H3,H4)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3)) ) ) )
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P) ) ) ).
% properI
tff(fact_304_properD2,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat),H5: heap_ext(product_unit)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> ( relH(As,H2,H5)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,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),As)) ) ) ) ) ).
% properD2
tff(fact_305_old_Oprod_Orec,axiom,
! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A3: B,B2: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F1,A3),B2) ).
% old.prod.rec
tff(fact_306_disjointI,axiom,
! [A: $tType,A3: set(A),B2: set(A)] :
( ! [X3: A] :
( member(A,X3,A3)
=> ~ member(A,X3,B2) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ).
% disjointI
tff(fact_307_wand__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [H4: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As3),As7) = bot_bot(set(nat)) )
& relH(As3,H3,H4)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As7)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As7))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_308_wand__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ~ ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [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)),As3),As6) = bot_bot(set(nat)) )
& relH(As3,H3,H7)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As6)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As6))) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
tff(fact_309_wand__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,wand_raw(X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( ( (Y)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
& ! [H6: heap_ext(product_unit),As5: set(nat)] :
( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As3),As5) = bot_bot(set(nat)) )
& relH(As3,H3,H6)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As3))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As5)) )
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As5))) ) ) )
=> ~ aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)))) ) ) ) ) ).
% wand_raw.pelims(1)
tff(fact_310_Set_Ois__empty__def,axiom,
! [A: $tType,Aa2: set(A)] :
( is_empty2(A,Aa2)
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% Set.is_empty_def
tff(fact_311_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A3: B,B2: C] : produc5280177257484947105e_prod(B,C,A,C2,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C2,A3),B2) ).
% internal_case_prod_conv
tff(fact_312_times__assn__raw_Opelims_I3_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ? [As13: set(nat),As24: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As13))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As24)) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
tff(fact_313_times__assn__raw_Opelims_I2_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ~ ? [As12: set(nat),As23: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As12))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As23)) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
tff(fact_314_times__assn__raw_Opelims_I1_J,axiom,
! [X: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,times_assn_raw(X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( ( (Y)
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As1))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As22)) ) )
=> ~ aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)))) ) ) ) ) ).
% times_assn_raw.pelims(1)
tff(fact_315_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_316_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
<=> ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_317_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_318_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_319_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_320_uminus__apply,axiom,
! [A: $tType,B: $tType] :
( uminus(A)
=> ! [Aa2: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),Aa2),X) = aa(A,A,uminus_uminus(A),aa(B,A,Aa2,X)) ) ).
% uminus_apply
tff(fact_321_merge__pure__and,axiom,
! [A3: $o,B2: $o] :
aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((A3))),pure_assn((B2))) = pure_assn(( (A3)
& (B2) )) ).
% merge_pure_and
tff(fact_322_Compl__disjoint2,axiom,
! [A: $tType,Aa2: 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)),Aa2)),Aa2) = bot_bot(set(A)) ).
% Compl_disjoint2
tff(fact_323_Compl__disjoint,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)) = bot_bot(set(A)) ).
% Compl_disjoint
tff(fact_324_mod__not__dist,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,uminus_uminus(assn),P)),H2)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,H2)
& ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2) ) ) ).
% mod_not_dist
tff(fact_325_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).
% boolean_algebra.compl_zero
tff(fact_326_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).
% boolean_algebra.compl_one
tff(fact_327_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_328_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_329_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_330_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_331_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_332_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).
% boolean_algebra.disj_cancel_right
tff(fact_333_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).
% boolean_algebra.disj_cancel_left
tff(fact_334_sup__compl__top__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).
% sup_compl_top_left2
tff(fact_335_sup__compl__top__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).
% sup_compl_top_left1
tff(fact_336_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_337_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% boolean_algebra.de_Morgan_conj
tff(fact_338_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(6)
tff(fact_339_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_340_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_341_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( uminus(B)
=> ! [Aa2: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),Aa2),X2) = aa(B,B,uminus_uminus(B),aa(A,B,Aa2,X2)) ) ).
% fun_Compl_def
tff(fact_342_Compl__UNIV__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).
% Compl_UNIV_eq
tff(fact_343_Compl__empty__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).
% Compl_empty_eq
tff(fact_344_mod__and__dist,axiom,
! [P: assn,Q: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q)),H2)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H2)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Q),H2) ) ) ).
% mod_and_dist
tff(fact_345_ent__conjE2,axiom,
! [Ba: assn,Ca: assn,Aa2: assn] :
( entails(Ba,Ca)
=> entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Aa2),Ba),Ca) ) ).
% ent_conjE2
tff(fact_346_ent__conjE1,axiom,
! [Aa2: assn,Ca: assn,Ba: assn] :
( entails(Aa2,Ca)
=> entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Aa2),Ba),Ca) ) ).
% ent_conjE1
tff(fact_347_ent__conjI,axiom,
! [Aa2: assn,Ba: assn,Ca: assn] :
( entails(Aa2,Ba)
=> ( entails(Aa2,Ca)
=> entails(Aa2,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),Ba),Ca)) ) ) ).
% ent_conjI
tff(fact_348_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_349_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_350_sup__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).
% sup_cancel_left2
tff(fact_351_sup__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).
% sup_cancel_left1
tff(fact_352_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
~ ! [F3: fun(A,B),A4: A,B3: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)) ).
% pairself.cases
tff(fact_353_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S)
=> ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S)
=> aa(B,$o,aa(A,fun(B,$o),P,A3),B2) )
=> ? [A4: A,B3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S)
& aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) ) ).
% bex2I
tff(fact_354_set__notEmptyE,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ~ ! [X3: A] : ~ member(A,X3,S) ) ).
% set_notEmptyE
tff(fact_355_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set(A)] :
( member(A,X,S)
=> ( S != bot_bot(set(A)) ) ) ).
% memb_imp_not_empty
tff(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_bot__empty__eq,axiom,
! [A: $tType,X2: A] :
( aa(A,$o,bot_bot(fun(A,$o)),X2)
<=> member(A,X2,bot_bot(set(A))) ) ).
% bot_empty_eq
tff(fact_363_Collect__empty__eq__bot,axiom,
! [A: $tType,P: fun(A,$o)] :
( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
<=> ( P = bot_bot(fun(A,$o)) ) ) ).
% Collect_empty_eq_bot
tff(fact_364_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_365_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_366_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_367_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_368_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_369_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : uncurry(B,C,A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% uncurry_apply
tff(fact_370_pairself_Opelims,axiom,
! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
=> ( aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),Xa))
=> ~ ! [A4: B,B3: B] :
( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3) )
=> ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A4)),aa(B,A,X,B3)) )
=> ~ aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),X),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3))) ) ) ) ) ).
% pairself.pelims
tff(fact_371_mod__pure,axiom,
! [B2: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,pure_assn((B2))),H2)
<=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) )
& (B2) ) ) ).
% mod_pure
tff(fact_372_type__definition__assn,axiom,
type_definition(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,abs_assn,aa(fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o),set(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),collect(fun(product_prod(heap_ext(product_unit),set(nat)),$o)),proper)) ).
% type_definition_assn
tff(fact_373_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod(A,set(B)),Y: $o] :
( ( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
<=> (Y) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H3: A,As3: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3) )
=> ( ( (Y)
<=> ( ( As3 = bot_bot(set(B)) )
& (X) ) )
=> ~ aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3))) ) ) ) ) ).
% pure_assn_raw.pelims(1)
tff(fact_374_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H3: A,As3: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3)))
=> ~ ( ( As3 = bot_bot(set(B)) )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
tff(fact_375_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod(A,set(B))] :
( ~ aa(product_prod(A,set(B)),$o,pure_assn_raw(A,B,(X)),Xa)
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),Xa))
=> ~ ! [H3: A,As3: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3) )
=> ( aa(product_prod($o,product_prod(A,set(B))),$o,accp(product_prod($o,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(X)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H3),As3)))
=> ( ( As3 = bot_bot(set(B)) )
& (X) ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
tff(fact_376_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).
% dbl_dec_simps(3)
tff(fact_377_mod__emp,axiom,
! [H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,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_378_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),X)),aa(A,A,aa(A,fun(A,A),Disj,Z2),X)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_379_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_380_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,Compl,X) = aa(A,A,Compl,Y) )
<=> ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_381_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,X),Z2)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
tff(fact_382_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
tff(fact_383_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
tff(fact_384_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,X)) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
tff(fact_385_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),X) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),X) = One )
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Y) = One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
tff(fact_386_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
tff(fact_387_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
tff(fact_388_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),Zero) = X ) ) ).
% abstract_boolean_algebra.disj_zero_right
tff(fact_389_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),Zero) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
tff(fact_390_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),One) = One ) ) ).
% abstract_boolean_algebra.disj_one_right
tff(fact_391_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,X),Y)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
tff(fact_392_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,X),Y)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
tff(fact_393_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,Zero),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
tff(fact_394_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),One) = X ) ) ).
% abstract_boolean_algebra.conj_one_right
tff(fact_395_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,One),X) = One ) ) ).
% abstract_boolean_algebra.disj_one_left
tff(fact_396_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,Compl,X)) = X ) ) ).
% abstract_boolean_algebra.double_compl
tff(fact_397_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,X),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,X),Y) = One )
=> ( aa(A,A,Compl,X) = Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
tff(fact_398_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,Zero) = One ) ) ).
% abstract_boolean_algebra.compl_zero
tff(fact_399_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,One) = Zero ) ) ).
% abstract_boolean_algebra.compl_one
tff(fact_400_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(B,$o)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),X))
=> aa(B,$o,P,B2) ) ) ).
% sndE
tff(fact_401_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
tff(fact_402_snd__conv,axiom,
! [B: $tType,A: $tType,X1: B,X22: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X22)) = X22 ).
% snd_conv
tff(fact_403_pairself_Oelims,axiom,
! [A: $tType,B: $tType,X: fun(B,A),Xa: product_prod(B,B),Y: product_prod(A,A)] :
( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,X),Xa) = Y )
=> ~ ! [A4: B,B3: B] :
( ( Xa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3) )
=> ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,X,A4)),aa(B,A,X,B3)) ) ) ) ).
% pairself.elims
tff(fact_404_pairself_Osimps,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A3: B,B2: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A3),B2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F,A3)),aa(B,A,F,B2)) ).
% pairself.simps
tff(fact_405_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z2: B] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z2 ) ) ).
% sndI
tff(fact_406_eq__snd__iff,axiom,
! [B: $tType,A: $tType,B2: A,P5: product_prod(B,A)] :
( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P5) )
<=> ? [A5: B] : P5 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2) ) ).
% eq_snd_iff
tff(fact_407_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_408_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_409_comm__monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
<=> ( abel_semigroup(A,F)
& comm_monoid_axioms(A,F,Z2) ) ) ).
% comm_monoid_def
tff(fact_410_comm__monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( abel_semigroup(A,F)
=> ( comm_monoid_axioms(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ) ).
% comm_monoid.intro
tff(fact_411_disjoint__eq__subset__Compl,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Ba)) ) ).
% disjoint_eq_subset_Compl
tff(fact_412_semilattice__neutr__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
<=> ( semilattice(A,F)
& comm_monoid(A,F,Z2) ) ) ).
% semilattice_neutr_def
tff(fact_413_semilattice__neutr_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice(A,F)
=> ( comm_monoid(A,F,Z2)
=> semilattice_neutr(A,F,Z2) ) ) ).
% semilattice_neutr.intro
tff(fact_414_minus__assn__def,axiom,
! [A3: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B2)) ).
% minus_assn_def
tff(fact_415_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra.axioms(3)
tff(fact_416_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X) ) ).
% order_refl
tff(fact_417_dual__order_Orefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).
% dual_order.refl
tff(fact_418_minus__apply,axiom,
! [A: $tType,B: $tType] :
( minus(A)
=> ! [Aa2: fun(B,A),Ba: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),Aa2),Ba),X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Aa2,X)),aa(B,A,Ba,X)) ) ).
% minus_apply
tff(fact_419_le__zero__eq,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),zero_zero(A))
<=> ( N = zero_zero(A) ) ) ) ).
% le_zero_eq
tff(fact_420_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_421_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_422_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_423_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_424_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_425_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).
% diff_self
tff(fact_426_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).
% diff_0_right
tff(fact_427_zero__diff,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% zero_diff
tff(fact_428_diff__zero,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).
% diff_zero
tff(fact_429_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_430_neg__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% neg_le_iff_le
tff(fact_431_compl__le__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% compl_le_compl_iff
tff(fact_432_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_433_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_434_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_435_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_436_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_437_le__inf__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).
% le_inf_iff
tff(fact_438_inf_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% inf.bounded_iff
tff(fact_439_le__sup__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% le_sup_iff
tff(fact_440_sup_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).
% sup.bounded_iff
tff(fact_441_minus__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).
% minus_diff_eq
tff(fact_442_empty__subsetI,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),Aa2) ).
% empty_subsetI
tff(fact_443_subset__empty,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),bot_bot(set(A)))
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% subset_empty
tff(fact_444_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% diff_ge_0_iff_ge
tff(fact_445_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% neg_0_le_iff_le
tff(fact_446_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).
% neg_le_0_iff_le
tff(fact_447_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% less_eq_neg_nonpos
tff(fact_448_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).
% neg_less_eq_nonneg
tff(fact_449_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_450_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_451_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_452_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_453_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).
% diff_numeral_special(9)
tff(fact_454_diff__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).
% diff_0
tff(fact_455_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,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_456_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> abel_semigroup(A,F) ) ).
% semilattice.axioms(1)
tff(fact_457_semilattice_Oidem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),A3) = A3 ) ) ).
% semilattice.idem
tff(fact_458_semilattice_Oleft__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.left_idem
tff(fact_459_semilattice_Oright__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.right_idem
tff(fact_460_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( minus(B)
=> ! [Aa2: fun(A,B),Ba: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),Aa2),Ba),X2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Aa2,X2)),aa(A,B,Ba,X2)) ) ).
% fun_diff_def
tff(fact_461_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).
% abel_semigroup.commute
tff(fact_462_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),B2: A,A3: A,C2: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C2)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).
% abel_semigroup.left_commute
tff(fact_463_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( ( A3 = B2 )
<=> ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_464_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).
% eq_iff_diff_eq_0
tff(fact_465_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).
% diff_mono
tff(fact_466_diff__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).
% diff_left_mono
tff(fact_467_diff__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).
% diff_right_mono
tff(fact_468_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A)) ) ) ).
% le_iff_diff_le_0
tff(fact_469_zero__le,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ).
% zero_le
tff(fact_470_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_471_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_472_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_473_nle__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& ( B2 != A3 ) ) ) ) ).
% nle_le
tff(fact_474_le__cases3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X) )
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ) ) ) ).
% le_cases3
tff(fact_475_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ( X = Y )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).
% order_class.order_eq_iff
tff(fact_476_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% ord_eq_le_trans
tff(fact_477_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( ( B2 = C2 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% ord_le_eq_trans
tff(fact_478_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( X = Y ) ) ) ) ).
% order_antisym
tff(fact_479_order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% order.trans
tff(fact_480_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2) ) ) ) ).
% order_trans
tff(fact_481_linorder__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
=> aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ).
% linorder_wlog
tff(fact_482_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% dual_order.eq_iff
tff(fact_483_dual__order_Oantisym,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
tff(fact_484_dual__order_Otrans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).
% dual_order.trans
tff(fact_485_antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( A3 = B2 ) ) ) ) ).
% antisym
tff(fact_486_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ) ).
% le_funD
tff(fact_487_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ) ).
% le_funE
tff(fact_488_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,G,X3))
=> aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G) ) ) ).
% le_funI
tff(fact_489_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
<=> ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,G,X4)) ) ) ).
% le_fun_def
tff(fact_490_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).
% Orderings.order_eq_iff
tff(fact_491_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,B2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% order_subst1
tff(fact_492_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% order_subst2
tff(fact_493_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( ( X = Y )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% order_eq_refl
tff(fact_494_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_linear
tff(fact_495_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% ord_eq_le_subst
tff(fact_496_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( ( aa(A,B,F,B2) = C2 )
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% ord_le_eq_subst
tff(fact_497_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_le_cases
tff(fact_498_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% order_antisym_conv
tff(fact_499_diff__shunt__var,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% diff_shunt_var
tff(fact_500_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_mono
tff(fact_501_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_mono'
tff(fact_502_zero__le__square,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).
% zero_le_square
tff(fact_503_split__mult__pos__le,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,B2: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ).
% split_mult_pos_le
tff(fact_504_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).
% mult_left_mono_neg
tff(fact_505_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% mult_nonpos_nonpos
tff(fact_506_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).
% mult_left_mono
tff(fact_507_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% mult_right_mono_neg
tff(fact_508_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% mult_right_mono
tff(fact_509_mult__le__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).
% mult_le_0_iff
tff(fact_510_split__mult__neg__le,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
=> aa(A,$o,aa(A,fun(A,$o),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_511_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% mult_nonneg_nonneg
tff(fact_512_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).
% mult_nonneg_nonpos
tff(fact_513_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_514_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ordered_semiring_0(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A)) ) ) ) ).
% mult_nonneg_nonpos2
tff(fact_515_zero__le__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).
% zero_le_mult_iff
tff(fact_516_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ordere2520102378445227354miring(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_517_not__one__le__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).
% not_one_le_zero
tff(fact_518_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_519_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).
% zero_less_one_class.zero_le_one
tff(fact_520_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
<=> ( ! [X4: A,Y3: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,aa(A,fun(A,A),Disj,Y3),Z4)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Conj,X4),Z4))
& ! [X4: A,Y3: A,Z4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,aa(A,fun(A,A),Conj,Y3),Z4)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X4),Y3)),aa(A,A,aa(A,fun(A,A),Disj,X4),Z4))
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),One) = X4
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),Zero) = X4
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,X4)) = Zero
& ! [X4: A] : aa(A,A,aa(A,fun(A,A),Disj,X4),aa(A,A,Compl,X4)) = One ) ) ).
% abstract_boolean_algebra_axioms_def
tff(fact_521_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),One: A,Zero: A,Compl: fun(A,A)] :
( ! [X3: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Conj,X3),aa(A,A,aa(A,fun(A,A),Disj,Y2),Z3)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X3),Y2)),aa(A,A,aa(A,fun(A,A),Conj,X3),Z3))
=> ( ! [X3: A,Y2: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Disj,X3),aa(A,A,aa(A,fun(A,A),Conj,Y2),Z3)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X3),Y2)),aa(A,A,aa(A,fun(A,A),Disj,X3),Z3))
=> ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),Conj,X3),One) = X3
=> ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),Disj,X3),Zero) = X3
=> ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),Conj,X3),aa(A,A,Compl,X3)) = Zero
=> ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),Disj,X3),aa(A,A,Compl,X3)) = One
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
tff(fact_522_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( abel_semigroup(A,Conj)
=> ( abel_semigroup(A,Disj)
=> ( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ) ) ).
% abstract_boolean_algebra.intro
tff(fact_523_abstract__boolean__algebra__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
<=> ( abel_semigroup(A,Conj)
& abel_semigroup(A,Disj)
& boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ).
% abstract_boolean_algebra_def
tff(fact_524_mult__left__le,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3) ) ) ) ).
% mult_left_le
tff(fact_525_mult__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)) ) ) ) ) ).
% mult_le_one
tff(fact_526_mult__right__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X) ) ) ) ) ).
% mult_right_le_one_le
tff(fact_527_mult__left__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X) ) ) ) ) ).
% mult_left_le_one_le
tff(fact_528_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).
% le_minus_one_simps(1)
tff(fact_529_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% le_minus_one_simps(3)
tff(fact_530_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),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_531_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),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_532_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),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_533_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),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_534_minus__diff__commute,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% minus_diff_commute
tff(fact_535_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).
% le_numeral_extra(4)
tff(fact_536_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_uniqueI
tff(fact_537_bot_Oextremum__unique,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
<=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_unique
tff(fact_538_bot_Oextremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A3) ) ).
% bot.extremum
tff(fact_539_subrelI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( ! [X3: A,Y2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),S2) )
=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).
% subrelI
tff(fact_540_le__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).
% le_minus_iff
tff(fact_541_minus__le__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).
% minus_le_iff
tff(fact_542_le__imp__neg__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).
% le_imp_neg_le
tff(fact_543_compl__mono,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X)) ) ) ).
% compl_mono
tff(fact_544_compl__le__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).
% compl_le_swap1
tff(fact_545_compl__le__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).
% compl_le_swap2
tff(fact_546_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_uniqueI
tff(fact_547_top_Oextremum__unique,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
<=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_unique
tff(fact_548_top__greatest,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),top_top(A)) ) ).
% top_greatest
tff(fact_549_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).
% inf_sup_ord(2)
tff(fact_550_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).
% inf_sup_ord(1)
tff(fact_551_inf__le1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X) ) ).
% inf_le1
tff(fact_552_inf__le2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) ) ).
% inf_le2
tff(fact_553_le__infE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2) ) ) ) ).
% le_infE
tff(fact_554_le__infI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% le_infI
tff(fact_555_inf__mono,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
=> aa(A,$o,aa(A,fun(A,$o),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_556_le__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).
% le_infI1
tff(fact_557_le__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).
% le_infI2
tff(fact_558_inf_OorderE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).
% inf.orderE
tff(fact_559_inf_OorderI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% inf.orderI
tff(fact_560_inf__unique,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X3: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X3),Y2)),X3)
=> ( ! [X3: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X3),Y2)),Y2)
=> ( ! [X3: A,Y2: A,Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)) ) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% inf_unique
tff(fact_561_le__iff__inf,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% le_iff_inf
tff(fact_562_inf_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb1
tff(fact_563_inf_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb2
tff(fact_564_inf__absorb1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% inf_absorb1
tff(fact_565_inf__absorb2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).
% inf_absorb2
tff(fact_566_inf_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% inf.boundedE
tff(fact_567_inf_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).
% inf.boundedI
tff(fact_568_inf__greatest,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ) ) ).
% inf_greatest
tff(fact_569_inf_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_570_inf_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3) ) ).
% inf.cobounded1
tff(fact_571_inf_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) ) ).
% inf.cobounded2
tff(fact_572_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_573_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_574_inf_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).
% inf.coboundedI1
tff(fact_575_inf_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).
% inf.coboundedI2
tff(fact_576_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).
% inf_sup_ord(4)
tff(fact_577_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).
% inf_sup_ord(3)
tff(fact_578_le__supE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X) ) ) ) ).
% le_supE
tff(fact_579_le__supI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,X: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X) ) ) ) ).
% le_supI
tff(fact_580_sup__ge1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).
% sup_ge1
tff(fact_581_sup__ge2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).
% sup_ge2
tff(fact_582_le__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% le_supI1
tff(fact_583_le__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% le_supI2
tff(fact_584_sup_Omono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,D3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_585_sup__mono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
=> aa(A,$o,aa(A,fun(A,$o),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_586_sup__least,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) ) ) ) ).
% sup_least
tff(fact_587_le__iff__sup,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% le_iff_sup
tff(fact_588_sup_OorderE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).
% sup.orderE
tff(fact_589_sup_OorderI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% sup.orderI
tff(fact_590_sup__unique,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X3: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F,X3),Y2))
=> ( ! [X3: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F,X3),Y2))
=> ( ! [X3: A,Y2: A,Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,Y2),Z3)),X3) ) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% sup_unique
tff(fact_591_sup_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb1
tff(fact_592_sup_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb2
tff(fact_593_sup__absorb1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).
% sup_absorb1
tff(fact_594_sup__absorb2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% sup_absorb2
tff(fact_595_sup_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).
% sup.boundedE
tff(fact_596_sup_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3) ) ) ) ).
% sup.boundedI
tff(fact_597_sup_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_598_sup_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).
% sup.cobounded1
tff(fact_599_sup_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ).
% sup.cobounded2
tff(fact_600_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_601_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_602_sup_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% sup.coboundedI1
tff(fact_603_sup_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% sup.coboundedI2
tff(fact_604_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_605_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_606_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_607_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_608_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_609_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_610_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_611_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> abel_semigroup(A,inf_inf(A)) ) ).
% inf.abel_semigroup_axioms
tff(fact_612_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> abel_semigroup(A,sup_sup(A)) ) ).
% sup.abel_semigroup_axioms
tff(fact_613_relH__subset,axiom,
! [Bs: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),As: set(nat)] :
( relH(Bs,H2,H5)
=> ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),Bs)
=> relH(As,H2,H5) ) ) ).
% relH_subset
tff(fact_614_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice(A,inf_inf(A)) ) ).
% inf.semilattice_axioms
tff(fact_615_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice(A,sup_sup(A)) ) ).
% sup.semilattice_axioms
tff(fact_616_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Disj) ) ).
% abstract_boolean_algebra.axioms(2)
tff(fact_617_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Conj) ) ).
% abstract_boolean_algebra.axioms(1)
tff(fact_618_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> abel_semigroup(A,F) ) ).
% comm_monoid.axioms(1)
tff(fact_619_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> semigroup(A,F) ) ).
% abel_semigroup.axioms(1)
tff(fact_620_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> semilattice(A,F) ) ).
% semilattice_neutr.axioms(1)
tff(fact_621_diff__eq,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,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_622_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).
% le_minus_one_simps(2)
tff(fact_623_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% le_minus_one_simps(4)
tff(fact_624_distrib__sup__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),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_625_distrib__inf__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,$o,aa(A,fun(A,$o),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_626_disjoint__mono,axiom,
! [A: $tType,A3: set(A),A6: set(A),B2: set(A),B5: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A6)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),B5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ) ) ).
% disjoint_mono
tff(fact_627_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_628_subset__Compl__self__eq,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Aa2))
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% subset_Compl_self_eq
tff(fact_629_in__range__subset,axiom,
! [As: set(nat),As2: set(nat),H2: heap_ext(product_unit)] :
( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),As),As2)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) ) ) ).
% in_range_subset
tff(fact_630_inf__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).
% inf_shunt
tff(fact_631_sup__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).
% sup_shunt
tff(fact_632_sup__neg__inf,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [P5: A,Q3: A,R2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P5),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P5),aa(A,A,uminus_uminus(A),Q3))),R2) ) ) ).
% sup_neg_inf
tff(fact_633_shunt2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ) ).
% shunt2
tff(fact_634_shunt1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),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_635_subset__emptyI,axiom,
! [A: $tType,Aa2: set(A)] :
( ! [X3: A] : ~ member(A,X3,Aa2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),bot_bot(set(A))) ) ).
% subset_emptyI
tff(fact_636_inf__period_I2_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
( ! [X3: A,K2: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
=> ( ! [X3: A,K2: A] :
( aa(A,$o,Q,X3)
<=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
=> ! [X2: A,K3: A] :
( ( aa(A,$o,P,X2)
| aa(A,$o,Q,X2) )
<=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))
| aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) ) ) ) ) ) ).
% inf_period(2)
tff(fact_637_inf__period_I1_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
( ! [X3: A,K2: A] :
( aa(A,$o,P,X3)
<=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
=> ( ! [X3: A,K2: A] :
( aa(A,$o,Q,X3)
<=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4))) )
=> ! [X2: A,K3: A] :
( ( aa(A,$o,P,X2)
& aa(A,$o,Q,X2) )
<=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))
& aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))) ) ) ) ) ) ).
% inf_period(1)
tff(fact_638_abel__semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
=> ( abel_s757365448890700780axioms(A,F)
=> abel_semigroup(A,F) ) ) ).
% abel_semigroup.intro
tff(fact_639_abel__semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
<=> ( semigroup(A,F)
& abel_s757365448890700780axioms(A,F) ) ) ).
% abel_semigroup_def
tff(fact_640_semilattice_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> ( semilattice_axioms(A,F)
=> semilattice(A,F) ) ) ).
% semilattice.intro
tff(fact_641_semilattice__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
<=> ( abel_semigroup(A,F)
& semilattice_axioms(A,F) ) ) ).
% semilattice_def
tff(fact_642_mod__star__trueE_H,axiom,
! [P: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn))),H2)
=> ~ ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H4) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H2) )
=> ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H4)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2))
=> ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),H4) ) ) ) ).
% mod_star_trueE'
tff(fact_643_divides__aux__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q3: A,R2: A] :
( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2))
<=> ( R2 = zero_zero(A) ) ) ) ).
% divides_aux_eq
tff(fact_644_mult__le__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).
% mult_le_cancel_left1
tff(fact_645_mult__le__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).
% mult_le_cancel_left2
tff(fact_646_Diff__empty,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),bot_bot(set(A))) = Aa2 ).
% Diff_empty
tff(fact_647_empty__Diff,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),Aa2) = bot_bot(set(A)) ).
% empty_Diff
tff(fact_648_Diff__cancel,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Aa2) = bot_bot(set(A)) ).
% Diff_cancel
tff(fact_649_not__gr__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N)
<=> ( N = zero_zero(A) ) ) ) ).
% not_gr_zero
tff(fact_650_compl__less__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% compl_less_compl_iff
tff(fact_651_neg__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% neg_less_iff_less
tff(fact_652_Diff__eq__empty__iff,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba) = bot_bot(set(A)) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba) ) ).
% Diff_eq_empty_iff
tff(fact_653_Diff__UNIV,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),top_top(set(A))) = bot_bot(set(A)) ).
% Diff_UNIV
tff(fact_654_Diff__disjoint,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ba),Aa2)) = bot_bot(set(A)) ).
% Diff_disjoint
tff(fact_655_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).
% diff_gt_0_iff_gt
tff(fact_656_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).
% neg_less_0_iff_less
tff(fact_657_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% neg_0_less_iff_less
tff(fact_658_neg__less__pos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).
% neg_less_pos
tff(fact_659_less__neg__neg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% less_neg_neg
tff(fact_660_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).
% prod.collapse
tff(fact_661_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F),G)
<=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
& ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F) ) ) ) ).
% less_fun_def
tff(fact_662_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_imp_not_less
tff(fact_663_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
tff(fact_664_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
tff(fact_665_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y )
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_less_linear
tff(fact_666_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,P: $o] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
=> (P) ) ) ) ).
% order_less_imp_triv
tff(fact_667_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_not_sym
tff(fact_668_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% order_less_subst2
tff(fact_669_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,B2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% order_less_subst1
tff(fact_670_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X) ) ).
% order_less_irrefl
tff(fact_671_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( ( aa(A,B,F,B2) = C2 )
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% ord_less_eq_subst
tff(fact_672_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% ord_eq_less_subst
tff(fact_673_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_less_trans
tff(fact_674_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).
% order_less_asym'
tff(fact_675_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_neq_iff
tff(fact_676_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% order_less_asym
tff(fact_677_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_neqE
tff(fact_678_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_679_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
tff(fact_680_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).
% dual_order.strict_trans
tff(fact_681_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_682_order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% order.strict_trans
tff(fact_683_linorder__less__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,$o)),A3: A,B2: A] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
=> ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
=> aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
=> aa(A,$o,aa(A,fun(A,$o),P,A3),B2) ) ) ) ) ).
% linorder_less_wlog
tff(fact_684_exists__least__iff,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o)] :
( ? [X_12: A] : aa(A,$o,P,X_12)
<=> ? [N2: A] :
( aa(A,$o,P,N2)
& ! [M: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),N2)
=> ~ aa(A,$o,P,M) ) ) ) ) ).
% exists_least_iff
tff(fact_685_dual__order_Oirrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).
% dual_order.irrefl
tff(fact_686_dual__order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% dual_order.asym
tff(fact_687_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( ( X != Y )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).
% linorder_cases
tff(fact_688_antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv3
tff(fact_689_less__induct,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A3: A] :
( ! [X3: A] :
( ! [Y4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X3)
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A3) ) ) ).
% less_induct
tff(fact_690_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( ( B2 = C2 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% ord_less_eq_trans
tff(fact_691_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% ord_eq_less_trans
tff(fact_692_order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ).
% order.asym
tff(fact_693_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( X != Y ) ) ) ).
% less_imp_neq
tff(fact_694_dense,axiom,
! [A: $tType] :
( dense_order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ? [Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).
% dense
tff(fact_695_gt__ex,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] :
? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_1) ) ).
% gt_ex
tff(fact_696_lt__ex,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] :
? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X) ) ).
% lt_ex
tff(fact_697_semilattice__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),A4) = A4
=> semilattice_axioms(A,F) ) ).
% semilattice_axioms.intro
tff(fact_698_semilattice__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice_axioms(A,F)
<=> ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),A5) = A5 ) ).
% semilattice_axioms_def
tff(fact_699_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),F,A4),B3) = aa(A,A,aa(A,fun(A,A),F,B3),A4)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup_axioms.intro
tff(fact_700_abel__semigroup__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F)
<=> ! [A5: A,B4: A] : aa(A,A,aa(A,fun(A,A),F,A5),B4) = aa(A,A,aa(A,fun(A,A),F,B4),A5) ) ).
% abel_semigroup_axioms_def
tff(fact_701_disjoint__alt__simp3,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba)),Aa2)
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp3
tff(fact_702_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),A3: A,B2: B,P: fun(A,$o)] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),X))
=> aa(A,$o,P,A3) ) ) ).
% fstE
tff(fact_703_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A3 )
=> ( X = A3 ) ) ).
% fst_eqD
tff(fact_704_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).
% fst_conv
tff(fact_705_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z2: B] :
( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).
% fstI
tff(fact_706_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P5: product_prod(A,B)] :
( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P5) )
<=> ? [B4: B] : P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4) ) ).
% eq_fst_iff
tff(fact_707_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_708_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_le_less_linear
tff(fact_709_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% order_less_le_subst2
tff(fact_710_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,B2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% order_less_le_subst1
tff(fact_711_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B2)),C2)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,A3)),C2) ) ) ) ) ).
% order_le_less_subst2
tff(fact_712_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F,B2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
=> ( ! [X3: B,Y2: B] :
( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,X3)),aa(B,A,F,Y2)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F,C2)) ) ) ) ) ).
% order_le_less_subst1
tff(fact_713_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_less_le_trans
tff(fact_714_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2) ) ) ) ).
% order_le_less_trans
tff(fact_715_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% order_neq_le_trans
tff(fact_716_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( ( A3 != B2 )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% order_le_neq_trans
tff(fact_717_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% order_less_imp_le
tff(fact_718_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% linorder_not_less
tff(fact_719_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).
% linorder_not_le
tff(fact_720_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& ( X != Y ) ) ) ) ).
% order_less_le
tff(fact_721_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
| ( X = Y ) ) ) ) ).
% order_le_less
tff(fact_722_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% dual_order.strict_implies_order
tff(fact_723_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% order.strict_implies_order
tff(fact_724_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% dual_order.strict_iff_not
tff(fact_725_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).
% dual_order.strict_trans2
tff(fact_726_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).
% dual_order.strict_trans1
tff(fact_727_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
tff(fact_728_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
tff(fact_729_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( ! [W: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),W)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% dense_le_bounded
tff(fact_730_dense__ge__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
=> ( ! [W: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),W)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% dense_ge_bounded
tff(fact_731_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).
% order.strict_iff_not
tff(fact_732_order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% order.strict_trans2
tff(fact_733_order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% order.strict_trans1
tff(fact_734_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
tff(fact_735_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
tff(fact_736_not__le__imp__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).
% not_le_imp_less
tff(fact_737_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
& ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).
% less_le_not_le
tff(fact_738_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,Z2: A] :
( ! [X3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).
% dense_le
tff(fact_739_dense__ge,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y: A] :
( ! [X3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ).
% dense_ge
tff(fact_740_antisym__conv2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv2
tff(fact_741_antisym__conv1,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
<=> ( X = Y ) ) ) ) ).
% antisym_conv1
tff(fact_742_nless__le,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
| ( A3 = B2 ) ) ) ) ).
% nless_le
tff(fact_743_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).
% leI
tff(fact_744_leD,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).
% leD
tff(fact_745_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N)
<=> ( N != zero_zero(A) ) ) ) ).
% zero_less_iff_neq_zero
tff(fact_746_gr__implies__not__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [M2: A,N: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N)
=> ( N != zero_zero(A) ) ) ) ).
% gr_implies_not_zero
tff(fact_747_not__less__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),N),zero_zero(A)) ) ).
% not_less_zero
tff(fact_748_gr__zeroI,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ( N != zero_zero(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),N) ) ) ).
% gr_zeroI
tff(fact_749_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).
% less_numeral_extra(4)
tff(fact_750_diff__strict__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).
% diff_strict_right_mono
tff(fact_751_diff__strict__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).
% diff_strict_left_mono
tff(fact_752_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3) ) ) ) ).
% diff_eq_diff_less
tff(fact_753_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).
% diff_strict_mono
tff(fact_754_bot_Oextremum__strict,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),bot_bot(A)) ) ).
% bot.extremum_strict
tff(fact_755_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( ( A3 != bot_bot(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A3) ) ) ).
% bot.not_eq_extremum
tff(fact_756_not__psubset__empty,axiom,
! [A: $tType,Aa2: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),bot_bot(set(A))) ).
% not_psubset_empty
tff(fact_757_compl__less__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).
% compl_less_swap2
tff(fact_758_compl__less__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),X))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).
% compl_less_swap1
tff(fact_759_minus__less__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3) ) ) ).
% minus_less_iff
tff(fact_760_less__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3)) ) ) ).
% less_minus_iff
tff(fact_761_top_Oextremum__strict,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A3) ) ).
% top.extremum_strict
tff(fact_762_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( ( A3 != top_top(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),top_top(A)) ) ) ).
% top.not_eq_extremum
tff(fact_763_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).
% inf.strict_coboundedI2
tff(fact_764_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) ) ) ).
% inf.strict_coboundedI1
tff(fact_765_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_766_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% inf.strict_boundedE
tff(fact_767_inf_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb4
tff(fact_768_inf_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb3
tff(fact_769_less__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).
% less_infI2
tff(fact_770_less__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X) ) ) ).
% less_infI1
tff(fact_771_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% sup.strict_coboundedI2
tff(fact_772_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% sup.strict_coboundedI1
tff(fact_773_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_774_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).
% sup.strict_boundedE
tff(fact_775_sup_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb4
tff(fact_776_sup_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb3
tff(fact_777_less__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% less_supI2
tff(fact_778_less__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ).
% less_supI1
tff(fact_779_less__eq__assn__def,axiom,
! [A3: assn,B2: assn] :
( aa(assn,$o,aa(assn,fun(assn,$o),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_780_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% prod.exhaust_sel
tff(fact_781_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).
% surjective_pairing
tff(fact_782_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord2810124833399127020strict(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),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_783_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_784_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% mult_strict_right_mono
tff(fact_785_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% mult_strict_right_mono_neg
tff(fact_786_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_787_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).
% mult_strict_left_mono
tff(fact_788_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).
% mult_strict_left_mono_neg
tff(fact_789_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% mult_less_cancel_left_pos
tff(fact_790_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).
% mult_less_cancel_left_neg
tff(fact_791_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).
% zero_less_mult_pos2
tff(fact_792_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).
% zero_less_mult_pos
tff(fact_793_zero__less__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).
% zero_less_mult_iff
tff(fact_794_mult__pos__neg2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A)) ) ) ) ).
% mult_pos_neg2
tff(fact_795_mult__pos__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% mult_pos_pos
tff(fact_796_mult__pos__neg,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).
% mult_pos_neg
tff(fact_797_mult__neg__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)) ) ) ) ).
% mult_neg_pos
tff(fact_798_mult__less__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).
% mult_less_0_iff
tff(fact_799_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),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_800_mult__neg__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% mult_neg_neg
tff(fact_801_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).
% less_numeral_extra(1)
tff(fact_802_zero__less__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).
% zero_less_one
tff(fact_803_not__one__less__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).
% not_one_less_zero
tff(fact_804_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A)) ) ) ).
% less_iff_diff_less_0
tff(fact_805_subset__minus__empty,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba) = bot_bot(set(A)) ) ) ).
% subset_minus_empty
tff(fact_806_less__1__mult,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M2: A,N: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M2),N)) ) ) ) ).
% less_1_mult
tff(fact_807_disjoint__alt__simp2,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba) != Aa2 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp2
tff(fact_808_disjoint__alt__simp1,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba) = Aa2 )
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) ) ) ).
% disjoint_alt_simp1
tff(fact_809_Int__Diff__disjoint,axiom,
! [A: $tType,Aa2: set(A),Ba: 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)),Aa2),Ba)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba)) = bot_bot(set(A)) ).
% Int_Diff_disjoint
tff(fact_810_Diff__triv,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba) = Aa2 ) ) ).
% Diff_triv
tff(fact_811_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% less_minus_one_simps(4)
tff(fact_812_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).
% less_minus_one_simps(2)
tff(fact_813_mult__le__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).
% mult_le_cancel_left
tff(fact_814_mult__le__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).
% mult_le_cancel_right
tff(fact_815_mult__left__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% mult_left_less_imp_less
tff(fact_816_mult__strict__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_strict_mono
tff(fact_817_mult__less__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).
% mult_less_cancel_left
tff(fact_818_mult__right__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% mult_right_less_imp_less
tff(fact_819_mult__strict__mono_H,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_strict_mono'
tff(fact_820_mult__less__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).
% mult_less_cancel_right
tff(fact_821_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).
% mult_le_cancel_left_neg
tff(fact_822_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% mult_le_cancel_left_pos
tff(fact_823_mult__left__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% mult_left_le_imp_le
tff(fact_824_mult__right__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% mult_right_le_imp_le
tff(fact_825_mult__le__less__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_826_mult__less__le__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_827_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% less_minus_one_simps(3)
tff(fact_828_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).
% less_minus_one_simps(1)
tff(fact_829_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R: set(product_prod(A,B)),S3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S2),R)
=> ( ( S3 = S2 )
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S3),R) ) ) ).
% ssubst_Pair_rhs
tff(fact_830_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup.axioms(2)
tff(fact_831_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> semilattice_axioms(A,F) ) ).
% semilattice.axioms(2)
tff(fact_832_mult__less__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).
% mult_less_cancel_right2
tff(fact_833_mult__less__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).
% mult_less_cancel_right1
tff(fact_834_mult__less__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).
% mult_less_cancel_left2
tff(fact_835_mult__less__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).
% mult_less_cancel_left1
tff(fact_836_mult__le__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).
% mult_le_cancel_right2
tff(fact_837_mult__le__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).
% mult_le_cancel_right1
tff(fact_838_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).
% field_le_mult_one_interval
tff(fact_839_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mult_le_cancel_iff2
tff(fact_840_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mult_le_cancel_iff1
tff(fact_841_exI__realizer,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Y: A,X: B] :
( aa(B,$o,aa(A,fun(B,$o),P,Y),X)
=> aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))) ) ).
% exI_realizer
tff(fact_842_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: fun(A,$o),P5: A,Q: fun(B,$o),Q3: B] :
( aa(A,$o,P,P5)
=> ( aa(B,$o,Q,Q3)
=> ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P5),Q3)))
& aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P5),Q3))) ) ) ) ).
% conjI_realizer
tff(fact_843_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,A3: product_prod(A,B)] :
( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3)) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_844_mult__less__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).
% mult_less_iff1
tff(fact_845_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_846_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_847_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_top(A)
=> ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).
% top.ordering_top_axioms
tff(fact_848_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_849_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).
% sgn_pos
tff(fact_850_less__assn__def,axiom,
! [A3: assn,B2: assn] :
( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A3),B2)
<=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A3),B2)
& ( A3 != B2 ) ) ) ).
% less_assn_def
tff(fact_851_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
tff(fact_852_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( Z2 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
tff(fact_853_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),Top) ) ).
% ordering_top.extremum
tff(fact_854_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,Top),A3) ) ).
% ordering_top.extremum_strict
tff(fact_855_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
<=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
tff(fact_856_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( ( A3 != Top )
<=> aa(A,$o,aa(A,fun(A,$o),Less,A3),Top) ) ) ).
% ordering_top.not_eq_extremum
tff(fact_857_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
tff(fact_858_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_859_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_860_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_order.axioms(1)
tff(fact_861_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).
% sgn_1_pos
tff(fact_862_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
aa(A,A,sgn_sgn(A),X) = $ite(
X = zero_zero(A),
zero_zero(A),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).
% sgn_if
tff(fact_863_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% sgn_1_neg
tff(fact_864_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_865_le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).
% le_minus_divide_eq
tff(fact_866_minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).
% minus_divide_le_eq
tff(fact_867_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),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_868_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),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_869_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),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_870_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),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_871_prod_Oswap__def,axiom,
! [A: $tType,B: $tType,P5: product_prod(B,A)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(B,A),A,product_snd(B,A),P5)),aa(product_prod(B,A),B,product_fst(B,A),P5)) ).
% prod.swap_def
tff(fact_872_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% le_divide_eq_1_pos
tff(fact_873_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_874_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_875_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% add_le_cancel_right
tff(fact_876_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% add_le_cancel_left
tff(fact_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% add_less_cancel_right
tff(fact_887_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% add_less_cancel_left
tff(fact_888_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).
% add_diff_cancel_right'
tff(fact_889_add__diff__cancel__right,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).
% add_diff_cancel_right
tff(fact_890_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),A3) = B2 ) ).
% add_diff_cancel_left'
tff(fact_891_add__diff__cancel__left,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).
% add_diff_cancel_left
tff(fact_892_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ).
% diff_add_cancel
tff(fact_893_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).
% add_diff_cancel
tff(fact_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_div__by__1,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : divide_divide(A,A3,one_one(A)) = A3 ) ).
% div_by_1
tff(fact_902_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) ).
% swap_simp
tff(fact_903_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_904_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_905_le__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).
% le_add_same_cancel2
tff(fact_906_le__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).
% le_add_same_cancel1
tff(fact_907_add__le__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% add_le_same_cancel2
tff(fact_908_add__le__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% add_le_same_cancel1
tff(fact_909_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_910_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_911_less__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).
% less_add_same_cancel2
tff(fact_912_less__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).
% less_add_same_cancel1
tff(fact_913_add__less__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% add_less_same_cancel2
tff(fact_914_add__less__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% add_less_same_cancel1
tff(fact_915_diff__add__zero,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).
% diff_add_zero
tff(fact_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: 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)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B2)) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_925_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_926_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_927_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_928_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_929_divide__self__if,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
divide_divide(A,A3,A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% divide_self_if
tff(fact_930_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_931_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_932_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_933_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_934_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).
% uminus_add_conv_diff
tff(fact_935_diff__minus__eq__add,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).
% diff_minus_eq_add
tff(fact_936_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_937_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_938_divide__le__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).
% divide_le_0_1_iff
tff(fact_939_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).
% zero_le_divide_1_iff
tff(fact_940_divide__less__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).
% divide_less_0_1_iff
tff(fact_941_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A3)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% divide_less_eq_1_neg
tff(fact_942_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A3)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).
% divide_less_eq_1_pos
tff(fact_943_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ).
% less_divide_eq_1_neg
tff(fact_944_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ).
% less_divide_eq_1_pos
tff(fact_945_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).
% zero_less_divide_1_iff
tff(fact_946_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_947_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_948_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_949_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_950_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ).
% divide_le_eq_1_neg
tff(fact_951_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).
% divide_le_eq_1_pos
tff(fact_952_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ).
% le_divide_eq_1_neg
tff(fact_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_group__cancel_Oadd2,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Ba: A,K: A,B2: A,A3: A] :
( ( Ba = 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),Ba) = 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_963_group__cancel_Oadd1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Aa2: A,K: A,A3: A,B2: A] :
( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),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_964_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_965_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_966_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_967_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_968_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_969_add__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: 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,W2,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),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% add_frac_eq
tff(fact_970_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,Z2: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B2,Z2)) = $ite(Z2 = zero_zero(A),A3,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_971_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,Z2: A,B2: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B2) = $ite(Z2 = zero_zero(A),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_972_gt__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),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_973_less__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_974_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% add_le_imp_le_right
tff(fact_975_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% add_le_imp_le_left
tff(fact_976_le__iff__add,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
<=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).
% le_iff_add
tff(fact_977_add__right__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_978_less__eqE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).
% less_eqE
tff(fact_979_add__left__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_980_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).
% add_mono
tff(fact_981_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_982_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_983_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
& ( K = L ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_984_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_985_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_986_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_987_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_988_divide__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W2: A] : divide_divide(A,divide_divide(A,X,Y),divide_divide(A,Z2,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),W2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ).
% divide_divide_times_eq
tff(fact_989_times__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,X,Y)),divide_divide(A,Z2,W2)) = 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),W2)) ) ).
% times_divide_times_eq
tff(fact_990_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% add_less_imp_less_right
tff(fact_991_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% add_less_imp_less_left
tff(fact_992_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_993_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_994_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).
% add_strict_mono
tff(fact_995_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
& ( K = L ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_996_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_997_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_998_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,E3: 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),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),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)),E3)),C2) ) ).
% combine_common_factor
tff(fact_999_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_1000_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_1001_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_1002_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_1003_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_1004_diff__diff__eq,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% diff_diff_eq
tff(fact_1005_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 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).
% add_implies_diff
tff(fact_1006_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) ) ).
% diff_add_eq_diff_diff_swap
tff(fact_1007_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% diff_add_eq
tff(fact_1008_diff__diff__eq2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% diff_diff_eq2
tff(fact_1009_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).
% add_diff_eq
tff(fact_1010_eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).
% eq_diff_eq
tff(fact_1011_diff__eq__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,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_1012_group__cancel_Osub1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Aa2: A,K: A,A3: A,B2: A] :
( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).
% group_cancel.sub1
tff(fact_1013_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_1014_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Aa2: A,K: A,A3: A] :
( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,uminus_uminus(A),Aa2) = 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_1015_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_1016_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_1017_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_1018_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,Z2: A,B2: A] :
aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = $ite(Z2 = zero_zero(A),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_1019_add_Oac__operator__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> syntax_ac_operator(A,plus_plus(A)) ) ).
% add.ac_operator_axioms
tff(fact_1020_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_1021_scaling__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [U: A,V: A,R2: A,S2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U)),S2))),V) ) ) ) ) ).
% scaling_mono
tff(fact_1022_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_1023_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_1024_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_1025_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_1026_eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = divide_divide(A,B2,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2,A3 = zero_zero(A)) ) ) ).
% eq_divide_eq
tff(fact_1027_divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( divide_divide(A,B2,C2) = A3 )
<=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).
% divide_eq_eq
tff(fact_1028_frac__eq__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( ( divide_divide(A,X,Y) = divide_divide(A,W2,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),W2),Y) ) ) ) ) ) ).
% frac_eq_eq
tff(fact_1029_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_1030_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_1031_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_1032_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).
% add_nonpos_nonpos
tff(fact_1033_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% add_nonneg_nonneg
tff(fact_1034_add__increasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).
% add_increasing2
tff(fact_1035_add__decreasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).
% add_decreasing2
tff(fact_1036_add__increasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).
% add_increasing
tff(fact_1037_add__decreasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ) ) ).
% add_decreasing
tff(fact_1038_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).
% add_less_le_mono
tff(fact_1039_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).
% add_le_less_mono
tff(fact_1040_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_1041_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_1042_pos__add__strict,axiom,
! [A: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).
% pos_add_strict
tff(fact_1043_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ~ ! [C3: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
=> ( C3 = zero_zero(A) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_1044_add__pos__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% add_pos_pos
tff(fact_1045_add__neg__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).
% add_neg_neg
tff(fact_1046_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( ( aa(A,A,aa(A,fun(A,A),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_1047_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1048_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1049_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1050_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),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_1051_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1052_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1053_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))
<=> aa(A,$o,aa(A,fun(A,$o),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_1054_le__add__diff,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3)) ) ) ).
% le_add_diff
tff(fact_1055_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1056_le__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).
% le_diff_eq
tff(fact_1057_diff__le__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).
% diff_le_eq
tff(fact_1058_less__add__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).
% less_add_one
tff(fact_1059_add__mono1,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_1060_less__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ) ).
% less_diff_eq
tff(fact_1061_diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).
% diff_less_eq
tff(fact_1062_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_1063_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_1064_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_1065_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_1066_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_1067_square__diff__square__factored,axiom,
! [A: $tType] :
( comm_ring(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ).
% square_diff_square_factored
tff(fact_1068_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E3: 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),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
<=> ( 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),minus_minus(A),B2),A3)),E3)),D3) ) ) ) ).
% eq_add_iff2
tff(fact_1069_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E3: 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),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2) = D3 ) ) ) ).
% eq_add_iff1
tff(fact_1070_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1071_diff__conv__add__uminus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% diff_conv_add_uminus
tff(fact_1072_group__cancel_Osub2,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Ba: A,K: A,B2: A,A3: A] :
( ( Ba = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).
% group_cancel.sub2
tff(fact_1073_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_1074_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_1075_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> groups_monoid_list(A,plus_plus(A),zero_zero(A)) ) ).
% sum_list.monoid_list_axioms
tff(fact_1076_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).
% sum_list.comm_monoid_list_axioms
tff(fact_1077_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).
% sum.comm_monoid_list_set_axioms
tff(fact_1078_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_1079_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1080_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B2)) ) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_1081_divide__strict__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B2)) ) ) ) ) ).
% divide_strict_left_mono
tff(fact_1082_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),divide_divide(A,X,Y)) ) ) ) ).
% mult_imp_less_div_pos
tff(fact_1083_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,X,Y)),Z2) ) ) ) ).
% mult_imp_div_pos_less
tff(fact_1084_pos__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B2,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).
% pos_less_divide_eq
tff(fact_1085_pos__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% pos_divide_less_eq
tff(fact_1086_neg__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B2,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% neg_less_divide_eq
tff(fact_1087_neg__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).
% neg_divide_less_eq
tff(fact_1088_less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% less_divide_eq
tff(fact_1089_divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A3)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).
% divide_less_eq
tff(fact_1090_divide__less__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A3)),one_one(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_less_eq_1
tff(fact_1091_less__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) ) ) ) ) ).
% less_divide_eq_1
tff(fact_1092_divide__diff__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),minus_minus(A),divide_divide(A,X,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,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_1093_diff__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),minus_minus(A),X),divide_divide(A,Y,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y),Z2) ) ) ) ).
% diff_divide_eq_iff
tff(fact_1094_diff__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% diff_frac_eq
tff(fact_1095_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,Z2: A] :
aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),divide_divide(A,B2,Z2)) = $ite(Z2 = zero_zero(A),A3,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2),Z2)) ) ).
% add_divide_eq_if_simps(4)
tff(fact_1096_add__strict__increasing2,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).
% add_strict_increasing2
tff(fact_1097_add__strict__increasing,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).
% add_strict_increasing
tff(fact_1098_add__pos__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% add_pos_nonneg
tff(fact_1099_add__nonpos__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).
% add_nonpos_neg
tff(fact_1100_add__nonneg__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% add_nonneg_pos
tff(fact_1101_add__neg__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A)) ) ) ) ).
% add_neg_nonpos
tff(fact_1102_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).
% sum_squares_ge_zero
tff(fact_1103_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_1104_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_1105_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 )
<=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).
% minus_divide_eq_eq
tff(fact_1106_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)) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2),A3 = zero_zero(A)) ) ) ).
% eq_minus_divide_eq
tff(fact_1107_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_1108_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).
% not_sum_squares_lt_zero
tff(fact_1109_discrete,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2) ) ) ).
% discrete
tff(fact_1110_zero__less__two,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).
% zero_less_two
tff(fact_1111_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2)),D3) ) ) ).
% le_add_iff1
tff(fact_1112_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3)) ) ) ).
% le_add_iff2
tff(fact_1113_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C2)),D3) ) ) ).
% less_add_iff1
tff(fact_1114_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3)) ) ) ).
% less_add_iff2
tff(fact_1115_square__diff__one__factored,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).
% square_diff_one_factored
tff(fact_1116_dbl__dec__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).
% dbl_dec_def
tff(fact_1117_divide__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B2)) ) ) ) ) ).
% divide_left_mono_neg
tff(fact_1118_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),divide_divide(A,X,Y)) ) ) ) ).
% mult_imp_le_div_pos
tff(fact_1119_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,X,Y)),Z2) ) ) ) ).
% mult_imp_div_pos_le
tff(fact_1120_pos__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B2,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).
% pos_le_divide_eq
tff(fact_1121_pos__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% pos_divide_le_eq
tff(fact_1122_neg__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B2,C2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% neg_le_divide_eq
tff(fact_1123_neg__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).
% neg_divide_le_eq
tff(fact_1124_divide__left__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C2,A3)),divide_divide(A,C2,B2)) ) ) ) ) ).
% divide_left_mono
tff(fact_1125_le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).
% le_divide_eq
tff(fact_1126_divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A3)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).
% divide_le_eq
tff(fact_1127_divide__le__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_le_eq_1
tff(fact_1128_le__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) )
| ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ) ) ).
% le_divide_eq_1
tff(fact_1129_frac__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A)) ) ) ) ) ).
% frac_le_eq
tff(fact_1130_frac__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A)) ) ) ) ) ).
% frac_less_eq
tff(fact_1131_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),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_1132_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),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_1133_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),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_1134_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> aa(A,$o,aa(A,fun(A,$o),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_1135_minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A3)
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).
% minus_divide_less_eq
tff(fact_1136_less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% less_minus_divide_eq
tff(fact_1137_minus__divide__diff__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),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,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_1138_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,Z2: A,B2: A] :
aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,Z2)),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),divide_divide(A,aa(A,A,aa(A,fun(A,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_1139_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,Z2: A,B2: A] :
aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = $ite(Z2 = zero_zero(A),aa(A,A,uminus_uminus(A),B2),divide_divide(A,aa(A,A,aa(A,fun(A,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_1140_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_1141_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_1142_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_1143_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_1144_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_1145_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_1146_div__mult__mult1__if,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: 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)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B2)) ) ).
% div_mult_mult1_if
tff(fact_1147_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_1148_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_1149_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_1150_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_1151_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_1152_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_1153_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_1154_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_1155_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_1156_crossproduct__eq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [W2: 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),W2),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),W2),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
<=> ( ( W2 = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
tff(fact_1157_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_1158_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_1159_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_1160_right__diff__distrib__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( ring(B)
=> ! [X: A,Y: A,A3: B,B2: B,C2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),A3)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_1161_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [V: num,W2: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2))),Z2) ) ).
% mult_numeral_left_semiring_numeral
tff(fact_1162_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_1163_distrib__left__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),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),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).
% distrib_left_numeral
tff(fact_1164_distrib__right__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).
% distrib_right_numeral
tff(fact_1165_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).
% right_diff_distrib_numeral
tff(fact_1166_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(A) )
=> ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).
% left_diff_distrib_numeral
tff(fact_1167_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_1168_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_1169_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_1170_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))) ) ) ).
% divide_le_eq_numeral1(1)
tff(fact_1171_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).
% le_divide_eq_numeral1(1)
tff(fact_1172_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W2: num,A3: A] :
( ( divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) = A3 )
<=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)),A3 = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_1173_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W2: num] :
( ( A3 = divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) )
<=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B2,A3 = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_1174_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))) ) ) ).
% divide_less_eq_numeral1(1)
tff(fact_1175_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).
% less_divide_eq_numeral1(1)
tff(fact_1176_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).
% le_divide_eq_numeral1(2)
tff(fact_1177_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2) ) ) ).
% divide_le_eq_numeral1(2)
tff(fact_1178_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W2: num] :
( ( A3 = divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
<=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != 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),W2))) = B2,A3 = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_1179_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W2: num,A3: A] :
( ( divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
<=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != 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),W2))),A3 = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_1180_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2) ) ) ).
% divide_less_eq_numeral1(2)
tff(fact_1181_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).
% less_divide_eq_numeral1(2)
tff(fact_1182_one__le__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) ) ).
% one_le_numeral
tff(fact_1183_not__numeral__less__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) ) ).
% not_numeral_less_one
tff(fact_1184_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_1185_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_1186_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_1187_neg__numeral__le__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A)) ) ).
% neg_numeral_le_one
tff(fact_1188_neg__one__le__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2)) ) ).
% neg_one_le_numeral
tff(fact_1189_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% neg_numeral_le_neg_one
tff(fact_1190_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),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_1191_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))) ) ).
% not_one_le_neg_numeral
tff(fact_1192_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W2: num] :
( ( divide_divide(A,B2,C2) = aa(num,A,numeral_numeral(A),W2) )
<=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_1193_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num,B2: A,C2: A] :
( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B2,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_1194_neg__numeral__less__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),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_1195_neg__one__less__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2)) ) ).
% neg_one_less_numeral
tff(fact_1196_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),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_1197_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))) ) ).
% not_one_less_neg_numeral
tff(fact_1198_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))) ) ).
% not_neg_one_less_neg_numeral
tff(fact_1199_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_1200_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_1201_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W2: num] :
( ( divide_divide(A,B2,C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
<=> $ite(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),W2))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_1202_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num,B2: A,C2: A] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B2,C2) )
<=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_1203_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_1204_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_1205_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C2))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_1206_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
<=> $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_1207_distrib__left__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( semiring(B)
=> ! [X: A,Y: A,A3: B,B2: B,C2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),A3)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).
% distrib_left_NO_MATCH
tff(fact_1208_distrib__right__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( semiring(B)
=> ! [X: A,Y: A,C2: B,A3: B,B2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),C2)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).
% distrib_right_NO_MATCH
tff(fact_1209_left__diff__distrib__NO__MATCH,axiom,
! [A: $tType,B: $tType] :
( ring(B)
=> ! [X: A,Y: A,C2: B,A3: B,B2: B] :
( nO_MATCH(A,B,divide_divide(A,X,Y),C2)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A3),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_1210_div__add__self2__no__field,axiom,
! [A: $tType,B: $tType] :
( ( euclid4440199948858584721cancel(B)
& field(A) )
=> ! [X: A,B2: B,A3: B] :
( nO_MATCH(A,B,X,B2)
=> ( ( B2 != zero_zero(B) )
=> ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A3,B2)),one_one(B)) ) ) ) ) ).
% div_add_self2_no_field
tff(fact_1211_div__add__self1__no__field,axiom,
! [A: $tType,B: $tType] :
( ( euclid4440199948858584721cancel(B)
& field(A) )
=> ! [X: A,B2: B,A3: B] :
( nO_MATCH(A,B,X,B2)
=> ( ( B2 != zero_zero(B) )
=> ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A3),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A3,B2)),one_one(B)) ) ) ) ) ).
% div_add_self1_no_field
tff(fact_1212_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2)))),Y) ) ).
% semiring_norm(170)
tff(fact_1213_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2)))),Y) ) ).
% semiring_norm(171)
tff(fact_1214_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2))),Y) ) ).
% semiring_norm(172)
tff(fact_1215_numeral__times__minus__swap,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [W2: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),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),W2))) ) ).
% numeral_times_minus_swap
tff(fact_1216_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),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_1217_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),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_1218_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_1219_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_1220_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_1221_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).
% diff_numeral_special(3)
tff(fact_1222_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),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_1223_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] :
( aa(A,$o,aa(A,fun(A,$o),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_1224_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M2: num] :
( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))
<=> ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_1225_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1226_power__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M2: nat,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2) ) ) ) ) ).
% power_decreasing_iff
tff(fact_1227_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_1228_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = divide_divide(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).
% inverse_eq_divide_neg_numeral
tff(fact_1229_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_1230_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_1231_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_1232_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_1233_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_1234_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_1235_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_1236_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),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_1237_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: nat,Y: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).
% power_strict_increasing_iff
tff(fact_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_numeral__le__one__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A))
<=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),N),one2) ) ) ).
% numeral_le_one_iff
tff(fact_1249_one__less__numeral__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N))
<=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),N) ) ) ).
% one_less_numeral_iff
tff(fact_1250_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M2: nat,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W2)) ) ).
% inverse_eq_divide_numeral
tff(fact_1259_power__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: nat,Y: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).
% power_increasing_iff
tff(fact_1260_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_1261_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_1262_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).
% group.inverse_distrib_swap
tff(fact_1263_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% group.group_left_neutral
tff(fact_1264_group_Oinverse__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,Z2) = Z2 ) ) ).
% group.inverse_neutral
tff(fact_1265_group_Oinverse__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).
% group.inverse_inverse
tff(fact_1266_group_Oinverse__unique,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
=> ( aa(A,A,Inverse,A3) = B2 ) ) ) ).
% group.inverse_unique
tff(fact_1267_group_Oright__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).
% group.right_inverse
tff(fact_1268_group_Oright__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,B2),A3) = aa(A,A,aa(A,fun(A,A),F,C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% group.right_cancel
tff(fact_1269_group_Oleft__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).
% group.left_inverse
tff(fact_1270_group_Oleft__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,A3),C2) )
<=> ( B2 = C2 ) ) ) ).
% group.left_cancel
tff(fact_1271_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_1272_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_1273_mod__mult__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,A6: A,B2: A,B5: A] :
( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A6,C2) )
=> ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,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),A6),B5),C2) ) ) ) ) ).
% mod_mult_cong
tff(fact_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_power__minus__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).
% power_minus_mult
tff(fact_1282_power__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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_mod__eqE,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,B2,C2) )
=> ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ).
% mod_eqE
tff(fact_1291_self__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).
% self_le_power
tff(fact_1292_one__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).
% one_le_power
tff(fact_1293_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N3: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),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),N3)) ) ) ) ).
% power_increasing
tff(fact_1294_power__0__left,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).
% power_0_left
tff(fact_1295_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).
% one_less_power
tff(fact_1296_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ) ) ).
% power_less_imp_less_exp
tff(fact_1297_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N3: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),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),N3)) ) ) ) ).
% power_strict_increasing
tff(fact_1298_power__eq__if,axiom,
! [A: $tType] :
( power(A)
=> ! [P5: A,M2: nat] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),P5),M2) = $ite(M2 = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P5),aa(nat,A,aa(A,fun(nat,A),power_power(A),P5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))) ) ).
% power_eq_if
tff(fact_1299_left__right__inverse__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).
% left_right_inverse_power
tff(fact_1300_power__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_1301_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> semigroup(A,F) ) ).
% group.axioms(1)
tff(fact_1302_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_1303_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_1304_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_1305_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).
% inverse_le_1_iff
tff(fact_1306_one__less__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).
% one_less_inverse
tff(fact_1307_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).
% one_less_inverse_iff
tff(fact_1308_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_1309_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_1310_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_1311_division__ring__inverse__diff,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),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)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% division_ring_inverse_diff
tff(fact_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)) = modulo_modulo(A,A3,B2) ) ).
% minus_div_mult_eq_mod
tff(fact_1322_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_1323_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,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_1324_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))) = modulo_modulo(A,A3,B2) ) ).
% minus_mult_div_eq_mod
tff(fact_1325_power__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)) ) ) ) ).
% power_le_one
tff(fact_1326_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N3: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).
% power_decreasing
tff(fact_1327_power__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M2: nat,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ) ).
% power_le_imp_le_exp
tff(fact_1328_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N3: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).
% power_strict_decreasing
tff(fact_1329_power__gt1__lemma,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% power_gt1_lemma
tff(fact_1330_power__less__power__Suc,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),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_1331_power__diff__power__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,M2: nat,N: nat] :
( ( A3 != zero_zero(A) )
=> ( 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)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ).
% power_diff_power_eq
tff(fact_1332_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_1333_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_1334_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_1335_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_1336_inverse__less__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ) ) ).
% inverse_less_iff
tff(fact_1337_inverse__le__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2))
<=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ) ) ).
% inverse_le_iff
tff(fact_1338_one__le__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).
% one_le_inverse
tff(fact_1339_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).
% inverse_less_1_iff
tff(fact_1340_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ) ).
% one_le_inverse_iff
tff(fact_1341_power__Suc__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1342_power__diff__conv__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M2: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),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_1343_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),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))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1344_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_1345_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_1346_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = divide_divide(A,A3,B2) ) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_1347_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_1348_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_1349_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_1350_mod__double__modulus,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,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_1351_one__add__one,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% one_add_one
tff(fact_1352_one__mod__two__eq__one,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% one_mod_two_eq_one
tff(fact_1353_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% bits_one_mod_two_eq_one
tff(fact_1354_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num] : unique8689654367752047608divmod(A,M2,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M2)),zero_zero(A)) ) ).
% divmod_algorithm_code(2)
tff(fact_1355_bits__1__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).
% bits_1_div_2
tff(fact_1356_one__div__two__eq__zero,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).
% one_div_two_eq_zero
tff(fact_1357_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).
% add_neg_numeral_special(9)
tff(fact_1358_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,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),aa(num,num,bit0,one2))) ) ) ).
% diff_numeral_special(10)
tff(fact_1359_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% diff_numeral_special(11)
tff(fact_1360_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),aa(num,num,bit0,one2))) != zero_zero(A) )
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_1361_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),aa(num,num,bit0,one2))) != one_one(A) )
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_1362_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),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% minus_1_div_2_eq
tff(fact_1363_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% minus_1_mod_2_eq
tff(fact_1364_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% bits_minus_1_mod_2_eq
tff(fact_1365_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),aa(num,num,bit0,one2))),N)) = one_one(A) ) ).
% power_minus1_even
tff(fact_1366_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(3)
tff(fact_1367_divmod_H__nat__def,axiom,
! [M2: num,N: num] : unique8689654367752047608divmod(nat,M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N))) ).
% divmod'_nat_def
tff(fact_1368_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),aa(num,num,bit0,aa(num,num,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_1369_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% power2_eq_square
tff(fact_1370_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% one_power2
tff(fact_1371_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),aa(num,num,bit0,one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2
tff(fact_1372_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2_right
tff(fact_1373_left__add__twice,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),B2) ) ).
% left_add_twice
tff(fact_1374_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),aa(num,num,bit0,one2))),N)) ) ).
% minus_power_mult_self
tff(fact_1375_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),aa(num,num,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_1376_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),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),X)),Y)) ) ).
% power2_sum
tff(fact_1377_square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).
% square_le_1
tff(fact_1378_divmod__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] : unique8689654367752047608divmod(A,M2,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N))) ) ).
% divmod_def
tff(fact_1379_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
=> ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).
% divmod_digit_0(2)
tff(fact_1380_power2__diff,axiom,
! [A: $tType] :
( comm_ring_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),minus_minus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),X)),Y)) ) ).
% power2_diff
tff(fact_1381_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,bit0,one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) ) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_1382_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = divide_divide(A,A3,B2) ) ) ) ) ).
% divmod_digit_0(1)
tff(fact_1383_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Q3: A,R2: A] :
unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),R2)) ) ).
% divmod_step_eq
tff(fact_1384_set__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% set_bit_0
tff(fact_1385_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),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).
% unset_bit_0
tff(fact_1386_divmod__divmod__step,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] :
unique8689654367752047608divmod(A,M2,N) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M2)),unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M2,aa(num,num,bit0,N)))) ) ).
% divmod_divmod_step
tff(fact_1387_signed__take__bit__rec,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] :
bit_ri4674362597316999326ke_bit(A,N,A3) = $ite(N = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))) ) ).
% signed_take_bit_rec
tff(fact_1388_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),aa(num,num,bit0,one2))) ) ) ).
% dbl_simps(4)
tff(fact_1389_dbl__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% dbl_simps(3)
tff(fact_1390_even__succ__mod__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,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),aa(num,num,bit0,one2))),N))) ) ) ) ) ).
% even_succ_mod_exp
tff(fact_1391_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( 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))
<=> dvd_dvd(A,B2,C2) ) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_1392_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( 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))
<=> dvd_dvd(A,B2,C2) ) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_1393_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,C2: A,B2: A] :
( 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) )
| dvd_dvd(A,A3,B2) ) ) ) ).
% dvd_mult_cancel_right
tff(fact_1394_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [C2: A,A3: A,B2: A] :
( 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) )
| dvd_dvd(A,A3,B2) ) ) ) ).
% dvd_mult_cancel_left
tff(fact_1395_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( 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))
<=> dvd_dvd(A,A3,B2) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_1396_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( 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)))
<=> dvd_dvd(A,A3,B2) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_1397_unit__prod,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( dvd_dvd(A,B2,one_one(A))
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A)) ) ) ) ).
% unit_prod
tff(fact_1398_dvd__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( 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_1399_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( 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_1400_unit__div__1__div__1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( 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_1401_unit__div__1__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( dvd_dvd(A,A3,one_one(A))
=> dvd_dvd(A,divide_divide(A,one_one(A),A3),one_one(A)) ) ) ).
% unit_div_1_unit
tff(fact_1402_unit__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( dvd_dvd(A,B2,one_one(A))
=> dvd_dvd(A,divide_divide(A,A3,B2),one_one(A)) ) ) ) ).
% unit_div
tff(fact_1403_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_1404_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_1405_unit__mult__div__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( 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_1406_unit__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( 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_1407_even__mult__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
| dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).
% even_mult_iff
tff(fact_1408_even__plus__one__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))
<=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3) ) ) ).
% even_plus_one_iff
tff(fact_1409_neg__one__even__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,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_1410_neg__one__odd__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,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_1411_even__succ__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).
% even_succ_div_2
tff(fact_1412_even__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).
% even_succ_div_two
tff(fact_1413_odd__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).
% odd_succ_div_two
tff(fact_1414_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A3 ) ) ) ).
% odd_two_times_div_two_succ
tff(fact_1415_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)))
<=> ( N = zero_zero(nat) ) ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_1416_even__succ__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,bit0,one2))),N)) = divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ) ) ) ).
% even_succ_div_exp
tff(fact_1417_dvd__triv__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) ) ).
% dvd_triv_right
tff(fact_1418_dvd__mult__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
=> dvd_dvd(A,B2,C2) ) ) ).
% dvd_mult_right
tff(fact_1419_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( dvd_dvd(A,A3,B2)
=> ( dvd_dvd(A,C2,D3)
=> 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_1420_dvd__triv__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% dvd_triv_left
tff(fact_1421_dvd__mult__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
=> dvd_dvd(A,A3,C2) ) ) ).
% dvd_mult_left
tff(fact_1422_dvd__mult2,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,A3,B2)
=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).
% dvd_mult2
tff(fact_1423_dvd__mult,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,C2: A,B2: A] :
( dvd_dvd(A,A3,C2)
=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).
% dvd_mult
tff(fact_1424_dvd__def,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( 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_1425_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) )
=> dvd_dvd(A,B2,A3) ) ) ).
% dvdI
tff(fact_1426_dvdE,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( dvd_dvd(A,B2,A3)
=> ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).
% dvdE
tff(fact_1427_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : dvd_dvd(A,one_one(A),A3) ) ).
% one_dvd
tff(fact_1428_unit__imp__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( dvd_dvd(A,B2,one_one(A))
=> dvd_dvd(A,B2,A3) ) ) ).
% unit_imp_dvd
tff(fact_1429_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,A3,B2)
=> ( dvd_dvd(A,B2,one_one(A))
=> dvd_dvd(A,A3,one_one(A)) ) ) ) ).
% dvd_unit_imp_unit
tff(fact_1430_not__is__unit__0,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ~ dvd_dvd(A,zero_zero(A),one_one(A)) ) ).
% not_is_unit_0
tff(fact_1431_is__unit__mult__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A))
<=> ( dvd_dvd(A,A3,one_one(A))
& dvd_dvd(A,B2,one_one(A)) ) ) ) ).
% is_unit_mult_iff
tff(fact_1432_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> dvd_dvd(A,A3,C2) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_1433_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
<=> dvd_dvd(A,A3,C2) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_1434_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> dvd_dvd(A,A3,C2) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_1435_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
<=> dvd_dvd(A,B2,C2) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_1436_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_1437_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_1438_dvd__div__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( 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_1439_div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( 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_1440_div__div__eq__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( dvd_dvd(A,C2,B2)
=> ( 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_1441_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( 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_1442_dvd__mult__imp__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A] :
( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2)
=> dvd_dvd(A,A3,divide_divide(A,B2,C2)) ) ) ).
% dvd_mult_imp_div
tff(fact_1443_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,D3: A,C2: A] :
( dvd_dvd(A,B2,A3)
=> ( 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_1444_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( dvd_dvd(A,A3,divide_divide(A,C2,B2))
<=> dvd_dvd(A,A3,C2) ) ) ) ).
% dvd_div_unit_iff
tff(fact_1445_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( dvd_dvd(A,divide_divide(A,A3,B2),C2)
<=> dvd_dvd(A,A3,C2) ) ) ) ).
% div_unit_dvd_iff
tff(fact_1446_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( ( divide_divide(A,B2,A3) = divide_divide(A,C2,A3) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
tff(fact_1447_unity__coeff__ex,axiom,
! [A: $tType] :
( ( dvd(A)
& semiring_0(A) )
=> ! [P: fun(A,$o),L: A] :
( ? [X4: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4))
<=> ? [X4: A] :
( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A)))
& aa(A,$o,P,X4) ) ) ) ).
% unity_coeff_ex
tff(fact_1448_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ).
% unit_dvdE
tff(fact_1449_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T2: A] :
( dvd_dvd(A,D3,D4)
=> ! [X2: A,K3: A] :
( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),T2))
<=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T2)) ) ) ) ).
% inf_period(3)
tff(fact_1450_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T2: A] :
( dvd_dvd(A,D3,D4)
=> ! [X2: A,K3: A] :
( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),T2))
<=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T2)) ) ) ) ).
% inf_period(4)
tff(fact_1451_dvd__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( 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_1452_div__dvd__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( dvd_dvd(A,B2,A3)
=> ( dvd_dvd(A,divide_divide(A,A3,B2),C2)
<=> dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).
% div_dvd_iff_mult
tff(fact_1453_dvd__div__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( dvd_dvd(A,C2,B2)
=> ( dvd_dvd(A,A3,divide_divide(A,B2,C2))
<=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),B2) ) ) ) ) ).
% dvd_div_iff_mult
tff(fact_1454_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) )
=> ( dvd_dvd(A,A3,B2)
=> ( 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_1455_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( 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_1456_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( 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_1457_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( 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_1458_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( dvd_dvd(A,C2,one_one(A))
=> ( 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_1459_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( 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_1460_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( 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_1461_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( 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_1462_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( dvd_dvd(A,B2,one_one(A))
=> ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).
% unit_imp_mod_eq_0
tff(fact_1463_is__unit__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,N: nat] :
( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),one_one(A))
<=> ( dvd_dvd(A,A3,one_one(A))
| ( N = zero_zero(nat) ) ) ) ) ).
% is_unit_power_iff
tff(fact_1464_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( 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_1465_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( 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_1466_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A] :
( dvd_dvd(A,A3,one_one(A))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [B3: A] :
( ( B3 != zero_zero(A) )
=> ( dvd_dvd(A,B3,one_one(A))
=> ( ( divide_divide(A,one_one(A),A3) = B3 )
=> ( ( divide_divide(A,one_one(A),B3) = A3 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
=> ( divide_divide(A,C2,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_1467_evenE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3) ) ) ).
% evenE
tff(fact_1468_odd__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),one_one(A)) ) ).
% odd_one
tff(fact_1469_dvd__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,M2: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))
<=> ( dvd_dvd(A,X,one_one(A))
| aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ) ) ).
% dvd_power_iff
tff(fact_1470_dvd__power,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] :
( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
| ( X = one_one(A) ) )
=> dvd_dvd(A,X,aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ).
% dvd_power
tff(fact_1471_even__two__times__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A3 ) ) ) ).
% even_two_times_div_two
tff(fact_1472_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
<=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_1473_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3)),one_one(A)) ) ) ).
% oddE
tff(fact_1474_mod2__eq__if,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),zero_zero(A),one_one(A)) ) ).
% mod2_eq_if
tff(fact_1475_parity__cases,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
=> ~ ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).
% parity_cases
tff(fact_1476_minus__one__power__iff,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))),N) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% minus_one_power_iff
tff(fact_1477_even__mask__div__iff_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M2: nat,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ).
% even_mask_div_iff'
tff(fact_1478_even__mask__div__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
| aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ) ) ).
% even_mask_div_iff
tff(fact_1479_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,M2: nat,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2)
| ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
| ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
& dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_1480_take__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
bit_se2584673776208193580ke_bit(A,N,A3) = $ite(N = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% take_bit_rec
tff(fact_1481_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] :
unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M2))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).
% divmod_algorithm_code(7)
tff(fact_1482_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] :
unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit1,N)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M2),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit0,aa(num,num,bit1,N))))) ) ).
% divmod_algorithm_code(8)
tff(fact_1483_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).
% one_mod_2_pow_eq
tff(fact_1484_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),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).
% push_bit_numeral_minus_1
tff(fact_1485_exp__div__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] :
divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,
aa(A,fun(A,A),times_times(A),
aa($o,A,zero_neq_one_of_bool(A),
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2) != zero_zero(A) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2) ))),
aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ).
% exp_div_exp_eq
tff(fact_1486_divmod__nat__def,axiom,
! [M2: nat,N: nat] : divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,M2,N)),modulo_modulo(nat,M2,N)) ).
% divmod_nat_def
tff(fact_1487_set__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5668285175392031749et_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% set_bit_Suc
tff(fact_1488_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: $o] :
( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
<=> (P) ) ) ).
% of_bool_eq_1_iff
tff(fact_1489_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).
% of_bool_eq(2)
tff(fact_1490_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_1491_of__bool__less__one__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: $o] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
<=> ~ (P) ) ) ).
% of_bool_less_one_iff
tff(fact_1492_of__bool__not__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).
% of_bool_not_iff
tff(fact_1493_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_1494_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_1495_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_1496_sgn__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).
% sgn_mult_self_eq
tff(fact_1497_take__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)) ) ).
% take_bit_of_1
tff(fact_1498_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),aa(num,num,bit1,one2)) ) ) ).
% dbl_inc_simps(3)
tff(fact_1499_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),aa(num,num,bit0,one2)))) ) ).
% push_bit_Suc
tff(fact_1500_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),aa(num,num,bit0,one2))),N) ) ).
% push_bit_of_1
tff(fact_1501_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(4)
tff(fact_1502_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),aa(num,num,bit1,one2))) ) ) ).
% dbl_dec_simps(4)
tff(fact_1503_bits__1__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).
% bits_1_div_exp
tff(fact_1504_one__div__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).
% one_div_2_pow_eq
tff(fact_1505_take__bit__of__exp,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M2: nat,N: nat] : bit_se2584673776208193580ke_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).
% take_bit_of_exp
tff(fact_1506_take__bit__of__2,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% take_bit_of_2
tff(fact_1507_of__bool__conj,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [P: $o,Q: $o] :
aa($o,A,zero_neq_one_of_bool(A),
( (P)
& (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).
% of_bool_conj
tff(fact_1508_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),aa(num,num,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),aa(num,num,bit0,one2)))),one_one(A)) ) ).
% take_bit_Suc_bit1
tff(fact_1509_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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).
% take_bit_Suc_bit0
tff(fact_1510_of__bool__less__eq__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).
% of_bool_less_eq_one
tff(fact_1511_split__of__bool__asm,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,$o),P5: $o] :
( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P5)))
<=> ~ ( ( (P5)
& ~ aa(A,$o,P,one_one(A)) )
| ( ~ (P5)
& ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).
% split_of_bool_asm
tff(fact_1512_split__of__bool,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,$o),P5: $o] :
( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P5)))
<=> ( ( (P5)
=> aa(A,$o,P,one_one(A)) )
& ( ~ (P5)
=> aa(A,$o,P,zero_zero(A)) ) ) ) ) ).
% split_of_bool
tff(fact_1513_of__bool__def,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P5: $o] :
aa($o,A,zero_neq_one_of_bool(A),(P5)) = $ite((P5),one_one(A),zero_zero(A)) ) ).
% of_bool_def
tff(fact_1514_xor__num_Ocases,axiom,
! [X: product_prod(num,num)] :
( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
=> ( ! [N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N4))
=> ( ! [N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N4))
=> ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)
=> ( ! [M3: num,N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N4))
=> ( ! [M3: num,N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N4))
=> ( ! [M3: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)
=> ( ! [M3: num,N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N4))
=> ~ ! [M3: num,N4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N4)) ) ) ) ) ) ) ) ) ).
% xor_num.cases
tff(fact_1515_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_1516_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_1517_numeral__Bit1,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).
% numeral_Bit1
tff(fact_1518_power__gt1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))) ) ) ).
% power_gt1
tff(fact_1519_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))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).
% take_bit_Suc_minus_1_eq
tff(fact_1520_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),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).
% take_bit_Suc
tff(fact_1521_power__Suc__le__self,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3) ) ) ) ).
% power_Suc_le_self
tff(fact_1522_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),aa(num,num,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_1523_power__Suc__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A)) ) ) ) ).
% power_Suc_less_one
tff(fact_1524_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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).
% push_bit_double
tff(fact_1525_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),aa(num,num,bit0,one2))),N)) ) ).
% push_bit_eq_mult
tff(fact_1526_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,$o),A3: A] :
( ! [A4: A] :
( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
=> aa(A,$o,P,A4) )
=> ( ! [A4: A,B3: $o] :
( aa(A,$o,P,A4)
=> ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4))) ) )
=> aa(A,$o,P,A3) ) ) ) ).
% bits_induct
tff(fact_1527_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),aa(num,num,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),aa(num,num,bit0,one2)))) ) ).
% power_odd_eq
tff(fact_1528_exp__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M2: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) ) ).
% exp_mod_exp
tff(fact_1529_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),aa(num,num,bit0,one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% power_minus1_odd
tff(fact_1530_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))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).
% take_bit_numeral_minus_1_eq
tff(fact_1531_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),aa(num,num,bit0,one2))) = A3 )
=> ( bit_se2584673776208193580ke_bit(A,N,A3) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_1532_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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% signed_take_bit_Suc
tff(fact_1533_unset__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2638667681897837118et_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% unset_bit_Suc
tff(fact_1534_flip__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% flip_bit_0
tff(fact_1535_flip__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% flip_bit_Suc
tff(fact_1536_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),aa(num,num,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),aa(num,num,bit0,one2)))),one_one(A)) ) ).
% take_bit_numeral_bit1
tff(fact_1537_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1538_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1539_one__xor__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% one_xor_eq
tff(fact_1540_xor__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% xor_one_eq
tff(fact_1541_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1542_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_1543_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_1544_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),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% or_numerals(2)
tff(fact_1545_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),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% or_numerals(8)
tff(fact_1546_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1547_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1548_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),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% or_numerals(5)
tff(fact_1549_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% or_numerals(1)
tff(fact_1550_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),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)) ) ).
% xor_numerals(8)
tff(fact_1551_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),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% xor_numerals(5)
tff(fact_1552_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),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y)) ) ).
% xor_numerals(2)
tff(fact_1553_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% xor_numerals(1)
tff(fact_1554_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1555_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1556_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1557_or_Osemigroup__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> semigroup(A,bit_se1065995026697491101ons_or(A)) ) ).
% or.semigroup_axioms
tff(fact_1558_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_1559_xor_Osemigroup__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> semigroup(A,bit_se5824344971392196577ns_xor(A)) ) ).
% xor.semigroup_axioms
tff(fact_1560_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.comm_monoid_axioms
tff(fact_1561_or_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.monoid_axioms
tff(fact_1562_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.semilattice_neutr_axioms
tff(fact_1563_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).
% xor.comm_monoid_axioms
tff(fact_1564_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).
% xor.monoid_axioms
tff(fact_1565_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_1566_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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).
% take_bit_numeral_bit0
tff(fact_1567_or__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% or_one_eq
tff(fact_1568_one__or__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% one_or_eq
tff(fact_1569_mask__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).
% mask_numeral
tff(fact_1570_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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N))) ) ).
% mask_Suc_double
tff(fact_1571_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_1572_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ).
% mask_eq_exp_minus_1
tff(fact_1573_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J2))
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),N)
<=> $ite(N = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ).
% bit_sum_mult_2_cases
tff(fact_1574_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),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_1575_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),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_1576_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),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_1577_bit__0__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).
% bit_0_eq
tff(fact_1578_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_1579_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_1580_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_1581_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),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).
% and_numerals(8)
tff(fact_1582_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),aa(num,num,bit1,Y))) = one_one(A) ) ).
% and_numerals(2)
tff(fact_1583_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_1584_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_1585_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),aa(num,num,bit0,Y))) = zero_zero(A) ) ).
% and_numerals(1)
tff(fact_1586_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),aa(num,num,bit0,X))),one_one(A)) = zero_zero(A) ) ).
% and_numerals(5)
tff(fact_1587_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1588_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M2: num] : aa(A,A,aa(A,fun(A,A),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_1589_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),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_1590_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),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1591_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),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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_1592_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_1593_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_1594_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_1595_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_1596_and_Osemigroup__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> semigroup(A,bit_se5824344872417868541ns_and(A)) ) ).
% and.semigroup_axioms
tff(fact_1597_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( aa(nat,$o,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_1598_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_1599_bit__1__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),N)
<=> ( N = zero_zero(nat) ) ) ) ).
% bit_1_iff
tff(fact_1600_not__bit__1__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N)) ) ).
% not_bit_1_Suc
tff(fact_1601_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N)) ) ).
% bit_numeral_simps(1)
tff(fact_1602_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_1603_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_1604_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_1605_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_1606_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),aa(num,num,bit0,one2))) ) ).
% one_and_eq
tff(fact_1607_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),aa(num,num,bit0,one2))) ) ).
% and_one_eq
tff(fact_1608_even__bit__succ__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
| ( N = zero_zero(nat) ) ) ) ) ) ).
% even_bit_succ_iff
tff(fact_1609_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)
=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N)
| ( N = zero_zero(nat) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_1610_signed__take__bit__def,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),bit_se2584673776208193580ke_bit(A,N,A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).
% signed_take_bit_def
tff(fact_1611_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_1612_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_1613_dvd__productE,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [P5: A,A3: A,B2: A] :
( dvd_dvd(A,P5,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
=> ~ ! [X3: A,Y2: A] :
( ( P5 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y2) )
=> ( dvd_dvd(A,X3,A3)
=> ~ dvd_dvd(A,Y2,B2) ) ) ) ) ).
% dvd_productE
tff(fact_1614_division__decomp,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
=> ? [B6: A,C5: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),C5) )
& dvd_dvd(A,B6,B2)
& dvd_dvd(A,C5,C2) ) ) ) ).
% division_decomp
tff(fact_1615_mask__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat,M2: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N))),one_one(A)) ) ).
% mask_mod_exp
tff(fact_1616_group_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( semigroup(A,F)
=> ( group_axioms(A,F,Z2,Inverse)
=> group(A,F,Z2,Inverse) ) ) ).
% group.intro
tff(fact_1617_group__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
<=> ( semigroup(A,F)
& group_axioms(A,F,Z2,Inverse) ) ) ).
% group_def
tff(fact_1618_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_1619_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_1620_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_1621_min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% min.bounded_iff
tff(fact_1622_min_Oabsorb2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb2
tff(fact_1623_min_Oabsorb1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb1
tff(fact_1624_min__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).
% min_less_iff_conj
tff(fact_1625_min_Oabsorb4,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb4
tff(fact_1626_min_Oabsorb3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb3
tff(fact_1627_min__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).
% min_bot
tff(fact_1628_min__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).
% min_bot2
tff(fact_1629_min__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).
% min_top
tff(fact_1630_min__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).
% min_top2
tff(fact_1631_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_1632_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_1633_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_1634_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_1635_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_1636_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_1637_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_1638_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_1639_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_1640_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_1641_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_1642_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_1643_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_1644_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),aa(num,num,bit0,one2))) ) ) ).
% not_one_eq
tff(fact_1645_group__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F,Z2,Inverse)
<=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
& ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A5)),A5) = Z2 ) ) ).
% group_axioms_def
tff(fact_1646_group__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A4) = A4
=> ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A4)),A4) = Z2
=> group_axioms(A,F,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_1647_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_1648_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_1649_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_1650_min__absorb2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).
% min_absorb2
tff(fact_1651_min__absorb1,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).
% min_absorb1
tff(fact_1652_min__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] :
aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),A3,B2) ) ).
% min_def
tff(fact_1653_min__le__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z2) ) ) ) ).
% min_le_iff_disj
tff(fact_1654_min_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).
% min.coboundedI2
tff(fact_1655_min_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).
% min.coboundedI1
tff(fact_1656_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1657_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1658_min_Ocobounded2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) ) ).
% min.cobounded2
tff(fact_1659_min_Ocobounded1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3) ) ).
% min.cobounded1
tff(fact_1660_min_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1661_min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).
% min.boundedI
tff(fact_1662_min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2) ) ) ) ).
% min.boundedE
tff(fact_1663_min_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% min.orderI
tff(fact_1664_min_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).
% min.orderE
tff(fact_1665_min_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
=> aa(A,$o,aa(A,fun(A,$o),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_1666_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).
% min.strict_coboundedI2
tff(fact_1667_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) ) ) ).
% min.strict_coboundedI1
tff(fact_1668_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_1669_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C2) ) ) ) ).
% min.strict_boundedE
tff(fact_1670_min__less__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).
% min_less_iff_disj
tff(fact_1671_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_1672_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_1673_min__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,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),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% min_diff_distrib_left
tff(fact_1674_inf__min,axiom,
! [A: $tType] :
( ( semilattice_inf(A)
& linorder(A) )
=> ( inf_inf(A) = ord_min(A) ) ) ).
% inf_min
tff(fact_1675_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_min(A)) ) ).
% min.abel_semigroup_axioms
tff(fact_1676_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_min(A)) ) ).
% min.semilattice_axioms
tff(fact_1677_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_min(A)) ) ).
% min.semigroup_axioms
tff(fact_1678_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_1679_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_1680_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_1681_not__eq__complement,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).
% not_eq_complement
tff(fact_1682_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).
% minus_eq_not_minus_1
tff(fact_1683_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> group_axioms(A,F,Z2,Inverse) ) ).
% group.axioms(2)
tff(fact_1684_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_1685_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_1686_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_1687_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_1688_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_1689_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_1690_abs__square__less__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).
% abs_square_less_1
tff(fact_1691_abs__square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).
% abs_square_le_1
tff(fact_1692_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( ~ dvd_dvd(A,A3,one_one(A))
=> aa(nat,$o,aa(nat,fun(nat,$o),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_1693_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
=> ( ! [F3: fun(nat,fun(A,A)),A4: nat,B3: nat,Acc: A] :
( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))))
=> ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A4)
=> aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F3,A4),Acc)) )
=> aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),A4),B3),Acc) ) )
=> aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).
% fold_atLeastAtMost_nat.pinduct
tff(fact_1694_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ) ) ).
% gbinomial_reduce_nat
tff(fact_1695_odd__card__imp__not__empty,axiom,
! [A: $tType,Aa2: set(A)] :
( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),finite_card(A,Aa2))
=> ( Aa2 != bot_bot(set(A)) ) ) ).
% odd_card_imp_not_empty
tff(fact_1696_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),aa(num,num,bit0,one2))) = one_one(A) )
<=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_1697_abs__idempotent,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_idempotent
tff(fact_1698_abs__0__eq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_0_eq
tff(fact_1699_abs__eq__0,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_eq_0
tff(fact_1700_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_1701_abs__add__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_add_abs
tff(fact_1702_abs__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% abs_mult_self_eq
tff(fact_1703_abs__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_1
tff(fact_1704_abs__minus__cancel,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_minus_cancel
tff(fact_1705_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_nonneg
tff(fact_1706_abs__le__self__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).
% abs_le_self_iff
tff(fact_1707_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_le_zero_iff
tff(fact_1708_zero__less__abs__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))
<=> ( A3 != zero_zero(A) ) ) ) ).
% zero_less_abs_iff
tff(fact_1709_abs__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).
% abs_neg_one
tff(fact_1710_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_1711_euclidean__size__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid6346220572633701492n_size(A,one_one(A)) = one_one(nat) ) ) ).
% euclidean_size_1
tff(fact_1712_abs__of__nonpos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_nonpos
tff(fact_1713_abs__sgn__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).
% abs_sgn_eq_1
tff(fact_1714_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_1715_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y))) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_1716_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),X) = Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_1717_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,One),X) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_1718_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,aa(A,fun(A,A),Xor,X),Y)) = Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_1719_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),One) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_1720_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_1721_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_1722_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_1723_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A,Z2: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_1724_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_1725_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z2: A,X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_1726_abs__le__D1,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% abs_le_D1
tff(fact_1727_abs__ge__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3)) ) ).
% abs_ge_self
tff(fact_1728_abs__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_mult
tff(fact_1729_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_1730_abs__minus__commute,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).
% abs_minus_commute
tff(fact_1731_abs__ge__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)) ) ).
% abs_ge_zero
tff(fact_1732_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)) ) ).
% abs_not_less_zero
tff(fact_1733_abs__of__pos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_pos
tff(fact_1734_abs__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) ) ).
% abs_triangle_ineq
tff(fact_1735_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ).
% abs_triangle_ineq2
tff(fact_1736_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ).
% abs_triangle_ineq3
tff(fact_1737_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))) ) ).
% abs_triangle_ineq2_sym
tff(fact_1738_abs__mult__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A,B2: A,D3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ) ).
% abs_mult_less
tff(fact_1739_abs__leI,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2) ) ) ) ).
% abs_leI
tff(fact_1740_abs__le__D2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ).
% abs_le_D2
tff(fact_1741_abs__le__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ) ) ).
% abs_le_iff
tff(fact_1742_abs__ge__minus__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3)) ) ).
% abs_ge_minus_self
tff(fact_1743_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
tff(fact_1744_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_1745_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)),aa(A,A,abs_abs(A),X)) = X ) ).
% mult_sgn_abs
tff(fact_1746_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)),aa(A,A,abs_abs(A),A3)) = A3 ) ).
% sgn_mult_abs
tff(fact_1747_abs__mult__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).
% abs_mult_sgn
tff(fact_1748_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).
% linordered_idom_class.abs_sgn
tff(fact_1749_euclidean__size__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) ) ) ) ).
% euclidean_size_unit
tff(fact_1750_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_1751_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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_addition_formula
tff(fact_1752_dense__eq0__I,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs(A)
& dense_linorder(A) )
=> ! [X: A] :
( ! [E2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2) )
=> ( X = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_1753_abs__eq__mult,axiom,
! [A: $tType] :
( ordered_ring_abs(A)
=> ! [A3: A,B2: A] :
( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
& ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
=> ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).
% abs_eq_mult
tff(fact_1754_abs__mult__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).
% abs_mult_pos
tff(fact_1755_abs__minus__le__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A)) ) ).
% abs_minus_le_zero
tff(fact_1756_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) ) ).
% abs_triangle_ineq4
tff(fact_1757_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).
% abs_diff_triangle_ineq
tff(fact_1758_abs__of__neg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_neg
tff(fact_1759_abs__sgn__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% abs_sgn_eq
tff(fact_1760_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( 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_1761_size__mult__mono,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> aa(nat,$o,aa(nat,fun(nat,$o),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_1762_size__mult__mono_H,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> aa(nat,$o,aa(nat,fun(nat,$o),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_1763_euclidean__size__times__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( 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_1764_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ).
% abs_add_one_gt_zero
tff(fact_1765_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
~ ! [F3: fun(nat,fun(A,A)),A4: nat,B3: nat,Acc: A] : X != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_1766_card_Oempty,axiom,
! [A: $tType] : finite_card(A,bot_bot(set(A))) = zero_zero(nat) ).
% card.empty
tff(fact_1767_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
=> ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,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))))
=> ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc))) )
=> ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,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_1768_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A3),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2))) ) ) ).
% fold_atLeastAtMost_nat.psimps
tff(fact_1769_divmod__cases,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,A3: A] :
( ( ( B2 != zero_zero(A) )
=> ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
=> ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2) ) ) )
=> ( ( ( B2 != zero_zero(A) )
=> ! [Q5: A,R3: A] :
( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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_1770_gbinomial__absorption_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).
% gbinomial_absorption'
tff(fact_1771_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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_1772_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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_1773_division__segment__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).
% division_segment_1
tff(fact_1774_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_1775_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_1776_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_1777_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_1778_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_1779_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_1780_is__unit__division__segment,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A] : dvd_dvd(A,euclid7384307370059645450egment(A,A3),one_one(A)) ) ).
% is_unit_division_segment
tff(fact_1781_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_1782_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),aa(A,A,aa(A,fun(A,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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_absorb_comp
tff(fact_1783_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_1784_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_1785_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_1786_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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_absorption
tff(fact_1787_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,M2: nat,A3: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K))) ) ) ) ).
% gbinomial_trinomial_revision
tff(fact_1788_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_1789_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_1790_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,aa(A,A,aa(A,fun(A,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,aa(A,A,aa(A,fun(A,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_1791_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,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).
% gbinomial_negated_upper
tff(fact_1792_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,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).
% gbinomial_minus
tff(fact_1793_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R2: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E3)
=> ~ ! [N4: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N4)))),E3) ) ) ).
% nat_approx_posE
tff(fact_1800_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
=> ? [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N4)),X)) ) ) ).
% ex_less_of_nat_mult
tff(fact_1801_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
% bot_nat_def
tff(fact_1802_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_1803_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),aa(num,num,bit0,one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),comm_s3205402744901411588hammer(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),aa(num,num,bit0,one2)))),N)) ) ).
% pochhammer_double
tff(fact_1804_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),aa(A,A,aa(A,fun(A,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_1805_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),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).
% pochhammer_minus
tff(fact_1806_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),aa(A,A,aa(A,fun(A,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_1807_fact__reduce,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).
% fact_reduce
tff(fact_1808_fact__num__eq__if,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M2: nat] :
semiring_char_0_fact(A,M2) = $ite(M2 = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))))) ) ).
% fact_num_eq_if
tff(fact_1809_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
=> boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
tff(fact_1810_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_1811_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_1812_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_1813_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_1814_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_1815_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_1816_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Xor: fun(A,fun(A,A))] :
( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
<=> ! [X4: A,Y3: A] : aa(A,A,aa(A,fun(A,A),Xor,X4),Y3) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X4),aa(A,A,Compl,Y3))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X4)),Y3)) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
tff(fact_1817_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Conj: fun(A,fun(A,A)),Compl: fun(A,A)] :
( ! [X3: A,Y2: A] : aa(A,A,aa(A,fun(A,A),Xor,X3),Y2) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X3),aa(A,A,Compl,Y2))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X3)),Y2))
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
tff(fact_1818_fact__ge__1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N)) ) ).
% fact_ge_1
tff(fact_1819_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_1820_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),aa(A,A,aa(A,fun(A,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_1821_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_1822_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,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_1823_pochhammer__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] :
comm_s3205402744901411588hammer(A,zero_zero(A),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).
% pochhammer_0_left
tff(fact_1824_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
tff(fact_1825_choose__dvd,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
=> 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,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))),semiring_char_0_fact(A,N)) ) ) ).
% choose_dvd
tff(fact_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))),semiring_char_0_fact(A,N)) ) ).
% fact_double
tff(fact_1832_pochhammer__product,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M2: nat,N: nat,Z2: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).
% pochhammer_product
tff(fact_1833_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
<=> ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
& boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff_def
tff(fact_1834_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( finite_card(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
<=> ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
tff(fact_1835_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
=> ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ).
% bezw_non_0
tff(fact_1836_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).
% neg_numeral_le_ceiling
tff(fact_1837_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).
% ceiling_less_neg_numeral
tff(fact_1838_binomial__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
=> ( aa(nat,A,semiring_1_of_nat(A),binomial(N,K)) = divide_divide(A,semiring_char_0_fact(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).
% binomial_fact
tff(fact_1839_fact__binomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),binomial(N,K))) = divide_divide(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).
% fact_binomial
tff(fact_1840_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).
% floor_le_neg_numeral
tff(fact_1841_singletonI,axiom,
! [A: $tType,A3: A] : member(A,A3,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ).
% singletonI
tff(fact_1842_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_1843_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_1844_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,Aa2: set(A)] :
( ( aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))) = aa(set(A),set(A),insert2(A,A3),Aa2) )
<=> ( ( A3 = B2 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ) ).
% singleton_insert_inj_eq
tff(fact_1845_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,Aa2: set(A),B2: A] :
( ( aa(set(A),set(A),insert2(A,A3),Aa2) = aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))) )
<=> ( ( A3 = B2 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ) ).
% singleton_insert_inj_eq'
tff(fact_1846_disjoint__insert_I2_J,axiom,
! [A: $tType,Aa2: set(A),B2: A,Ba: set(A)] :
( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),Ba)) )
<=> ( ~ member(A,B2,Aa2)
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) ) ) ) ).
% disjoint_insert(2)
tff(fact_1847_disjoint__insert_I1_J,axiom,
! [A: $tType,Ba: set(A),A3: A,Aa2: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Ba),aa(set(A),set(A),insert2(A,A3),Aa2)) = bot_bot(set(A)) )
<=> ( ~ member(A,A3,Ba)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Ba),Aa2) = bot_bot(set(A)) ) ) ) ).
% disjoint_insert(1)
tff(fact_1848_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,Aa2: set(A),Ba: 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),insert2(A,A3),Aa2)),Ba) )
<=> ( ~ member(A,A3,Ba)
& ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) ) ) ) ).
% insert_disjoint(2)
tff(fact_1849_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),insert2(A,A3),Aa2)),Ba) = bot_bot(set(A)) )
<=> ( ~ member(A,A3,Ba)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) ) ) ) ).
% insert_disjoint(1)
tff(fact_1850_insert__Diff__single,axiom,
! [A: $tType,A3: A,Aa2: set(A)] : aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = aa(set(A),set(A),insert2(A,A3),Aa2) ).
% insert_Diff_single
tff(fact_1851_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_1852_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_1853_one__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).
% one_le_floor
tff(fact_1854_zero__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).
% zero_less_floor
tff(fact_1855_floor__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).
% floor_le_zero
tff(fact_1856_floor__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).
% floor_less_one
tff(fact_1857_subset__Compl__singleton,axiom,
! [A: $tType,Aa2: set(A),B2: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))
<=> ~ member(A,B2,Aa2) ) ).
% subset_Compl_singleton
tff(fact_1858_ceiling__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ).
% ceiling_le_one
tff(fact_1859_one__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ).
% one_less_ceiling
tff(fact_1860_floor__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ).
% floor_diff_one
tff(fact_1861_ceiling__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).
% ceiling_diff_one
tff(fact_1862_ceiling__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).
% ceiling_less_zero
tff(fact_1863_zero__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).
% zero_le_ceiling
tff(fact_1864_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).
% numeral_less_floor
tff(fact_1865_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).
% floor_le_numeral
tff(fact_1866_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).
% ceiling_less_numeral
tff(fact_1867_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).
% numeral_le_ceiling
tff(fact_1868_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).
% neg_numeral_less_floor
tff(fact_1869_divmod__int__def,axiom,
! [M2: num,N: num] : unique8689654367752047608divmod(int,M2,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N))) ).
% divmod_int_def
tff(fact_1870_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) = aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))) )
=> ( A3 = B2 ) ) ).
% singleton_inject
tff(fact_1871_insert__not__empty,axiom,
! [A: $tType,A3: A,Aa2: set(A)] : aa(set(A),set(A),insert2(A,A3),Aa2) != bot_bot(set(A)) ).
% insert_not_empty
tff(fact_1872_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C2: A,D3: A] :
( ( aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,C2),aa(set(A),set(A),insert2(A,D3),bot_bot(set(A)))) )
<=> ( ( ( A3 = C2 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
tff(fact_1873_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( member(A,B2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))
<=> ( B2 = A3 ) ) ).
% singleton_iff
tff(fact_1874_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( member(A,B2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))
=> ( B2 = A3 ) ) ).
% singletonD
tff(fact_1875_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_1876_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( Y = $ite(
( member(int,X,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).
% and_int.elims
tff(fact_1877_and__int_Osimps,axiom,
! [K: int,L: int] :
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
( member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).
% and_int.simps
tff(fact_1878_subset__singletonD,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))
=> ( ( Aa2 = bot_bot(set(A)) )
| ( Aa2 = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ) ).
% subset_singletonD
tff(fact_1879_subset__singleton__iff,axiom,
! [A: $tType,X5: set(A),A3: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))
<=> ( ( X5 = bot_bot(set(A)) )
| ( X5 = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) ) ) ).
% subset_singleton_iff
tff(fact_1880_insert__is__Un,axiom,
! [A: $tType,A3: A,Aa2: set(A)] : aa(set(A),set(A),insert2(A,A3),Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))),Aa2) ).
% insert_is_Un
tff(fact_1881_Un__singleton__iff,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A),X: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
<=> ( ( ( Aa2 = bot_bot(set(A)) )
& ( Ba = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) )
| ( ( Aa2 = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
& ( Ba = bot_bot(set(A)) ) )
| ( ( Aa2 = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
& ( Ba = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ) ) ).
% Un_singleton_iff
tff(fact_1882_singleton__Un__iff,axiom,
! [A: $tType,X: A,Aa2: set(A),Ba: set(A)] :
( ( aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba) )
<=> ( ( ( Aa2 = bot_bot(set(A)) )
& ( Ba = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) )
| ( ( Aa2 = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
& ( Ba = bot_bot(set(A)) ) )
| ( ( Aa2 = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
& ( Ba = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ) ) ).
% singleton_Un_iff
tff(fact_1883_Diff__insert,axiom,
! [A: $tType,Aa2: set(A),A3: A,Ba: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),Ba)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ).
% Diff_insert
tff(fact_1884_insert__Diff,axiom,
! [A: $tType,A3: A,Aa2: set(A)] :
( member(A,A3,Aa2)
=> ( aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = Aa2 ) ) ).
% insert_Diff
tff(fact_1885_Diff__insert2,axiom,
! [A: $tType,Aa2: set(A),A3: A,Ba: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),Ba)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),Ba) ).
% Diff_insert2
tff(fact_1886_Diff__insert__absorb,axiom,
! [A: $tType,X: A,Aa2: set(A)] :
( ~ member(A,X,Aa2)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert2(A,X),Aa2)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = Aa2 ) ) ).
% Diff_insert_absorb
tff(fact_1887_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X5: set(A)] :
( ~ member(A,X,X5)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X5),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X5 ) ) ).
% set_minus_singleton_eq
tff(fact_1888_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,Aa2: set(A)] :
( ( X != Y )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert2(A,X),Aa2)),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))) ) ) ).
% insert_minus_eq
tff(fact_1889_le__mult__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(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_1890_mult__ceiling__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),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_1891_subset__insert__iff,axiom,
! [A: $tType,Aa2: set(A),X: A,Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),Ba))
<=> $ite(member(A,X,Aa2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),Ba),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)) ) ).
% subset_insert_iff
tff(fact_1892_Diff__single__insert,axiom,
! [A: $tType,Aa2: set(A),X: A,Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),Ba)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),Ba)) ) ).
% Diff_single_insert
tff(fact_1893_card__1__singletonE,axiom,
! [A: $tType,Aa2: set(A)] :
( ( finite_card(A,Aa2) = one_one(nat) )
=> ~ ! [X3: A] : Aa2 != aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_1894_remove__subset,axiom,
! [A: $tType,X: A,S: set(A)] :
( member(A,X,S)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),S) ) ).
% remove_subset
tff(fact_1895_Compl__insert,axiom,
! [A: $tType,X: A,Aa2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Aa2)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% Compl_insert
tff(fact_1896_card__Suc__eq,axiom,
! [A: $tType,Aa2: set(A),K: nat] :
( ( finite_card(A,Aa2) = aa(nat,nat,suc,K) )
<=> ? [B4: A,B7: set(A)] :
( ( Aa2 = aa(set(A),set(A),insert2(A,B4),B7) )
& ~ member(A,B4,B7)
& ( finite_card(A,B7) = K )
& ( ( K = zero_zero(nat) )
=> ( B7 = bot_bot(set(A)) ) ) ) ) ).
% card_Suc_eq
tff(fact_1897_card__eq__SucD,axiom,
! [A: $tType,Aa2: set(A),K: nat] :
( ( finite_card(A,Aa2) = aa(nat,nat,suc,K) )
=> ? [B3: A,B8: set(A)] :
( ( Aa2 = aa(set(A),set(A),insert2(A,B3),B8) )
& ~ member(A,B3,B8)
& ( finite_card(A,B8) = K )
& ( ( K = zero_zero(nat) )
=> ( B8 = bot_bot(set(A)) ) ) ) ) ).
% card_eq_SucD
tff(fact_1898_card__1__singleton__iff,axiom,
! [A: $tType,Aa2: set(A)] :
( ( finite_card(A,Aa2) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ? [X4: A] : Aa2 = aa(set(A),set(A),insert2(A,X4),bot_bot(set(A))) ) ).
% card_1_singleton_iff
tff(fact_1899_card__Diff1__le,axiom,
! [A: $tType,Aa2: set(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))),finite_card(A,Aa2)) ).
% card_Diff1_le
tff(fact_1900_psubset__insert__iff,axiom,
! [A: $tType,Aa2: set(A),X: A,Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),Ba))
<=> $ite(
member(A,X,Ba),
aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),Ba),
$ite(member(A,X,Aa2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),Ba),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)) ) ) ).
% psubset_insert_iff
tff(fact_1901_bezw_Osimps,axiom,
! [X: nat,Y: nat] :
bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y)))))) ).
% bezw.simps
tff(fact_1902_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),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_1903_card__2__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
<=> ? [X4: A,Y3: A] :
( ( S = aa(set(A),set(A),insert2(A,X4),aa(set(A),set(A),insert2(A,Y3),bot_bot(set(A)))) )
& ( X4 != Y3 ) ) ) ).
% card_2_iff
tff(fact_1904_card__3__iff,axiom,
! [A: $tType,S: set(A)] :
( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
<=> ? [X4: A,Y3: A,Z4: A] :
( ( S = aa(set(A),set(A),insert2(A,X4),aa(set(A),set(A),insert2(A,Y3),aa(set(A),set(A),insert2(A,Z4),bot_bot(set(A))))) )
& ( X4 != Y3 )
& ( Y3 != Z4 )
& ( X4 != Z4 ) ) ) ).
% card_3_iff
tff(fact_1905_card__Diff__singleton__if,axiom,
! [A: $tType,Aa2: set(A),X: A] :
finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = $ite(member(A,X,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),one_one(nat)),finite_card(A,Aa2)) ).
% card_Diff_singleton_if
tff(fact_1906_card__Diff__singleton,axiom,
! [A: $tType,X: A,Aa2: set(A)] :
( member(A,X,Aa2)
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,Aa2)),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_1907_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa) = Y )
=> ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
=> ~ ( ( Y = $ite(Xa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa)))))) )
=> ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).
% bezw.pelims
tff(fact_1908_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
=> ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),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),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)),one_one(int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_1909_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),aa(num,num,bit0,one2))))) ) ).
% round_def
tff(fact_1910_the__elem__eq,axiom,
! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ).
% the_elem_eq
tff(fact_1911_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R2: int] :
( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),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),aa(num,num,bit0,one2))),R2)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_1912_is__singletonI,axiom,
! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% is_singletonI
tff(fact_1913_divmod__BitM__2__eq,axiom,
! [M2: num] : unique8689654367752047608divmod(int,bitM(M2),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int))),one_one(int)) ).
% divmod_BitM_2_eq
tff(fact_1914_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_1915_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> ( modulo_modulo(int,K,L) = R2 ) ) ).
% mod_int_unique
tff(fact_1916_eucl__rel__int,axiom,
! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,K,L)),modulo_modulo(int,K,L))) ).
% eucl_rel_int
tff(fact_1917_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
=> ( divide_divide(int,K,L) = Q3 ) ) ).
% div_int_unique
tff(fact_1918_unique__quotient,axiom,
! [A3: int,B2: int,Q3: int,R2: int,Q6: int,R4: 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))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
=> ( Q3 = Q6 ) ) ) ).
% unique_quotient
tff(fact_1919_unique__remainder,axiom,
! [A3: int,B2: int,Q3: int,R2: int,Q6: int,R4: 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))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
=> ( R2 = R4 ) ) ) ).
% unique_remainder
tff(fact_1920_eucl__rel__int__by0,axiom,
! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).
% eucl_rel_int_by0
tff(fact_1921_is__singleton__the__elem,axiom,
! [A: $tType,Aa2: set(A)] :
( is_singleton(A,Aa2)
<=> ( Aa2 = aa(set(A),set(A),insert2(A,the_elem(A,Aa2)),bot_bot(set(A))) ) ) ).
% is_singleton_the_elem
tff(fact_1922_is__singletonI_H,axiom,
! [A: $tType,Aa2: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> ( X3 = Y2 ) ) )
=> is_singleton(A,Aa2) ) ) ).
% is_singletonI'
tff(fact_1923_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L != zero_zero(int) )
=> ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_1924_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K: int,Q3: int] :
( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R2) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_1925_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
<=> ( ? [K4: int] :
( ( A1 = K4 )
& ( A22 = zero_zero(int) )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K4) ) )
| ? [L2: int,K4: int,Q7: int] :
( ( A1 = K4 )
& ( A22 = L2 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),zero_zero(int)) )
& ( L2 != zero_zero(int) )
& ( K4 = aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L2) ) )
| ? [R5: int,L2: int,K4: int,Q7: int] :
( ( A1 = K4 )
& ( A22 = L2 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R5) )
& ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L2) )
& aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L2))
& ( K4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L2)),R5) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_1926_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
=> ( ( ( A22 = zero_zero(int) )
=> ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
=> ( ! [Q5: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
=> ( ( A22 != zero_zero(int) )
=> ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22) ) ) )
=> ~ ! [R3: int,Q5: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3) )
=> ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A22) )
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22))
=> ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22)),R3) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_1927_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),
$ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
$ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).
% zminus1_lemma
tff(fact_1928_numeral__BitM,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))),one_one(A)) ) ).
% numeral_BitM
tff(fact_1929_is__singleton__def,axiom,
! [A: $tType,Aa2: set(A)] :
( is_singleton(A,Aa2)
<=> ? [X4: A] : Aa2 = aa(set(A),set(A),insert2(A,X4),bot_bot(set(A))) ) ).
% is_singleton_def
tff(fact_1930_is__singletonE,axiom,
! [A: $tType,Aa2: set(A)] :
( is_singleton(A,Aa2)
=> ~ ! [X3: A] : Aa2 != aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))) ) ).
% is_singletonE
tff(fact_1931_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R2: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
<=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R2) )
& $ite(
aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
& aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
$ite(
aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
& aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
Q3 = zero_zero(int) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_1932_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).
% Divides.adjust_div_eq
tff(fact_1933_round__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
archimedean_round(A,X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).
% round_altdef
tff(fact_1934_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
=> ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
=> ~ ( ( Y = $ite(
( member(int,X,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
=> ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).
% and_int.pelims
tff(fact_1935_and__int_Opsimps,axiom,
! [K: int,L: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
( member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
aa(int,int,uminus_uminus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
aa(int,int,
aa(int,fun(int,int),plus_plus(int),
aa($o,int,zero_neq_one_of_bool(int),
( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)
& ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ))),
aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).
% and_int.psimps
tff(fact_1936_round__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
=> ( archimedean_round(A,X) = Y ) ) ) ) ).
% round_unique
tff(fact_1937_normalize__negative,axiom,
! [Q3: int,P5: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q3),zero_zero(int))
=> ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P5)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).
% normalize_negative
tff(fact_1938_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
=> ( archimedean_round(A,X) = N ) ) ) ).
% round_unique'
tff(fact_1939_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_1940_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_1941_of__int__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_mult
tff(fact_1942_normalize__denom__zero,axiom,
! [P5: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),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_1943_of__int__le__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),one_one(int)) ) ) ).
% of_int_le_1_iff
tff(fact_1944_of__int__1__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z2) ) ) ).
% of_int_1_le_iff
tff(fact_1945_of__int__less__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),one_one(int)) ) ) ).
% of_int_less_1_iff
tff(fact_1946_of__int__1__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2))
<=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z2) ) ) ).
% of_int_1_less_iff
tff(fact_1947_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_1948_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_1949_frac__lt__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,X)),one_one(A)) ) ).
% frac_lt_1
tff(fact_1950_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_1951_of__int__leD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X)
=> ( ( N = zero_zero(int) )
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).
% of_int_leD
tff(fact_1952_of__int__lessD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X)
=> ( ( N = zero_zero(int) )
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).
% of_int_lessD
tff(fact_1953_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,$o),T2: A] :
( aa(int,$o,P,archim6421214686448440834_floor(A,T2))
<=> ! [I2: int] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),T2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))) )
=> aa(int,$o,P,I2) ) ) ) ).
% floor_split
tff(fact_1954_floor__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archim6421214686448440834_floor(A,X) = A3 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))) ) ) ) ).
% floor_eq_iff
tff(fact_1955_floor__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
=> ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).
% floor_unique
tff(fact_1956_ceiling__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).
% ceiling_correct
tff(fact_1957_ceiling__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))
=> ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).
% ceiling_unique
tff(fact_1958_ceiling__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archimedean_ceiling(A,X) = A3 )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).
% ceiling_eq_iff
tff(fact_1959_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,$o),T2: A] :
( aa(int,$o,P,archimedean_ceiling(A,T2))
<=> ! [I2: int] :
( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))),T2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I2)) )
=> aa(int,$o,P,I2) ) ) ) ).
% ceiling_split
tff(fact_1960_less__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),archim6421214686448440834_floor(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X) ) ) ).
% less_floor_iff
tff(fact_1961_floor__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).
% floor_le_iff
tff(fact_1962_ceiling__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),Z2)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))) ) ) ).
% ceiling_less_iff
tff(fact_1963_le__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z2),archimedean_ceiling(A,X))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X) ) ) ).
% le_ceiling_iff
tff(fact_1964_normalize__denom__pos,axiom,
! [R2: product_prod(int,int),P5: int,Q3: int] :
( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).
% normalize_denom_pos
tff(fact_1965_normalize__crossproduct,axiom,
! [Q3: int,S2: int,P5: int,R2: int] :
( ( Q3 != zero_zero(int) )
=> ( ( S2 != zero_zero(int) )
=> ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S2)) )
=> ( aa(int,int,aa(int,fun(int,int),times_times(int),P5),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q3) ) ) ) ) ).
% normalize_crossproduct
tff(fact_1966_frac__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( archimedean_frac(A,X) = X )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).
% frac_eq
tff(fact_1967_frac__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A))) ) ).
% frac_add
tff(fact_1968_floor__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P5: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P5,Q3)))),Q3)),P5) ) ) ).
% floor_divide_lower
tff(fact_1969_ceiling__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P5: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P5),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,P5,Q3)))),Q3)) ) ) ).
% ceiling_divide_upper
tff(fact_1970_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P5: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),P5),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,P5,Q3)))),one_one(A))),Q3)) ) ) ).
% floor_divide_upper
tff(fact_1971_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P5: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P5,Q3)))),one_one(A))),Q3)),P5) ) ) ).
% ceiling_divide_lower
tff(fact_1972_of__int__round__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% of_int_round_le
tff(fact_1973_of__int__round__ge,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).
% of_int_round_ge
tff(fact_1974_of__int__round__gt,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).
% of_int_round_gt
tff(fact_1975_floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).
% floor_add
tff(fact_1976_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
=> ( ! [K2: int,L3: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L3))
=> ( ( ~ ( member(int,K2,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
& member(int,L3,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
=> aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
=> aa(int,$o,aa(int,fun(int,$o),P,K2),L3) ) )
=> aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).
% and_int.pinduct
tff(fact_1977_of__int__round__abs__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).
% of_int_round_abs_le
tff(fact_1978_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
=> ( ! [I3: int,J2: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
=> ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J2)
=> aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J2) )
=> aa(int,$o,aa(int,fun(int,$o),P,I3),J2) ) )
=> aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).
% upto.pinduct
tff(fact_1979_mult__ceiling__le__Ints,axiom,
! [B: $tType,A: $tType] :
( ( archim2362893244070406136eiling(A)
& linordered_idom(B) )
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( member(A,A3,ring_1_Ints(A))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).
% mult_ceiling_le_Ints
tff(fact_1980_le__mult__floor__Ints,axiom,
! [B: $tType,A: $tType] :
( ( archim2362893244070406136eiling(A)
& linordered_idom(B) )
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( member(A,A3,ring_1_Ints(A))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).
% le_mult_floor_Ints
tff(fact_1981_frac__unique__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: A] :
( ( archimedean_frac(A,X) = A3 )
<=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3),ring_1_Ints(A))
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) ) ) ) ).
% frac_unique_iff
tff(fact_1982_Gcd__0__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
( ( gcd_Gcd(A,Aa2) = zero_zero(A) )
<=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,zero_zero(A)),bot_bot(set(A)))) ) ) ).
% Gcd_0_iff
tff(fact_1983_remove__def,axiom,
! [A: $tType,X: A,Aa2: set(A)] : remove(A,X,Aa2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% remove_def
tff(fact_1984_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_1985_Gcd__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_empty
tff(fact_1986_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_1987_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_1988_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_1989_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_1990_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_1991_Ints__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,B2: A] :
( member(A,A3,ring_1_Ints(A))
=> ( member(A,B2,ring_1_Ints(A))
=> member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),ring_1_Ints(A)) ) ) ) ).
% Ints_mult
tff(fact_1992_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> member(A,one_one(A),ring_1_Ints(A)) ) ).
% Ints_1
tff(fact_1993_quotient__of__div,axiom,
! [R2: rat,N: int,D3: int] :
( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),D3) )
=> ( R2 = divide_divide(rat,aa(int,rat,ring_1_of_int(rat),N),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).
% quotient_of_div
tff(fact_1994_Gcd__1,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
( member(A,one_one(A),Aa2)
=> ( gcd_Gcd(A,Aa2) = one_one(A) ) ) ) ).
% Gcd_1
tff(fact_1995_Gcd__eq__1__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,Aa2: set(A)] :
( dvd_dvd(A,A3,one_one(A))
=> ( member(A,A3,Aa2)
=> ( gcd_Gcd(A,Aa2) = one_one(A) ) ) ) ) ).
% Gcd_eq_1_I
tff(fact_1996_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [A3: A] :
( member(A,A3,ring_1_Ints(A))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).
% Ints_odd_nonzero
tff(fact_1997_quotient__of__denom__pos,axiom,
! [R2: rat,P5: int,Q3: int] :
( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).
% quotient_of_denom_pos
tff(fact_1998_Ints__odd__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( member(A,A3,ring_1_Ints(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).
% Ints_odd_less_0
tff(fact_1999_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( member(A,X,ring_1_Ints(A))
=> ( ( X != zero_zero(A) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X)) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_2000_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( member(A,X,ring_1_Ints(A))
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))
=> ( X = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_2001_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( member(A,X,ring_1_Ints(A))
=> ( member(A,Y,ring_1_Ints(A))
=> ( ( X = Y )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),one_one(A)) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_2002_rat__sgn__code,axiom,
! [P5: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P5)) = 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(P5)))),one_one(int)) ).
% rat_sgn_code
tff(fact_2003_frac__neg,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = $ite(member(A,X,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X))) ) ).
% frac_neg
tff(fact_2004_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_2005_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_2006_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_2007_Frct__code__post_I6_J,axiom,
! [K: num,L: 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)),aa(num,int,numeral_numeral(int),L))) = divide_divide(rat,aa(num,rat,numeral_numeral(rat),K),aa(num,rat,numeral_numeral(rat),L)) ).
% Frct_code_post(6)
tff(fact_2008_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).
% Frct_code_post(8)
tff(fact_2009_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A3)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).
% Frct_code_post(7)
tff(fact_2010_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
aa(num,A,numeral_numeral(A),num_of_nat(N)) = $ite(N = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% numeral_num_of_nat_unfold
tff(fact_2011_Frct__code__post_I1_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A3)) = zero_zero(rat) ).
% Frct_code_post(1)
tff(fact_2012_Frct__code__post_I2_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),zero_zero(int))) = zero_zero(rat) ).
% Frct_code_post(2)
tff(fact_2013_Frct__code__post_I3_J,axiom,
frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).
% Frct_code_post(3)
tff(fact_2014_zero__rat__def,axiom,
zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).
% zero_rat_def
tff(fact_2015_zero__rat_Otransfer,axiom,
aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat)) ).
% zero_rat.transfer
tff(fact_2016_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_2017_simp__from__to,axiom,
! [I: int,J: int] :
set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),insert2(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).
% simp_from_to
tff(fact_2018_one__int_Otransfer,axiom,
aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int)) ).
% one_int.transfer
tff(fact_2019_card__Un__disjoint,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,Aa2)),finite_card(A,Ba)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_2020_card__Diff1__less,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))),finite_card(A,Aa2)) ) ) ).
% card_Diff1_less
tff(fact_2021_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% atLeastatMost_empty_iff2
tff(fact_2022_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2) ) ) ).
% atLeastatMost_empty_iff
tff(fact_2023_atLeastatMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastatMost_empty
tff(fact_2024_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) ).
% atLeastAtMost_singleton
tff(fact_2025_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A,C2: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert2(A,C2),bot_bot(set(A))) )
<=> ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_2026_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_2027_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_2028_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_2029_card__0__eq,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( finite_card(A,Aa2) = zero_zero(nat) )
<=> ( Aa2 = bot_bot(set(A)) ) ) ) ).
% card_0_eq
tff(fact_2030_of__rat__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),field_char_0_of_rat(A,R2)),one_one(A))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).
% of_rat_le_1_iff
tff(fact_2031_one__le__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R2))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).
% one_le_of_rat_iff
tff(fact_2032_of__rat__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),one_one(A))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).
% of_rat_less_1_iff
tff(fact_2033_one__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R2: rat] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),field_char_0_of_rat(A,R2))
<=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).
% one_less_of_rat_iff
tff(fact_2034_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_2035_finite_OemptyI,axiom,
! [A: $tType] : aa(set(A),$o,finite_finite(A),bot_bot(set(A))) ).
% finite.emptyI
tff(fact_2036_infinite__imp__nonempty,axiom,
! [A: $tType,S: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),S)
=> ( S != bot_bot(set(A)) ) ) ).
% infinite_imp_nonempty
tff(fact_2037_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) ) ) ).
% atLeastAtMost_singleton'
tff(fact_2038_finite__has__minimal,axiom,
! [A: $tType] :
( order(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,Aa2)
& ! [Xa3: A] :
( member(A,Xa3,Aa2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa3),X3)
=> ( X3 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_2039_finite__has__maximal,axiom,
! [A: $tType] :
( order(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,Aa2)
& ! [Xa3: A] :
( member(A,Xa3,Aa2)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa3)
=> ( X3 = Xa3 ) ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_2040_finite_Ocases,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite(A),A3)
=> ( ( A3 != bot_bot(set(A)) )
=> ~ ! [A7: set(A)] :
( ? [A4: A] : A3 = aa(set(A),set(A),insert2(A,A4),A7)
=> ~ aa(set(A),$o,finite_finite(A),A7) ) ) ) ).
% finite.cases
tff(fact_2041_finite_Osimps,axiom,
! [A: $tType,A3: set(A)] :
( aa(set(A),$o,finite_finite(A),A3)
<=> ( ( A3 = bot_bot(set(A)) )
| ? [A8: set(A),A5: A] :
( ( A3 = aa(set(A),set(A),insert2(A,A5),A8) )
& aa(set(A),$o,finite_finite(A),A8) ) ) ) ).
% finite.simps
tff(fact_2042_finite__induct,axiom,
! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite(A),F5)
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X3),F5)) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ).
% finite_induct
tff(fact_2043_finite__ne__induct,axiom,
! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( ( F4 != bot_bot(set(A)) )
=> ( ! [X3: A] : aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite(A),F5)
=> ( ( F5 != bot_bot(set(A)) )
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X3),F5)) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_ne_induct
tff(fact_2044_infinite__finite__induct,axiom,
! [A: $tType,P: fun(set(A),$o),Aa2: set(A)] :
( ! [A7: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),A7)
=> aa(set(A),$o,P,A7) )
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,F5: set(A)] :
( aa(set(A),$o,finite_finite(A),F5)
=> ( ~ member(A,X3,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X3),F5)) ) ) )
=> aa(set(A),$o,P,Aa2) ) ) ) ).
% infinite_finite_induct
tff(fact_2045_Gcd__remove0__nat,axiom,
! [M4: set(nat)] :
( aa(set(nat),$o,finite_finite(nat),M4)
=> ( gcd_Gcd(nat,M4) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M4),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).
% Gcd_remove0_nat
tff(fact_2046_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( bounded_lattice(A)
=> ! [X: A,Y: A] :
( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
<=> ( ( X = bot_bot(A) )
& ( Y = top_top(A) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_2047_finite__subset__induct_H,axiom,
! [A: $tType,F4: set(A),Aa2: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),Aa2)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A4: A,F5: set(A)] :
( aa(set(A),$o,finite_finite(A),F5)
=> ( member(A,A4,Aa2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),Aa2)
=> ( ~ member(A,A4,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,A4),F5)) ) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_subset_induct'
tff(fact_2048_finite__subset__induct,axiom,
! [A: $tType,F4: set(A),Aa2: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),Aa2)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A4: A,F5: set(A)] :
( aa(set(A),$o,finite_finite(A),F5)
=> ( member(A,A4,Aa2)
=> ( ~ member(A,A4,F5)
=> ( aa(set(A),$o,P,F5)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,A4),F5)) ) ) ) )
=> aa(set(A),$o,P,F4) ) ) ) ) ).
% finite_subset_induct
tff(fact_2049_finite__empty__induct,axiom,
! [A: $tType,Aa2: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,P,Aa2)
=> ( ! [A4: A,A7: set(A)] :
( aa(set(A),$o,finite_finite(A),A7)
=> ( member(A,A4,A7)
=> ( aa(set(A),$o,P,A7)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert2(A,A4),bot_bot(set(A))))) ) ) )
=> aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).
% finite_empty_induct
tff(fact_2050_infinite__coinduct,axiom,
! [A: $tType,X5: fun(set(A),$o),Aa2: set(A)] :
( aa(set(A),$o,X5,Aa2)
=> ( ! [A7: set(A)] :
( aa(set(A),$o,X5,A7)
=> ? [X2: A] :
( member(A,X2,A7)
& ( aa(set(A),$o,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))))
| ~ aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) ) ) )
=> ~ aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% infinite_coinduct
tff(fact_2051_infinite__remove,axiom,
! [A: $tType,S: set(A),A3: A] :
( ~ aa(set(A),$o,finite_finite(A),S)
=> ~ aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) ) ).
% infinite_remove
tff(fact_2052_card__eq__0__iff,axiom,
! [A: $tType,Aa2: set(A)] :
( ( finite_card(A,Aa2) = zero_zero(nat) )
<=> ( ( Aa2 = bot_bot(set(A)) )
| ~ aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% card_eq_0_iff
tff(fact_2053_zero__int_Otransfer,axiom,
aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int)) ).
% zero_int.transfer
tff(fact_2054_remove__induct,axiom,
! [A: $tType,P: fun(set(A),$o),Ba: set(A)] :
( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ( ~ aa(set(A),$o,finite_finite(A),Ba)
=> aa(set(A),$o,P,Ba) )
=> ( ! [A7: set(A)] :
( aa(set(A),$o,finite_finite(A),A7)
=> ( ( A7 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),Ba)
=> ( ! [X2: A] :
( member(A,X2,A7)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) )
=> aa(set(A),$o,P,A7) ) ) ) )
=> aa(set(A),$o,P,Ba) ) ) ) ).
% remove_induct
tff(fact_2055_finite__remove__induct,axiom,
! [A: $tType,Ba: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),Ba)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [A7: set(A)] :
( aa(set(A),$o,finite_finite(A),A7)
=> ( ( A7 != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),Ba)
=> ( ! [X2: A] :
( member(A,X2,A7)
=> aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) )
=> aa(set(A),$o,P,A7) ) ) ) )
=> aa(set(A),$o,P,Ba) ) ) ) ).
% finite_remove_induct
tff(fact_2056_card__gt__0__iff,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,Aa2))
<=> ( ( Aa2 != bot_bot(set(A)) )
& aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% card_gt_0_iff
tff(fact_2057_card__1__singletonI,axiom,
! [A: $tType,S: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( finite_card(A,S) = one_one(nat) )
=> ( member(A,X,S)
=> ( S = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ) ) ).
% card_1_singletonI
tff(fact_2058_finite__induct__select,axiom,
! [A: $tType,S: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [T3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T3),S)
=> ( aa(set(A),$o,P,T3)
=> ? [X2: A] :
( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T3))
& aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X2),T3)) ) ) )
=> aa(set(A),$o,P,S) ) ) ) ).
% finite_induct_select
tff(fact_2059_one__rat_Otransfer,axiom,
aa(rat,$o,aa(product_prod(int,int),fun(rat,$o),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat)) ).
% one_rat.transfer
tff(fact_2060_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_2061_card__Suc__Diff1,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) = finite_card(A,Aa2) ) ) ) ).
% card_Suc_Diff1
tff(fact_2062_card_Oinsert__remove,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( finite_card(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ).
% card.insert_remove
tff(fact_2063_card_Oremove,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( finite_card(A,Aa2) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% card.remove
tff(fact_2064_card__Diff1__less__iff,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))),finite_card(A,Aa2))
<=> ( aa(set(A),$o,finite_finite(A),Aa2)
& member(A,X,Aa2) ) ) ).
% card_Diff1_less_iff
tff(fact_2065_card__Diff2__less,axiom,
! [A: $tType,Aa2: set(A),X: A,Y: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( member(A,Y,Aa2)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))))),finite_card(A,Aa2)) ) ) ) ).
% card_Diff2_less
tff(fact_2066_finite__linorder__max__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [B3: A,A7: set(A)] :
( aa(set(A),$o,finite_finite(A),A7)
=> ( ! [X2: A] :
( member(A,X2,A7)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),B3) )
=> ( aa(set(A),$o,P,A7)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,B3),A7)) ) ) )
=> aa(set(A),$o,P,Aa2) ) ) ) ) ).
% finite_linorder_max_induct
tff(fact_2067_finite__linorder__min__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),P: fun(set(A),$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [B3: A,A7: set(A)] :
( aa(set(A),$o,finite_finite(A),A7)
=> ( ! [X2: A] :
( member(A,X2,A7)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),X2) )
=> ( aa(set(A),$o,P,A7)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,B3),A7)) ) ) )
=> aa(set(A),$o,P,Aa2) ) ) ) ) ).
% finite_linorder_min_induct
tff(fact_2068_finite__ranking__induct,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S: set(A),P: fun(set(A),$o),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),$o,P,bot_bot(set(A)))
=> ( ! [X3: A,S4: set(A)] :
( aa(set(A),$o,finite_finite(A),S4)
=> ( ! [Y4: A] :
( member(A,Y4,S4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y4)),aa(A,B,F,X3)) )
=> ( aa(set(A),$o,P,S4)
=> aa(set(A),$o,P,aa(set(A),set(A),insert2(A,X3),S4)) ) ) )
=> aa(set(A),$o,P,S) ) ) ) ) ).
% finite_ranking_induct
tff(fact_2069_ex__min__if__finite,axiom,
! [A: $tType] :
( order(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ? [X3: A] :
( member(A,X3,S)
& ~ ? [Xa3: A] :
( member(A,Xa3,S)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X3) ) ) ) ) ) ).
% ex_min_if_finite
tff(fact_2070_infinite__growing,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X5: set(A)] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa3) ) )
=> ~ aa(set(A),$o,finite_finite(A),X5) ) ) ) ).
% infinite_growing
tff(fact_2071_uminus__rat_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
=> ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).
% uminus_rat.abs_eq
tff(fact_2072_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod(int,int),X: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
=> ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_2073_one__rat_Orsp,axiom,
aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).
% one_rat.rsp
tff(fact_2074_zero__rat_Orsp,axiom,
aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).
% zero_rat.rsp
tff(fact_2075_times__rat_Oabs__eq,axiom,
! [Xa: product_prod(int,int),X: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
=> ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
=> ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% times_rat.abs_eq
tff(fact_2076_inverse__rat_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,X),X)
=> ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,
$ite(aa(product_prod(int,int),int,product_fst(int,int),X) = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_2077_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ~ ? [X2: A] :
( member(A,X2,S)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S))) ) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_2078_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S: set(A),Y: A,F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( member(A,Y,S)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S))),aa(A,B,F,Y)) ) ) ) ) ).
% arg_min_least
tff(fact_2079_finite__transitivity__chain,axiom,
! [A: $tType,Aa2: set(A),R: fun(A,fun(A,$o))] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
=> ( ! [X3: A,Y2: A,Z3: A] :
( aa(A,$o,aa(A,fun(A,$o),R,X3),Y2)
=> ( aa(A,$o,aa(A,fun(A,$o),R,Y2),Z3)
=> aa(A,$o,aa(A,fun(A,$o),R,X3),Z3) ) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> ? [Y4: A] :
( member(A,Y4,Aa2)
& aa(A,$o,aa(A,fun(A,$o),R,X3),Y4) ) )
=> ( Aa2 = bot_bot(set(A)) ) ) ) ) ) ).
% finite_transitivity_chain
tff(fact_2080_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2) = zero_zero(A) )
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,zero_zero(A)),bot_bot(set(A))))
& aa(set(A),$o,finite_finite(A),Aa2) ) ) ) ).
% Gcd_fin_0_iff
tff(fact_2081_greaterThan__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).
% greaterThan_Suc
tff(fact_2082_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ).
% Sup_fin.insert_remove
tff(fact_2083_Sup__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ) ).
% Sup_fin.remove
tff(fact_2084_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_fin.empty
tff(fact_2085_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% Sup_fin.singleton
tff(fact_2086_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_2087_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
( dvd_dvd(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2),one_one(A))
<=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2) = one_one(A) ) ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_2088_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) ) ) ) ) ).
% Sup_fin.insert
tff(fact_2089_greaterThan__non__empty,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).
% greaterThan_non_empty
tff(fact_2090_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) = aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2) ) ) ) ) ).
% Sup_fin.in_idem
tff(fact_2091_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),X)
=> ! [A9: A] :
( member(A,A9,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),X) ) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_2092_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),X) ) ) ) ) ).
% Sup_fin.boundedI
tff(fact_2093_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),X)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_2094_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(7)
tff(fact_2095_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),aa(set(A),A,lattic5882676163264333800up_fin(A),Ba)) ) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_2096_Sup__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Ba)),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) = aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2) ) ) ) ) ) ).
% Sup_fin.subset
tff(fact_2097_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2),Aa2) ) ) ) ) ).
% Sup_fin.closed
tff(fact_2098_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_2099_Sup__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),aa(set(A),A,lattic5882676163264333800up_fin(A),Ba)) ) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_2100_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> member(A,lattic7623131987881927897min_on(A,B,F,S),S) ) ) ) ).
% arg_min_if_finite(1)
tff(fact_2101_enumerate__Suc_H,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,infini527867602293511546merate(A,S,zero_zero(nat))),bot_bot(set(A)))),N) ) ).
% enumerate_Suc'
tff(fact_2102_Inf__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( lattic7752659483105999362nf_fin(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ) ).
% Inf_fin.remove
tff(fact_2103_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ).
% Inf_fin.insert_remove
tff(fact_2104_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).
% Inf_fin.insert
tff(fact_2105_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)),bot_bot(set(A))) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_2106_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_atMost_eq_remove0
tff(fact_2107_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_2108_image__is__empty,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: set(B)] :
( ( aa(set(B),set(A),image2(B,A,F),Aa2) = bot_bot(set(A)) )
<=> ( Aa2 = bot_bot(set(B)) ) ) ).
% image_is_empty
tff(fact_2109_empty__is__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: set(B)] :
( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F),Aa2) )
<=> ( Aa2 = bot_bot(set(B)) ) ) ).
% empty_is_image
tff(fact_2110_image__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(set(B),set(A),image2(B,A,F),bot_bot(set(B))) = bot_bot(set(A)) ).
% image_empty
tff(fact_2111_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S)
=> member(A,A3,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)) ) ).
% img_fst
tff(fact_2112_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S)
=> member(B,B2,aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S)) ) ).
% img_snd
tff(fact_2113_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,Aa2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),Aa2))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),Aa2) ) ).
% pair_in_swap_image
tff(fact_2114_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
=> ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanAtMost_empty
tff(fact_2115_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).
% greaterThanAtMost_empty_iff
tff(fact_2116_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_2117_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% Inf_fin.singleton
tff(fact_2118_sup__Inf__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Aa2: set(A),A3: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,A3,Aa2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,Aa2)),A3) = A3 ) ) ) ) ).
% sup_Inf_absorb
tff(fact_2119_atMost__0,axiom,
set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))) ).
% atMost_0
tff(fact_2120_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D3: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,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_2121_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_2122_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [H2: A] : bot_bot(set(A)) != set_ord_atMost(A,H2) ) ).
% not_empty_eq_Iic_eq_empty
tff(fact_2123_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [H2: fun(A,A),N3: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H2,X3)),aa(A,A,H2,Y2))
=> ( aa(set(A),$o,finite_finite(A),N3)
=> ( ( N3 != bot_bot(set(A)) )
=> ( aa(A,A,H2,lattic7752659483105999362nf_fin(A,N3)) = lattic7752659483105999362nf_fin(A,aa(set(A),set(A),image2(A,A,H2),N3)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_2124_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
( member(A,X,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))
=> ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),S) ) ).
% in_fst_imageE
tff(fact_2125_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
( member(A,Y,aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))
=> ~ ! [X3: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y),S) ) ).
% in_snd_imageE
tff(fact_2126_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_2127_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] :
( ( set_ord_atMost(A,X) = top_top(set(A)) )
<=> ( X = top_top(A) ) ) ) ).
% atMost_eq_UNIV_iff
tff(fact_2128_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,B),X: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [Y2: A] :
( member(A,Y2,Aa2)
=> ( aa(A,B,F,Y2) = aa(A,B,F,X) ) )
=> ( the_elem(B,aa(set(A),set(B),image2(A,B,F),Aa2)) = aa(A,B,F,X) ) ) ) ).
% the_elem_image_unique
tff(fact_2129_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A3: A,X: B] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) )
=> ( aa(B,A,F,X) = A3 ) ) ).
% range_eq_singletonD
tff(fact_2130_fst__image__mp,axiom,
! [B: $tType,A: $tType,Aa2: set(product_prod(A,B)),Ba: set(A),X: A,Y: B] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),Aa2)),Ba)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),Aa2)
=> member(A,X,Ba) ) ) ).
% fst_image_mp
tff(fact_2131_snd__image__mp,axiom,
! [B: $tType,A: $tType,Aa2: set(product_prod(B,A)),Ba: set(A),X: B,Y: A] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),Aa2)),Ba)
=> ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),Aa2)
=> member(A,Y,Ba) ) ) ).
% snd_image_mp
tff(fact_2132_in__image__insert__iff,axiom,
! [A: $tType,Ba: set(set(A)),X: A,Aa2: set(A)] :
( ! [C6: set(A)] :
( member(set(A),C6,Ba)
=> ~ member(A,X,C6) )
=> ( member(set(A),Aa2,aa(set(set(A)),set(set(A)),image2(set(A),set(A),insert2(A,X)),Ba))
<=> ( member(A,X,Aa2)
& member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))),Ba) ) ) ) ).
% in_image_insert_iff
tff(fact_2133_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A3) ) ) ) ).
% Ioc_disjoint
tff(fact_2134_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_2135_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,Aa2))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_2136_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).
% Inf_fin.boundedI
tff(fact_2137_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,Aa2))
=> ! [A9: A] :
( member(A,A9,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A9) ) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_2138_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(5)
tff(fact_2139_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [H2: fun(A,A),N3: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H2,X3)),aa(A,A,H2,Y2))
=> ( aa(set(A),$o,finite_finite(A),N3)
=> ( ( N3 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic5882676163264333800up_fin(A),N3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H2),N3)) ) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_2140_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_2141_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,Ba)),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_2142_Inf__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,Ba)),lattic7752659483105999362nf_fin(A,Aa2)) = lattic7752659483105999362nf_fin(A,Aa2) ) ) ) ) ) ).
% Inf_fin.subset
tff(fact_2143_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,lattic7752659483105999362nf_fin(A,Aa2),Aa2) ) ) ) ) ).
% Inf_fin.closed
tff(fact_2144_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),lattic7752659483105999362nf_fin(A,Aa2)) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_2145_Inf__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic7752659483105999362nf_fin(A,Aa2)),lattic7752659483105999362nf_fin(A,Ba)) ) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_2146_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,Aa2)),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) ) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_2147_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_2148_prod_Ocl__ivl__Suc,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,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,N)),M2),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ).
% prod.cl_ivl_Suc
tff(fact_2149_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K: num] :
set_or7035219750837199246ssThan(nat,M2,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),pred_numeral(K)),aa(set(nat),set(nat),insert2(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M2,pred_numeral(K))),bot_bot(set(nat))) ).
% atLeastLessThan_nat_numeral
tff(fact_2150_atLeast__Suc,axiom,
! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),insert2(nat,K),bot_bot(set(nat)))) ).
% atLeast_Suc
tff(fact_2151_Min_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( lattic643756798350308766er_Min(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ) ).
% Min.remove
tff(fact_2152_Min_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ).
% Min.insert_remove
tff(fact_2153_gbinomial__code,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] :
aa(nat,A,gbinomial(A,A3),K) = $ite(K = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_aa(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K))) ) ).
% gbinomial_code
tff(fact_2154_bool__assn__proper_I2_J,axiom,
aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_ab(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
% bool_assn_proper(2)
tff(fact_2155_bool__assn__proper_I3_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ac(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).
% bool_assn_proper(3)
tff(fact_2156_bool__assn__proper_I4_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> ( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,Q)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ad(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ) ) ).
% bool_assn_proper(4)
tff(fact_2157_singleton__conv2,axiom,
! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A3)) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ).
% singleton_conv2
tff(fact_2158_singleton__conv,axiom,
! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ae(A,fun(A,$o),A3)) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ).
% singleton_conv
tff(fact_2159_Collect__const,axiom,
! [A: $tType,P: $o] :
aa(fun(A,$o),set(A),collect(A),aTP_Lamp_af($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).
% Collect_const
tff(fact_2160_bool__assn__proper_I5_J,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o)] :
( aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,P)
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_ag(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ) ).
% bool_assn_proper(5)
tff(fact_2161_atLeastLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastLessThan_empty
tff(fact_2162_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% atLeastLessThan_empty_iff2
tff(fact_2163_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2) ) ) ).
% atLeastLessThan_empty_iff
tff(fact_2164_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% greaterThanLessThan_empty_iff2
tff(fact_2165_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% greaterThanLessThan_empty_iff
tff(fact_2166_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
=> ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanLessThan_empty
tff(fact_2167_Min__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : lattic643756798350308766er_Min(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% Min_singleton
tff(fact_2168_Min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,Aa2))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).
% Min.bounded_iff
tff(fact_2169_Min__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),lattic643756798350308766er_Min(A,Aa2))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X4) ) ) ) ) ) ).
% Min_gr_iff
tff(fact_2170_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_2171_range__constant,axiom,
! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),X)),top_top(set(B))) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).
% range_constant
tff(fact_2172_atLeastLessThan__singleton,axiom,
! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,M2)) = aa(set(nat),set(nat),insert2(nat,M2),bot_bot(set(nat))) ).
% atLeastLessThan_singleton
tff(fact_2173_Min__const,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Aa2: set(A),C2: B] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_ai(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ) ).
% Min_const
tff(fact_2174_Min__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).
% Min_insert
tff(fact_2175_prod_Oop__ivl__Suc,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_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ).
% prod.op_ivl_Suc
tff(fact_2176_lambda__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ( aTP_Lamp_aj(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).
% lambda_zero
tff(fact_2177_top__empty__eq2,axiom,
! [B: $tType,A: $tType,X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),top_top(set(product_prod(A,B)))) ) ).
% top_empty_eq2
tff(fact_2178_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),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_2179_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( ! [X4: A,Xa2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),R)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),S) )
<=> ( R = S ) ) ).
% pred_equals_eq2
tff(fact_2180_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S)) ) ).
% sup_Un_eq2
tff(fact_2181_Set_Oempty__def,axiom,
! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_al(A,$o)) ).
% Set.empty_def
tff(fact_2182_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),bot_bot(set(product_prod(A,B)))) ) ).
% bot_empty_eq2
tff(fact_2183_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_am(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_2184_Collect__conv__if2,axiom,
! [A: $tType,A3: A,P: fun(A,$o)] :
aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).
% Collect_conv_if2
tff(fact_2185_Collect__conv__if,axiom,
! [A: $tType,A3: A,P: fun(A,$o)] :
aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).
% Collect_conv_if
tff(fact_2186_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),S))
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S) ) ).
% pred_subset_eq2
tff(fact_2187_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,P5: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),P5)
=> ( 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,P5))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,P5)) ) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_2188_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_2189_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_2190_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_2191_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_ap(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_2192_Min__add__commute,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [S: set(A),F: fun(A,B),K: B] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_aq(fun(A,B),fun(B,fun(A,B)),F),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F),S))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_2193_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).
% prod.atMost_Suc_shift
tff(fact_2194_bot__assn__def,axiom,
bot_bot(assn) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_ab(product_prod(heap_ext(product_unit),set(nat)),$o)) ).
% bot_assn_def
tff(fact_2195_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ar(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_2196_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A] : bot_bot(set(A)) != set_ord_atLeast(A,L) ) ).
% not_empty_eq_Ici_eq_empty
tff(fact_2197_image__constant,axiom,
! [A: $tType,B: $tType,X: A,Aa2: set(A),C2: B] :
( member(A,X,Aa2)
=> ( aa(set(A),set(B),image2(A,B,aTP_Lamp_as(B,fun(A,B),C2)),Aa2) = aa(set(B),set(B),insert2(B,C2),bot_bot(set(B))) ) ) ).
% image_constant
tff(fact_2198_image__constant__conv,axiom,
! [B: $tType,A: $tType,C2: A,Aa2: set(B)] :
aa(set(B),set(A),image2(B,A,aTP_Lamp_ah(A,fun(B,A),C2)),Aa2) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert2(A,C2),bot_bot(set(A)))) ).
% image_constant_conv
tff(fact_2199_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_at(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact
tff(fact_2200_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_at(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact'
tff(fact_2201_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_2202_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2203_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: nat,B2: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2204_prod_Olast__plus,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2205_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,X: nat,Y: nat] :
aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_au(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
$ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_2206_sup__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_av(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).
% sup_assn_def
tff(fact_2207_inf__assn__def,axiom,
! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P),Q) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_aw(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),P),Q)) ).
% inf_assn_def
tff(fact_2208_numeral__code_I3_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] :
aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = $let(
m: A,
m:= aa(num,A,numeral_numeral(A),N),
aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).
% numeral_code(3)
tff(fact_2209_power__numeral__even,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W2: num] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W2))) = $let(
w: A,
w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)),
aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).
% power_numeral_even
tff(fact_2210_power__numeral__odd,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W2: num] :
aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = $let(
w: A,
w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)),
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),w)),w) ) ) ).
% power_numeral_odd
tff(fact_2211_prod_Ohead__if,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,M2,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G,N))) ) ).
% prod.head_if
tff(fact_2212_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_bot(A)
=> ordering_top(A,aTP_Lamp_ax(A,fun(A,$o)),aTP_Lamp_ay(A,fun(A,$o)),bot_bot(A)) ) ).
% bot.ordering_top_axioms
tff(fact_2213_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_2214_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),aa(num,num,bit0,one2))),M2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_az(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).
% prod.in_pairs
tff(fact_2215_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),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_az(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% prod.in_pairs_0
tff(fact_2216_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_2217_atLeastLessThan0,axiom,
! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,zero_zero(nat)) = bot_bot(set(nat)) ).
% atLeastLessThan0
tff(fact_2218_Min__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> member(A,lattic643756798350308766er_Min(A,Aa2),Aa2) ) ) ) ).
% Min_in
tff(fact_2219_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_ba(A,fun(A,$o)),aTP_Lamp_bb(A,fun(A,$o))) ) ).
% sup_bot.semilattice_neutr_order_axioms
tff(fact_2220_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] :
( ( set_ord_atLeast(A,X) = top_top(set(A)) )
<=> ( X = bot_bot(A) ) ) ) ).
% atLeast_eq_UNIV_iff
tff(fact_2221_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P5: nat,K: nat,G: fun(nat,A),H2: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P5)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P5)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_bc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,P5)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_bd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P5),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% prod.zero_middle
tff(fact_2222_uminus__assn__def,axiom,
! [P: assn] : aa(assn,assn,uminus_uminus(assn),P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_be(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ).
% uminus_assn_def
tff(fact_2223_fact__split,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).
% fact_split
tff(fact_2224_prod_Ohead,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2225_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_2226_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
aa(set(A),set(A),image2(A,A,aTP_Lamp_bf(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2))),
bot_bot(set(A)) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_2227_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M2: A,C2: A,A3: A,B2: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_bg(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),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)),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_2228_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M2: A,C2: A,A3: A,B2: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_bh(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C2))) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_2229_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M2: A,C2: A,A3: A,B2: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_bi(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C2))) ) ) ).
% image_affinity_atLeastAtMost_div
tff(fact_2230_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M2: A,C2: A,A3: A,B2: A] :
aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_bj(A,fun(A,fun(A,A)),M2),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = $ite(
set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)),
bot_bot(set(A)),
$ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C2))) ) ) ).
% image_affinity_atLeastAtMost_div_diff
tff(fact_2231_Min__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),M2: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ( lattic643756798350308766er_Min(A,Aa2) = M2 )
<=> ( member(A,M2,Aa2)
& ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),X4) ) ) ) ) ) ) ).
% Min_eq_iff
tff(fact_2232_Min__le__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,Aa2)),X)
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).
% Min_le_iff
tff(fact_2233_eq__Min__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),M2: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ( M2 = lattic643756798350308766er_Min(A,Aa2) )
<=> ( member(A,M2,Aa2)
& ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M2),X4) ) ) ) ) ) ) ).
% eq_Min_iff
tff(fact_2234_Min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,Aa2))
=> ! [A9: A] :
( member(A,A9,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A9) ) ) ) ) ) ).
% Min.boundedE
tff(fact_2235_Min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A4) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).
% Min.boundedI
tff(fact_2236_Min__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798350308766er_Min(A,Aa2)),X)
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X) ) ) ) ) ) ).
% Min_less_iff
tff(fact_2237_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_2238_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_2239_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_2240_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_2241_signed__take__bit__code,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] :
bit_ri4674362597316999326ke_bit(A,N,A3) = $let(
l: A,
l:= bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3),
$ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).
% signed_take_bit_code
tff(fact_2242_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_2243_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2244_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_2245_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_2246_Min_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,Ba)),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ).
% Min.subset_imp
tff(fact_2247_Min__antimono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M4: set(A),N3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N3)
=> ( ( M4 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),N3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798350308766er_Min(A,N3)),lattic643756798350308766er_Min(A,M4)) ) ) ) ) ).
% Min_antimono
tff(fact_2248_hom__Min__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H2: fun(A,A),N3: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H2,X3)),aa(A,A,H2,Y2))
=> ( aa(set(A),$o,finite_finite(A),N3)
=> ( ( N3 != bot_bot(set(A)) )
=> ( aa(A,A,H2,lattic643756798350308766er_Min(A,N3)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,H2),N3)) ) ) ) ) ) ).
% hom_Min_commute
tff(fact_2249_pochhammer__code,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] :
comm_s3205402744901411588hammer(A,A3,N) = $ite(N = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_bk(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A))) ) ).
% pochhammer_code
tff(fact_2250_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N),aa(set(nat),set(nat),insert2(nat,N),set_or7035219750837199246ssThan(nat,M2,N)),bot_bot(set(nat))) ).
% atLeastLessThanSuc
tff(fact_2251_Min_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,Ba)),lattic643756798350308766er_Min(A,Aa2)) = lattic643756798350308766er_Min(A,Aa2) ) ) ) ) ) ).
% Min.subset
tff(fact_2252_Min_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,lattic643756798350308766er_Min(A,Aa2),Aa2) ) ) ) ) ).
% Min.closed
tff(fact_2253_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),lattic643756798350308766er_Min(A,Aa2)) ) ) ) ) ) ).
% Min.insert_not_elem
tff(fact_2254_Min_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),ord_min(A),lattic643756798350308766er_Min(A,Aa2)),lattic643756798350308766er_Min(A,Ba)) ) ) ) ) ) ) ).
% Min.union
tff(fact_2255_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),insert2(A,N),bot_bot(set(A))) ) ).
% atMost_Int_atLeast
tff(fact_2256_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).
% atLeastAtMost_diff_ends
tff(fact_2257_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = set_ord_atLeast(A,L) ) ).
% ivl_disj_un_singleton(1)
tff(fact_2258_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),aa(num,num,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_bl(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),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).
% pochhammer_times_pochhammer_half
tff(fact_2259_prod_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: nat,N: nat,G: fun(nat,A),P5: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),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),P5))) = 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),P5)))) ) ) ) ).
% prod.ub_add_nat
tff(fact_2260_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(6)
tff(fact_2261_prod_Oinsert,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)) ) ) ) ) ).
% prod.insert
tff(fact_2262_prod_Odelta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A3: A,B2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_bm(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(member(A,A3,S),aa(A,B,B2,A3),one_one(B)) ) ) ) ).
% prod.delta'
tff(fact_2263_prod_Odelta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A3: A,B2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_bn(A,fun(fun(A,B),fun(A,B)),A3),B2)),S) = $ite(member(A,A3,S),aa(A,B,B2,A3),one_one(B)) ) ) ) ).
% prod.delta
tff(fact_2264_prod_Oinfinite,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = one_one(B) ) ) ) ).
% prod.infinite
tff(fact_2265_prod__gen__delta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A3: A,B2: fun(A,B),C2: B] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_bo(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B2),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,S)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),finite_card(A,S))) ) ) ) ).
% prod_gen_delta
tff(fact_2266_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_2267_prod__Un,axiom,
! [B: $tType,A: $tType] :
( field(B)
=> ! [Aa2: set(A),Ba: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))
=> ( aa(A,B,F,X3) != zero_zero(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Ba)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))) ) ) ) ) ) ).
% prod_Un
tff(fact_2268_atLeast__empty__triv,axiom,
! [A: $tType] : set_ord_atLeast(set(A),bot_bot(set(A))) = top_top(set(set(A))) ).
% atLeast_empty_triv
tff(fact_2269_predicate2I,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
( ! [X3: A,Y2: B] :
( aa(B,$o,aa(A,fun(B,$o),P,X3),Y2)
=> aa(B,$o,aa(A,fun(B,$o),Q,X3),Y2) )
=> aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).
% predicate2I
tff(fact_2270_inf2I,axiom,
! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),X: A,Y: B,Ba: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),Aa2,X),Y)
=> ( aa(B,$o,aa(A,fun(B,$o),Ba,X),Y)
=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),Aa2),Ba),X),Y) ) ) ).
% inf2I
tff(fact_2271_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Aa2: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_bp(B,A)),Aa2) = one_one(A) ) ).
% prod.neutral_const
tff(fact_2272_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Y) ).
% bot2E
tff(fact_2273_inf2D2,axiom,
! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),Ba: fun(A,fun(B,$o)),X: A,Y: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),Aa2),Ba),X),Y)
=> aa(B,$o,aa(A,fun(B,$o),Ba,X),Y) ) ).
% inf2D2
tff(fact_2274_inf2D1,axiom,
! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),Ba: fun(A,fun(B,$o)),X: A,Y: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),Aa2),Ba),X),Y)
=> aa(B,$o,aa(A,fun(B,$o),Aa2,X),Y) ) ).
% inf2D1
tff(fact_2275_inf2E,axiom,
! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),Ba: fun(A,fun(B,$o)),X: A,Y: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),Aa2),Ba),X),Y)
=> ~ ( aa(B,$o,aa(A,fun(B,$o),Aa2,X),Y)
=> ~ aa(B,$o,aa(A,fun(B,$o),Ba,X),Y) ) ) ).
% inf2E
tff(fact_2276_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,Q: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
=> aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).
% rev_predicate2D
tff(fact_2277_predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),X: A,Y: B] :
( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
=> ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
=> aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).
% predicate2D
tff(fact_2278_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Aa2: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),Aa2) != one_one(A) )
=> ~ ! [A4: B] :
( member(B,A4,Aa2)
=> ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_2279_prod_Oneutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,Aa2)
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = one_one(B) ) ) ) ).
% prod.neutral
tff(fact_2280_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H2: fun(B,A),Aa2: 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_bq(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),Aa2) = 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),Aa2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),Aa2)) ) ).
% prod.distrib
tff(fact_2281_prod__ge__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)) ) ) ).
% prod_ge_1
tff(fact_2282_prod_Ofinite__Collect__op,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I4: set(A),X: fun(A,B),Y: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),I4),X)))
=> ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),I4),Y)))
=> aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bs(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I4),X),Y))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_2283_prod_Ointer__filter,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,$o),fun(A,$o)),Aa2),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_bu(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),Aa2) ) ) ) ).
% prod.inter_filter
tff(fact_2284_prod__le__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,Aa2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X3))
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),one_one(B)) ) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)),one_one(B)) ) ) ).
% prod_le_1
tff(fact_2285_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R: fun(A,fun(A,$o)),S: set(B),H2: fun(B,A),G: fun(B,A)] :
( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
=> ( ! [X12: A,Y12: A,X23: A,Y23: A] :
( ( aa(A,$o,aa(A,fun(A,$o),R,X12),X23)
& aa(A,$o,aa(A,fun(A,$o),R,Y12),Y23) )
=> aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
=> ( aa(set(B),$o,finite_finite(B),S)
=> ( ! [X3: B] :
( member(B,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H2,X3)),aa(B,A,G,X3)) )
=> aa(A,$o,aa(A,fun(A,$o),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_2286_prod_Oinsert__if,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(member(A,X,Aa2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2))) ) ) ) ).
% prod.insert_if
tff(fact_2287_prod_Oreindex__bij__witness__not__neutral,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [S5: set(A),T4: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T5: set(B),G: fun(A,C),H2: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),S5)
=> ( aa(set(B),$o,finite_finite(B),T4)
=> ( ! [A4: A] :
( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5))
=> ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
=> ( ! [A4: A] :
( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5))
=> member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4)) )
=> ( ! [B3: B] :
( member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
=> ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
=> ( ! [B3: B] :
( member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
=> member(A,aa(B,A,I,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5)) )
=> ( ! [A4: A] :
( member(A,A4,S5)
=> ( aa(A,C,G,A4) = one_one(C) ) )
=> ( ! [B3: B] :
( member(B,B3,T4)
=> ( aa(B,C,H2,B3) = one_one(C) ) )
=> ( ! [A4: A] :
( member(A,A4,S)
=> ( aa(B,C,H2,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H2),T5) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_2288_prod_Ointer__restrict,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_bv(fun(A,B),fun(set(A),fun(A,B)),G),Ba)),Aa2) ) ) ) ).
% prod.inter_restrict
tff(fact_2289_prod_Osetdiff__irrelevant,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bw(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) ) ) ) ).
% prod.setdiff_irrelevant
tff(fact_2290_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),I: A,F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( member(A,I,I4)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I))
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4)) ) ) ) ) ) ).
% less_1_prod2
tff(fact_2291_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4)) ) ) ) ) ).
% less_1_prod
tff(fact_2292_prod_Osubset__diff,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Ba: set(A),Aa2: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ba)) ) ) ) ) ).
% prod.subset_diff
tff(fact_2293_prod_Osame__carrier,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Ca: set(A),Aa2: set(A),Ba: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Ca)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ca)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Ca)
=> ( ! [A4: A] :
( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ca),Aa2))
=> ( aa(A,B,G,A4) = one_one(B) ) )
=> ( ! [B3: A] :
( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ca),Ba))
=> ( aa(A,B,H2,B3) = one_one(B) ) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),Ba) )
<=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ca) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),Ca) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_2294_prod_Osame__carrierI,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Ca: set(A),Aa2: set(A),Ba: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Ca)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ca)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Ca)
=> ( ! [A4: A] :
( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ca),Aa2))
=> ( aa(A,B,G,A4) = one_one(B) ) )
=> ( ! [B3: A] :
( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ca),Ba))
=> ( aa(A,B,H2,B3) = one_one(B) ) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ca) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),Ca) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),Ba) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_2295_prod_Omono__neutral__left,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T5: set(A),S: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),T5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) ) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_2296_prod_Omono__neutral__right,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T5: set(A),S: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),T5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_2297_prod_Omono__neutral__cong__left,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T5: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),T5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,H2,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,H2,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T5) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_2298_prod_Omono__neutral__cong__right,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T5: set(A),S: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),T5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,H2,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),S) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_2299_prod_Ounion__inter,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),Ba: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ba)) ) ) ) ) ).
% prod.union_inter
tff(fact_2300_prod_OInt__Diff,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba))) ) ) ) ).
% prod.Int_Diff
tff(fact_2301_prod_Omono__neutral__cong,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [T5: set(A),S: set(A),H2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),T5)
=> ( aa(set(A),$o,finite_finite(A),S)
=> ( ! [I3: A] :
( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,H2,I3) = one_one(B) ) )
=> ( ! [I3: A] :
( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T5))
=> ( aa(A,B,G,I3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T5))
=> ( aa(A,B,G,X3) = aa(A,B,H2,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T5) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_2302_prod_OIf__cases,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),P: fun(A,$o),H2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_bx(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H2),G)),Aa2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).
% prod.If_cases
tff(fact_2303_prod__mono__strict,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(B)
=> ! [Aa2: set(A),F: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ! [I3: A] :
( member(A,I3,Aa2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3))
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I3)),aa(A,B,G,I3)) ) )
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)) ) ) ) ) ).
% prod_mono_strict
tff(fact_2304_prod_Oremove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% prod.remove
tff(fact_2305_prod_Oinsert__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% prod.insert_remove
tff(fact_2306_prod_Ounion__inter__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),Ba: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ba)) ) ) ) ) ) ).
% prod.union_inter_neutral
tff(fact_2307_prod_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),Ba: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),Ba)) ) ) ) ) ) ).
% prod.union_disjoint
tff(fact_2308_prod_Ounion__diff2,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: set(A),Ba: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ba),Aa2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))) ) ) ) ) ).
% prod.union_diff2
tff(fact_2309_prod_Odelta__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_by(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).
% prod.delta_remove
tff(fact_2310_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [Ba: set(A),Aa2: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Ba)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ! [B3: A] :
( member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ba),Aa2))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,B3)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A4)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Ba)) ) ) ) ) ) ).
% prod_mono2
tff(fact_2311_prod__le__power,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(B)
=> ! [Aa2: set(A),F: fun(A,B),N: B,K: nat] :
( ! [I3: A] :
( member(A,I3,Aa2)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3))
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),N) ) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Aa2)),K)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),N)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),N),K)) ) ) ) ) ).
% prod_le_power
tff(fact_2312_prod__diff1,axiom,
! [B: $tType,A: $tType] :
( semidom_divide(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( aa(A,B,F,A3) != zero_zero(B) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2),aa(A,B,F,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),Aa2)) ) ) ) ) ).
% prod_diff1
tff(fact_2313_choose__even__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bz(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),aa(num,num,bit0,one2))),N) ) ) ) ).
% choose_even_sum
tff(fact_2314_choose__odd__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ca(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),aa(num,num,bit0,one2))),N) ) ) ) ).
% choose_odd_sum
tff(fact_2315_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_cb(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),aa(num,num,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_2316_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),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M2)) ) ).
% gbinomial_r_part_sum
tff(fact_2317_sum__gp,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,N: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2),
zero_zero(A),
$ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M2)),divide_divide(A,aa(A,A,aa(A,fun(A,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))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X))) ) ) ).
% sum_gp
tff(fact_2318_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_cc(A,A),N,zero_zero(A)) ) ).
% of_nat_code
tff(fact_2319_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty
tff(fact_2320_sum__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,Aa2: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cd(A,fun(B,A),Y)),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,Aa2))),Y) ) ).
% sum_constant
tff(fact_2321_sum__of__bool__mult__eq,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Aa2: set(A),P: fun(A,$o),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ce(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F)),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).
% sum_of_bool_mult_eq
tff(fact_2322_sum__mult__of__bool__eq,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Aa2: set(A),F: fun(A,B),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_cf(fun(A,B),fun(fun(A,$o),fun(A,B)),F),P)),Aa2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).
% sum_mult_of_bool_eq
tff(fact_2323_sum__zero__power,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: fun(nat,A),Aa2: set(nat)] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cg(fun(nat,A),fun(nat,A),C2)),Aa2) = $ite(
( aa(set(nat),$o,finite_finite(nat),Aa2)
& member(nat,zero_zero(nat),Aa2) ),
aa(nat,A,C2,zero_zero(nat)),
zero_zero(A) ) ) ).
% sum_zero_power
tff(fact_2324_sum__zero__power_H,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: fun(nat,A),D3: fun(nat,A),Aa2: set(nat)] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ch(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),Aa2) = $ite(
( aa(set(nat),$o,finite_finite(nat),Aa2)
& member(nat,zero_zero(nat),Aa2) ),
divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))),
zero_zero(A) ) ) ).
% sum_zero_power'
tff(fact_2325_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [R2: A,F: fun(B,A),Aa2: 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),F),Aa2)) = 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_ci(A,fun(fun(B,A),fun(B,A)),R2),F)),Aa2) ) ).
% sum_distrib_left
tff(fact_2326_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),Aa2: 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),F),Aa2)),R2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_cj(fun(B,A),fun(A,fun(B,A)),F),R2)),Aa2) ) ).
% sum_distrib_right
tff(fact_2327_sum__product,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),Aa2: set(B),G: fun(C,A),Ba: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),Aa2)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),Ba)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cl(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),Ba)),Aa2) ) ).
% sum_product
tff(fact_2328_sum__strict__mono,axiom,
! [B: $tType,A: $tType] :
( strict7427464778891057005id_add(B)
=> ! [Aa2: set(A),F: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2)) ) ) ) ) ).
% sum_strict_mono
tff(fact_2329_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,I4: 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_cm(A,fun(nat,fun(nat,A)),X),M2)),I4) = 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)),I4)) ) ).
% sum_power_add
tff(fact_2330_sum__pos,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [I4: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),I4)) ) ) ) ) ).
% sum_pos
tff(fact_2331_sum__bounded__above,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [Aa2: set(A),F: fun(A,B),K5: B] :
( ! [I3: A] :
( member(A,I3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),K5) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K5)) ) ) ).
% sum_bounded_above
tff(fact_2332_sum__bounded__below,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [Aa2: set(A),K5: B,F: fun(A,B)] :
( ! [I3: A] :
( member(A,I3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F,I3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)) ) ) ).
% sum_bounded_below
tff(fact_2333_sum_Oinsert__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Aa2: set(A),G: fun(A,B),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% sum.insert_remove
tff(fact_2334_sum_Oremove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Aa2: set(A),X: A,G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% sum.remove
tff(fact_2335_sum__diff1,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)),aa(A,B,F,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)) ) ) ) ).
% sum_diff1
tff(fact_2336_sum_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Aa2: set(A),Ba: set(A),G: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Aa2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),Ba)) ) ) ) ) ) ).
% sum.union_disjoint
tff(fact_2337_sum_Odelta__remove,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [S: set(A),A3: A,B2: fun(A,B),C2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cn(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B2),C2)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).
% sum.delta_remove
tff(fact_2338_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_co(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_2339_member__le__sum,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [I: A,Aa2: set(A),F: fun(A,B)] :
( member(A,I,Aa2)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,I),bot_bot(set(A)))))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X3)) )
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)) ) ) ) ) ).
% member_le_sum
tff(fact_2340_sum__bounded__above__divide,axiom,
! [A: $tType,B: $tType] :
( linordered_field(B)
=> ! [Aa2: set(A),F: fun(A,B),K5: B] :
( ! [I3: A] :
( member(A,I3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),divide_divide(B,K5,aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2)))) )
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)),K5) ) ) ) ) ).
% sum_bounded_above_divide
tff(fact_2341_sum__bounded__above__strict,axiom,
! [A: $tType,B: $tType] :
( ( ordere8940638589300402666id_add(B)
& semiring_1(B) )
=> ! [Aa2: set(A),F: fun(A,B),K5: B] :
( ! [I3: A] :
( member(A,I3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,I3)),K5) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,Aa2))
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),Aa2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,Aa2))),K5)) ) ) ) ).
% sum_bounded_above_strict
tff(fact_2342_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3)) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I4) = one_one(B) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I3)),B2))),Delta) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_cp(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I4)),B2))),Delta) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_2343_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S: set(A),R: set(B),G: fun(A,B),F: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(B),$o,finite_finite(B),R)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),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_cq(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_cs(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F)),R) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_2344_sum__gp__basic,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),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(A,A,aa(A,fun(A,A),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_2345_sum__power__shift,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M2: nat,N: nat,X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ).
% sum_power_shift
tff(fact_2346_mask__eq__sum__exp,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ct(nat,fun(nat,$o),N))) ) ).
% mask_eq_sum_exp
tff(fact_2347_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M2: nat,N: nat,X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),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),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_2348_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_cu(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,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M2)) ) ).
% gbinomial_sum_lower_neg
tff(fact_2349_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_cv(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% binomial_ring
tff(fact_2350_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_cw(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% pochhammer_binomial_sum
tff(fact_2351_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_cx(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_2352_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_cy(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_cz(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_2353_double__arith__series,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,D3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_da(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_2354_double__gauss__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_2355_sum__gp0,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,N)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X))) ) ).
% sum_gp0
tff(fact_2356_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_cy(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_db(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_2357_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_dc(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_2358_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,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_2359_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),aa(num,num,bit0,one2))) ) ).
% gauss_sum
tff(fact_2360_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_dd(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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).
% arith_series
tff(fact_2361_sum__gp__offset,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M2: nat,N: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = $ite(X = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X))) ) ).
% sum_gp_offset
tff(fact_2362_choose__alternating__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_de(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_2363_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),aa(num,num,bit0,one2))) ) ).
% gauss_sum_from_Suc_0
tff(fact_2364_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_cb(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),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M2))) ) ).
% gchoose_row_sum_weighted
tff(fact_2365_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_df(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).
% divmod_algorithm_code(6)
tff(fact_2366_divmod__step__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_dg(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_2367_normalize__def,axiom,
! [P5: product_prod(int,int)] :
normalize(P5) = $ite(
aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P5)),
$let(
a3: int,
a3:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P5)),aa(product_prod(int,int),int,product_snd(int,int),P5)),
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),P5),a3)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P5),a3)) ),
$ite(
aa(product_prod(int,int),int,product_snd(int,int),P5) = zero_zero(int),
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
$let(
a3: int,
a3:= 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),P5)),aa(product_prod(int,int),int,product_snd(int,int),P5))),
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),P5),a3)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P5),a3)) ) ) ) ).
% normalize_def
tff(fact_2368_interval__cases,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A)] :
( ! [A4: A,B3: A,X3: A] :
( member(A,A4,S)
=> ( member(A,B3,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B3)
=> member(A,X3,S) ) ) ) )
=> ? [A4: A,B3: A] :
( ( S = bot_bot(set(A)) )
| ( S = top_top(set(A)) )
| ( S = set_ord_lessThan(A,B3) )
| ( S = set_ord_atMost(A,B3) )
| ( S = aa(A,set(A),set_ord_greaterThan(A),A4) )
| ( S = set_ord_atLeast(A,A4) )
| ( S = set_or5935395276787703475ssThan(A,A4,B3) )
| ( S = set_or3652927894154168847AtMost(A,A4,B3) )
| ( S = set_or7035219750837199246ssThan(A,A4,B3) )
| ( S = set_or1337092689740270186AtMost(A,A4,B3) ) ) ) ) ).
% interval_cases
tff(fact_2369_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_dh(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_2370_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [F4: set(A),I4: set(A),F: fun(A,B),I: A] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_di(set(A),fun(fun(A,B),fun(A,$o)),I4),F))),F4)
=> ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = $ite(member(A,I,I4),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F,I4)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I4)) ) ) ) ) ).
% sum_diff1'_aux
tff(fact_2371_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_2372_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_2373_lessThan__0,axiom,
set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).
% lessThan_0
tff(fact_2374_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% case_prod_conv
tff(fact_2375_is__unit__gcd__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( 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_2376_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_2377_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [P5: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P5,bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty'
tff(fact_2378_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,Aa2: set(product_prod(A,B)),F: fun(A,fun(B,C))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),Aa2)
=> member(C,aa(B,C,aa(A,fun(B,C),F,A3),B2),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),Aa2)) ) ).
% pair_imageI
tff(fact_2379_single__Diff__lessThan,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert2(A,K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),insert2(A,K),bot_bot(set(A))) ) ).
% single_Diff_lessThan
tff(fact_2380_Gcd__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).
% Gcd_2
tff(fact_2381_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M2: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M2),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_dj(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).
% divmod_algorithm_code(5)
tff(fact_2382_old_Oprod_Ocase,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),X1: B,X22: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X22)) = aa(C,A,aa(B,fun(C,A),F,X1),X22) ).
% old.prod.case
tff(fact_2383_gcd_Osemigroup__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> semigroup(A,gcd_gcd(A)) ) ).
% gcd.semigroup_axioms
tff(fact_2384_lessThan__non__empty,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] : set_ord_lessThan(A,X) != bot_bot(set(A)) ) ).
% lessThan_non_empty
tff(fact_2385_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
( ! [X3: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F,X3),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).
% cond_case_prod_eta
tff(fact_2386_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_dk(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).
% case_prod_eta
tff(fact_2387_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2))
=> ~ ! [X3: B,Y2: C] :
( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X3),Y2)) ) ) ).
% case_prodE2
tff(fact_2388_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_2389_gcd__dvd__prod,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,K: A] : 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_2390_lessThan__empty__iff,axiom,
! [N: nat] :
( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
<=> ( N = zero_zero(nat) ) ) ).
% lessThan_empty_iff
tff(fact_2391_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( linorder(A)
& order_bot(A) )
=> ! [N: A] :
( ( set_ord_lessThan(A,N) = bot_bot(set(A)) )
<=> ( N = bot_bot(A) ) ) ) ).
% Iio_eq_empty_iff
tff(fact_2392_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_2393_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_2394_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_2395_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_2396_prod_Osplit__sel__asm,axiom,
! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),Prod))
<=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
& ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).
% prod.split_sel_asm
tff(fact_2397_prod_Osplit__sel,axiom,
! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),Prod))
<=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
=> aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).
% prod.split_sel
tff(fact_2398_rat__abs__code,axiom,
! [P5: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P5)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_dl(int,fun(int,product_prod(int,int)))),quotient_of(P5)) ).
% rat_abs_code
tff(fact_2399_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_2400_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_2401_sum__diff1__nat,axiom,
! [A: $tType,F: fun(A,nat),Aa2: set(A),A3: A] :
aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),Aa2)),aa(A,nat,F,A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),Aa2)) ).
% sum_diff1_nat
tff(fact_2402_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,Aa2: set(A)] :
( member(A,A3,Aa2)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).
% Gcd_fin.remove
tff(fact_2403_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,Aa2: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),insert2(A,A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ).
% Gcd_fin.insert_remove
tff(fact_2404_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_ap(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.lessThan_Suc_shift
tff(fact_2405_Iio__Int__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [K: A,X: A] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),K),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),bot_bot(set(A))) ) ).
% Iio_Int_singleton
tff(fact_2406_rat__uminus__code,axiom,
! [P5: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P5)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_dm(int,fun(int,product_prod(int,int)))),quotient_of(P5)) ).
% rat_uminus_code
tff(fact_2407_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).
% ivl_disj_un_singleton(2)
tff(fact_2408_one__diff__power__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),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_2409_power__diff__1__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),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_2410_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,aa(A,A,aa(A,fun(A,A),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),minus_minus(A),X),one_one(A))) ) ) ) ).
% geometric_sum
tff(fact_2411_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_ap(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.atMost_shift
tff(fact_2412_rat__minus__code,axiom,
! [P5: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P5),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_do(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P5)) ).
% rat_minus_code
tff(fact_2413_rat__divide__code,axiom,
! [P5: rat,Q3: rat] : quotient_of(divide_divide(rat,P5,Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_dq(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P5)) ).
% rat_divide_code
tff(fact_2414_rat__times__code,axiom,
! [P5: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P5),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ds(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P5)) ).
% rat_times_code
tff(fact_2415_sum__gp__strict,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: nat] :
aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),N),divide_divide(A,aa(A,A,aa(A,fun(A,A),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),minus_minus(A),one_one(A)),X))) ) ).
% sum_gp_strict
tff(fact_2416_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,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),aa(A,A,aa(A,fun(A,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_dt(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_2417_power__diff__sumr2,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,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),aa(A,A,aa(A,fun(A,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_du(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,N))) ) ).
% power_diff_sumr2
tff(fact_2418_divmod__nat__if,axiom,
! [M2: nat,N: nat] :
divmod_nat(M2,N) = $ite(
( ( N = zero_zero(nat) )
| aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ),
aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M2),
aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_dv(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ).
% divmod_nat_if
tff(fact_2419_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_lessThan_eq_remove0
tff(fact_2420_rat__plus__code,axiom,
! [P5: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P5),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_dx(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P5)) ).
% rat_plus_code
tff(fact_2421_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dy(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq'
tff(fact_2422_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [I4: set(A),F: fun(A,B),I: A] :
( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_di(set(A),fun(fun(A,B),fun(A,$o)),I4),F)))
=> ( groups1027152243600224163dd_sum(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = $ite(member(A,I,I4),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F,I4)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I4)) ) ) ) ).
% sum_diff1'
tff(fact_2423_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_dz(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_2424_rat__inverse__code,axiom,
! [P5: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P5)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ea(int,fun(int,product_prod(int,int)))),quotient_of(P5)) ).
% rat_inverse_code
tff(fact_2425_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P5) = P5 ).
% case_prod_Pair_iden
tff(fact_2426_divmod__step__integer__def,axiom,
! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_eb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_2427_inverse__rat_Otransfer,axiom,
aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_ec(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat)) ).
% inverse_rat.transfer
tff(fact_2428_less__by__empty,axiom,
! [A: $tType,Aa2: set(product_prod(A,A)),Ba: set(product_prod(A,A))] :
( ( Aa2 = bot_bot(set(product_prod(A,A))) )
=> aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Aa2),Ba) ) ).
% less_by_empty
tff(fact_2429_enumerate__Suc,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,ord_Least(A,aTP_Lamp_ed(set(A),fun(A,$o),S))),bot_bot(set(A)))),N) ) ).
% enumerate_Suc
tff(fact_2430_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P5: product_prod(A,B)] :
( ! [X3: A,Y2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) = Q3 )
=> ( aa(B,C,aa(A,fun(B,C),F,X3),Y2) = aa(B,C,aa(A,fun(B,C),G,X3),Y2) ) )
=> ( ( P5 = Q3 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),P5) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q3) ) ) ) ).
% split_cong
tff(fact_2431_case__prodI,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A3: A,B2: B] :
( aa(B,$o,aa(A,fun(B,$o),F,A3),B2)
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ).
% case_prodI
tff(fact_2432_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod(A,B),C2: fun(A,fun(B,$o))] :
( ! [A4: A,B3: B] :
( ( P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> aa(B,$o,aa(A,fun(B,$o),C2,A4),B3) )
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P5) ) ).
% case_prodI2
tff(fact_2433_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
( member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B2))
=> member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2))) ) ).
% mem_case_prodI
tff(fact_2434_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
( ! [A4: A,B3: B] :
( ( P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> member(C,Z2,aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3)) )
=> member(C,Z2,aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P5)) ) ).
% mem_case_prodI2
tff(fact_2435_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),X: C] :
( ! [A4: A,B3: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P5 )
=> aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B3),X) )
=> aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P5),X) ) ).
% case_prodI2'
tff(fact_2436_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ee($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).
% Collect_const_case_prod
tff(fact_2437_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( power(B)
& power(A) )
=> ! [R: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),times_times(A)),times_times(B))
=> aa(fun(B,fun(nat,B)),$o,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),$o),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R,bNF_rel_fun(nat,nat,A,B,fequal(nat),R)),power_power(A)),power_power(B)) ) ) ) ).
% power_transfer
tff(fact_2438_divmod__integer_H__def,axiom,
! [M2: num,N: num] : unique8689654367752047608divmod(code_integer,M2,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M2),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M2),aa(num,code_integer,numeral_numeral(code_integer),N))) ).
% divmod_integer'_def
tff(fact_2439_LeastI2__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),Q: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_ex
tff(fact_2440_LeastI__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% LeastI_ex
tff(fact_2441_LeastI2,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
( aa(A,$o,P,A3)
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,Q,X3) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2
tff(fact_2442_LeastI,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),K: A] :
( aa(A,$o,P,K)
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% LeastI
tff(fact_2443_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& semiring_numeral(B)
& monoid_add(A)
& semiring_numeral(A) )
=> ! [R: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
=> aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).
% transfer_rule_numeral
tff(fact_2444_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).
% gcd_nat.semilattice_neutr_axioms
tff(fact_2445_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),P5: product_prod(B,C)] :
( member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P5))
=> ~ ! [X3: B,Y2: C] :
( ( P5 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,X3),Y2)) ) ) ).
% mem_case_prodE
tff(fact_2446_case__prodD,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A3: A,B2: B] :
( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))
=> aa(B,$o,aa(A,fun(B,$o),F,A3),B2) ) ).
% case_prodD
tff(fact_2447_case__prodE,axiom,
! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P5: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P5)
=> ~ ! [X3: A,Y2: B] :
( ( P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ aa(B,$o,aa(A,fun(B,$o),C2,X3),Y2) ) ) ).
% case_prodE
tff(fact_2448_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ring_1(B)
& ring_1(A) )
=> ! [R: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
=> ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R,R),uminus_uminus(A)),uminus_uminus(B))
=> aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).
% transfer_rule_of_int
tff(fact_2449_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A3: A,B2: B,C2: C] :
( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C2)
=> aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A3),B2),C2) ) ).
% case_prodD'
tff(fact_2450_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P5: product_prod(A,B),Z2: C] :
( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P5),Z2)
=> ~ ! [X3: A,Y2: B] :
( ( P5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X3),Y2),Z2) ) ) ).
% case_prodE'
tff(fact_2451_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),Q: fun(A,$o)] :
( ? [X_13: A] : aa(A,$o,P,X_13)
=> ( ! [A4: A] :
( aa(A,$o,P,A4)
=> ( ! [B9: A] :
( aa(A,$o,P,B9)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B9) )
=> aa(A,$o,Q,A4) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_wellorder_ex
tff(fact_2452_LeastI2__wellorder,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
( aa(A,$o,P,A3)
=> ( ! [A4: A] :
( aa(A,$o,P,A4)
=> ( ! [B9: A] :
( aa(A,$o,P,B9)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B9) )
=> aa(A,$o,Q,A4) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).
% LeastI2_wellorder
tff(fact_2453_Least__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
=> ( ord_Least(A,P) = X ) ) ) ) ).
% Least_equality
tff(fact_2454_LeastI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y2) )
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> ( ! [Y4: A] :
( aa(A,$o,P,Y4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y4) )
=> aa(A,$o,Q,X3) ) )
=> aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).
% LeastI2_order
tff(fact_2455_Least1__le,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),Z2: A] :
( ? [X2: A] :
( aa(A,$o,P,X2)
& ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2) )
& ! [Y2: A] :
( ( aa(A,$o,P,Y2)
& ! [Ya: A] :
( aa(A,$o,P,Ya)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya) ) )
=> ( Y2 = X2 ) ) )
=> ( aa(A,$o,P,Z2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z2) ) ) ) ).
% Least1_le
tff(fact_2456_Least1I,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o)] :
( ? [X2: A] :
( aa(A,$o,P,X2)
& ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2) )
& ! [Y2: A] :
( ( aa(A,$o,P,Y2)
& ! [Ya: A] :
( aa(A,$o,P,Ya)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Ya) ) )
=> ( Y2 = X2 ) ) )
=> aa(A,$o,P,ord_Least(A,P)) ) ) ).
% Least1I
tff(fact_2457_Least__le,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,$o),K: A] :
( aa(A,$o,P,K)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).
% Least_le
tff(fact_2458_not__less__Least,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [K: A,P: fun(A,$o)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
=> ~ aa(A,$o,P,K) ) ) ).
% not_less_Least
tff(fact_2459_Gcd__in,axiom,
! [Aa2: set(nat)] :
( ! [A4: nat,B3: nat] :
( member(nat,A4,Aa2)
=> ( member(nat,B3,Aa2)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B3),Aa2) ) )
=> ( ( Aa2 != bot_bot(set(nat)) )
=> member(nat,gcd_Gcd(nat,Aa2),Aa2) ) ) ).
% Gcd_in
tff(fact_2460_uminus__rat_Otransfer,axiom,
aa(fun(rat,rat),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),$o),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_ef(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat)) ).
% uminus_rat.transfer
tff(fact_2461_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y )
=> ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
=> ~ ( ( Y = $ite(Xa = zero_zero(nat),X,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa))) )
=> ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).
% gcd_nat.pelims
tff(fact_2462_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& semiring_1(A) )
=> ! [R: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
=> aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).
% transfer_rule_of_nat
tff(fact_2463_plus__rat_Otransfer,axiom,
aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_eg(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat)) ).
% plus_rat.transfer
tff(fact_2464_times__rat_Otransfer,axiom,
aa(fun(rat,fun(rat,rat)),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),$o),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_eh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat)) ).
% times_rat.transfer
tff(fact_2465_times__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).
% times_int.abs_eq
tff(fact_2466_prod_Oinsert_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I4: set(A),P5: fun(A,B),I: A] :
( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),I4),P5)))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P5),aa(set(A),set(A),insert2(A,I),I4)) = $ite(member(A,I,I4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P5),I4),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P5,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),P5),I4))) ) ) ) ).
% prod.insert'
tff(fact_2467_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P5: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P5),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty'
tff(fact_2468_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y2: nat] : Z2 != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y2)) ).
% eq_Abs_Integ
tff(fact_2469_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ek(fun(B,A),fun(set(B),fun(B,$o)),G),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I4) ) ).
% prod.non_neutral'
tff(fact_2470_Fract_Orsp,axiom,
aa(fun(int,fun(int,product_prod(int,int))),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_el(int,fun(int,product_prod(int,int)))),aTP_Lamp_el(int,fun(int,product_prod(int,int)))) ).
% Fract.rsp
tff(fact_2471_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( zero_neq_one(B)
& zero_neq_one(A) )
=> ! [R: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
=> ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
=> aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).
% transfer_rule_of_bool
tff(fact_2472_times__rat_Orsp,axiom,
aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_eh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_eh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat.rsp
tff(fact_2473_plus__int_Otransfer,axiom,
aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int)) ).
% plus_int.transfer
tff(fact_2474_minus__int_Otransfer,axiom,
aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int)) ).
% minus_int.transfer
tff(fact_2475_plus__rat_Orsp,axiom,
aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_eg(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_eg(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat.rsp
tff(fact_2476_times__int_Otransfer,axiom,
aa(fun(int,fun(int,int)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int)) ).
% times_int.transfer
tff(fact_2477_prod_Odistrib__triv_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_eq(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),I4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),I4)) ) ) ) ).
% prod.distrib_triv'
tff(fact_2478_prod_Omono__neutral__cong__right_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T5: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,H2,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),S) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_2479_prod_Omono__neutral__cong__left_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T5: set(A),H2: fun(A,B),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [I3: A] :
( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,H2,I3) = one_one(B) ) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> ( aa(A,B,G,X3) = aa(A,B,H2,X3) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),T5) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_2480_prod_Omono__neutral__right_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T5: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) ) ) ) ) ).
% prod.mono_neutral_right'
tff(fact_2481_prod_Omono__neutral__left_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [S: set(A),T5: set(A),G: fun(A,B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T5)
=> ( ! [X3: A] :
( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T5),S))
=> ( aa(A,B,G,X3) = one_one(B) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),T5) ) ) ) ) ).
% prod.mono_neutral_left'
tff(fact_2482_zero__int__def,axiom,
zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).
% zero_int_def
tff(fact_2483_int__def,axiom,
! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).
% int_def
tff(fact_2484_prod_Odistrib_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [I4: set(A),G: fun(A,B),H2: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),I4),G)))
=> ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),I4),H2)))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_eq(fun(A,B),fun(fun(A,B),fun(A,B)),G),H2)),I4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),G),I4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1962203154675924110t_prod(A,B),H2),I4)) ) ) ) ) ).
% prod.distrib'
tff(fact_2485_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P5: fun(B,A),I4: set(B)] :
aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P5),I4) = $ite(aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ek(fun(B,A),fun(set(B),fun(B,$o)),P5),I4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P5),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ek(fun(B,A),fun(set(B),fun(B,$o)),P5),I4))),one_one(A)) ) ).
% prod.G_def
tff(fact_2486_int__transfer,axiom,
aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_er(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).
% int_transfer
tff(fact_2487_uminus__int_Otransfer,axiom,
aa(fun(int,int),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),$o),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int)) ).
% uminus_int.transfer
tff(fact_2488_uminus__int_Oabs__eq,axiom,
! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat)))),X)) ).
% uminus_int.abs_eq
tff(fact_2489_one__int__def,axiom,
one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).
% one_int_def
tff(fact_2490_uminus__rat_Orsp,axiom,
aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_ef(product_prod(int,int),product_prod(int,int))),aTP_Lamp_ef(product_prod(int,int),product_prod(int,int))) ).
% uminus_rat.rsp
tff(fact_2491_plus__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).
% plus_int.abs_eq
tff(fact_2492_minus__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).
% minus_int.abs_eq
tff(fact_2493_inverse__rat_Orsp,axiom,
aa(fun(product_prod(int,int),product_prod(int,int)),$o,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_ec(product_prod(int,int),product_prod(int,int))),aTP_Lamp_ec(product_prod(int,int),product_prod(int,int))) ).
% inverse_rat.rsp
tff(fact_2494_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),divide_divide(code_integer,K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)),K)) ).
% bit_cut_integer_def
tff(fact_2495_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,K,L)),modulo_modulo(code_integer,K,L)) ).
% divmod_integer_def
tff(fact_2496_times__int_Orsp,axiom,
aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int.rsp
tff(fact_2497_mod__h__bot__iff_I8_J,axiom,
! [A: $tType,R: fun(A,assn),H2: heap_ext(product_unit)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,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_2498_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I4: set(A),Aa2: fun(A,set(B))] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ! [X3: A] :
( member(A,X3,I4)
=> aa(set(B),$o,finite_finite(B),aa(A,set(B),Aa2,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I4)
=> ! [Xa4: A] :
( member(A,Xa4,I4)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X3)),aa(A,set(B),Aa2,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( finite_card(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),I4))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_et(fun(A,set(B)),fun(A,nat),Aa2)),I4) ) ) ) ) ).
% card_UN_disjoint
tff(fact_2499_minus__Min__eq__Max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).
% minus_Min_eq_Max
tff(fact_2500_ex__assn__const,axiom,
! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_eu(assn,fun(A,assn),C2)) = C2 ).
% ex_assn_const
tff(fact_2501_mod__ex__dist,axiom,
! [A: $tType,P: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,P)),H2)
<=> ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,P,X4)),H2) ) ).
% mod_ex_dist
tff(fact_2502_cSup__singleton,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% cSup_singleton
tff(fact_2503_Max__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : lattic643756798349783984er_Max(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% Max_singleton
tff(fact_2504_Sup__atLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),set_ord_atLeast(A,X)) = top_top(A) ) ).
% Sup_atLeast
tff(fact_2505_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).
% intrel_iff
tff(fact_2506_cSUP__const,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),C2: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ev(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ).
% cSUP_const
tff(fact_2507_Max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),X)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),X) ) ) ) ) ) ).
% Max.bounded_iff
tff(fact_2508_Max__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),lattic643756798349783984er_Max(A,Aa2)),X)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X) ) ) ) ) ) ).
% Max_less_iff
tff(fact_2509_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).
% Sup_greaterThanAtLeast
tff(fact_2510_Max__const,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Aa2: set(A),C2: B] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aTP_Lamp_ai(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ) ).
% Max_const
tff(fact_2511_minus__Max__eq__Min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S: set(A)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).
% minus_Max_eq_Min
tff(fact_2512_cSup__eq__Max,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = lattic643756798349783984er_Max(A,X5) ) ) ) ) ).
% cSup_eq_Max
tff(fact_2513_Max__Sup,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,Aa2) = aa(set(A),A,complete_Sup_Sup(A),Aa2) ) ) ) ) ).
% Max_Sup
tff(fact_2514_cSup__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A3) )
=> ( ! [Y2: A] :
( ! [X2: A] :
( member(A,X2,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y2) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = A3 ) ) ) ) ) ).
% cSup_eq_non_empty
tff(fact_2515_cSup__least,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z2) ) ) ) ).
% cSup_least
tff(fact_2516_less__cSupE,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [Y: A,X5: set(A)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
=> ( ( X5 != bot_bot(set(A)) )
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X3) ) ) ) ) ).
% less_cSupE
tff(fact_2517_less__cSupD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X5))
=> ? [X3: A] :
( member(A,X3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X3) ) ) ) ) ).
% less_cSupD
tff(fact_2518_insert__partition,axiom,
! [A: $tType,X: set(A),F4: set(set(A))] :
( ~ member(set(A),X,F4)
=> ( ! [X3: set(A)] :
( member(set(A),X3,aa(set(set(A)),set(set(A)),insert2(set(A),X),F4))
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,aa(set(set(A)),set(set(A)),insert2(set(A),X),F4))
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).
% insert_partition
tff(fact_2519_Max__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> member(A,lattic643756798349783984er_Max(A,Aa2),Aa2) ) ) ) ).
% Max_in
tff(fact_2520_ex__one__point__gen,axiom,
! [A: $tType,P: fun(A,assn),V: A] :
( ! [H3: product_prod(heap_ext(product_unit),set(nat)),X3: A] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,P,X3)),H3)
=> ( X3 = V ) )
=> ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).
% ex_one_point_gen
tff(fact_2521_mod__exI,axiom,
! [A: $tType,P: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
( ? [X2: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,P,X2)),H2)
=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,P)),H2) ) ).
% mod_exI
tff(fact_2522_mod__exE,axiom,
! [A: $tType,P: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,ex_assn(A,P)),H2)
=> ~ ! [X3: A] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,P,X3)),H2) ) ).
% mod_exE
tff(fact_2523_ent__ex__preI,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] :
( ! [X3: A] : entails(aa(A,assn,P,X3),Q)
=> entails(ex_assn(A,P),Q) ) ).
% ent_ex_preI
tff(fact_2524_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: fun(A,assn),X: A] :
( entails(P,aa(A,assn,Q,X))
=> entails(P,ex_assn(A,Q)) ) ).
% ent_ex_postI
tff(fact_2525_ex__distrib__star,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_ew(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P)),Q) ).
% ex_distrib_star
tff(fact_2526_zero__int_Orsp,axiom,
aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).
% zero_int.rsp
tff(fact_2527_ex__distrib__or,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_ex(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_2528_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_ey(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ez(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P),Q)) ).
% ex_join_or
tff(fact_2529_ex__distrib__and,axiom,
! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_fa(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_2530_cSUP__least,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),M4: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),M4) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),M4) ) ) ) ).
% cSUP_least
tff(fact_2531_finite__Sup__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A3: A] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A3)
<=> ! [X4: A] :
( member(A,X4,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A3) ) ) ) ) ) ).
% finite_Sup_less_iff
tff(fact_2532_finite__Sup__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y2),Aa2) ) )
=> member(A,aa(set(A),A,complete_Sup_Sup(A),Aa2),Aa2) ) ) ) ) ).
% finite_Sup_in
tff(fact_2533_cSup__abs__le,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3) ) ) ) ).
% cSup_abs_le
tff(fact_2534_Sup__fin__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2) = aa(set(A),A,complete_Sup_Sup(A),Aa2) ) ) ) ) ).
% Sup_fin_Sup
tff(fact_2535_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic5882676163264333800up_fin(A),X5) ) ) ) ) ).
% cSup_eq_Sup_fin
tff(fact_2536_Max__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),M2: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ( lattic643756798349783984er_Max(A,Aa2) = M2 )
<=> ( member(A,M2,Aa2)
& ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M2) ) ) ) ) ) ) ).
% Max_eq_iff
tff(fact_2537_Max__ge__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),lattic643756798349783984er_Max(A,Aa2))
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X4) ) ) ) ) ) ).
% Max_ge_iff
tff(fact_2538_eq__Max__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),M2: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ( M2 = lattic643756798349783984er_Max(A,Aa2) )
<=> ( member(A,M2,Aa2)
& ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M2) ) ) ) ) ) ) ).
% eq_Max_iff
tff(fact_2539_Max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),X)
=> ! [A9: A] :
( member(A,A9,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),X) ) ) ) ) ) ).
% Max.boundedE
tff(fact_2540_Max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),X) ) ) ) ) ).
% Max.boundedI
tff(fact_2541_Max__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),lattic643756798349783984er_Max(A,Aa2))
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X4) ) ) ) ) ) ).
% Max_gr_iff
tff(fact_2542_uminus__int_Orsp,axiom,
aa(fun(product_prod(nat,nat),product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat))))),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat))))) ).
% uminus_int.rsp
tff(fact_2543_one__int_Orsp,axiom,
aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).
% one_int.rsp
tff(fact_2544_cSup__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S)),L))),E3) ) ) ) ).
% cSup_asclose
tff(fact_2545_Max_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,Aa2)),lattic643756798349783984er_Max(A,Ba)) ) ) ) ) ).
% Max.subset_imp
tff(fact_2546_Max__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M4: set(A),N3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N3)
=> ( ( M4 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),N3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,M4)),lattic643756798349783984er_Max(A,N3)) ) ) ) ) ).
% Max_mono
tff(fact_2547_Max__add__commute,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [S: set(A),F: fun(A,B),K: B] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_aq(fun(A,B),fun(B,fun(A,B)),F),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F),S))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_2548_card__partition,axiom,
! [A: $tType,Ca: set(set(A)),K: nat] :
( aa(set(set(A)),$o,finite_finite(set(A)),Ca)
=> ( aa(set(A),$o,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca))
=> ( ! [C3: set(A)] :
( member(set(A),C3,Ca)
=> ( finite_card(A,C3) = K ) )
=> ( ! [C1: set(A),C22: set(A)] :
( member(set(A),C1,Ca)
=> ( member(set(A),C22,Ca)
=> ( ( 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),Ca)) = finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca)) ) ) ) ) ) ).
% card_partition
tff(fact_2549_dvd__partition,axiom,
! [A: $tType,Ca: set(set(A)),K: nat] :
( aa(set(A),$o,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca))
=> ( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> dvd_dvd(nat,K,finite_card(A,X3)) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,Ca)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> dvd_dvd(nat,K,finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca))) ) ) ) ).
% dvd_partition
tff(fact_2550_sum_OUNION__disjoint,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_add(C)
=> ! [I4: set(A),Aa2: fun(A,set(B)),G: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ! [X3: A] :
( member(A,X3,I4)
=> aa(set(B),$o,finite_finite(B),aa(A,set(B),Aa2,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I4)
=> ! [Xa4: A] :
( member(A,Xa4,I4)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X3)),aa(A,set(B),Aa2,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),I4))) = 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_fb(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Aa2),G)),I4) ) ) ) ) ) ).
% sum.UNION_disjoint
tff(fact_2551_prod_OUNION__disjoint,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [I4: set(A),Aa2: fun(A,set(B)),G: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ! [X3: A] :
( member(A,X3,I4)
=> aa(set(B),$o,finite_finite(B),aa(A,set(B),Aa2,X3)) )
=> ( ! [X3: A] :
( member(A,X3,I4)
=> ! [Xa4: A] :
( member(A,Xa4,I4)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Aa2,X3)),aa(A,set(B),Aa2,Xa4)) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),I4))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_fc(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Aa2),G)),I4) ) ) ) ) ) ).
% prod.UNION_disjoint
tff(fact_2552_minus__int_Orsp,axiom,
aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% minus_int.rsp
tff(fact_2553_plus__int_Orsp,axiom,
aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% plus_int.rsp
tff(fact_2554_UN__simps_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fd(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca)))) ).
% UN_simps(3)
tff(fact_2555_UN__simps_I2_J,axiom,
! [B: $tType,A: $tType,Aa2: fun(B,set(A)),Ba: set(A),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fe(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba)) ).
% UN_simps(2)
tff(fact_2556_UN__singleton,axiom,
! [A: $tType,Aa2: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_ff(A,set(A))),Aa2)) = Aa2 ).
% UN_singleton
tff(fact_2557_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,A3: A,Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fg(A,fun(fun(B,set(A)),fun(B,set(A))),A3),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca)))) ).
% UN_simps(1)
tff(fact_2558_SUP__eq__top__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),Aa2: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),Aa2)) = top_top(A) )
<=> ! [X4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
=> ? [Xa2: B] :
( member(B,Xa2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),aa(B,A,F,Xa2)) ) ) ) ) ).
% SUP_eq_top_iff
tff(fact_2559_UN__constant,axiom,
! [B: $tType,A: $tType,C2: set(A),Aa2: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fh(set(A),fun(B,set(A)),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),C2) ).
% UN_constant
tff(fact_2560_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),Aa2) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ( X4 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(2)
tff(fact_2561_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = bot_bot(A) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ( X4 = bot_bot(A) ) ) ) ) ).
% Sup_bot_conv(1)
tff(fact_2562_Sup__nat__empty,axiom,
aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).
% Sup_nat_empty
tff(fact_2563_Sup__eq__top__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = top_top(A) )
<=> ! [X4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
=> ? [Xa2: A] :
( member(A,Xa2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa2) ) ) ) ) ).
% Sup_eq_top_iff
tff(fact_2564_Sup__empty,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).
% Sup_empty
tff(fact_2565_Sup__UNIV,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) = top_top(A) ) ) ).
% Sup_UNIV
tff(fact_2566_Sup__insert,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A3: A,Aa2: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,A3),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ).
% Sup_insert
tff(fact_2567_SUP__bot__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Ba: fun(B,A),Aa2: set(B)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Ba),Aa2)) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,A,Ba,X4) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(2)
tff(fact_2568_SUP__bot__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Ba: fun(B,A),Aa2: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Ba),Aa2)) = bot_bot(A) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,A,Ba,X4) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(1)
tff(fact_2569_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_fi(B,A)),Aa2)) = bot_bot(A) ) ).
% SUP_bot
tff(fact_2570_SUP__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_fj(B,fun(A,B),F)),Aa2)) = F ) ) ) ).
% SUP_const
tff(fact_2571_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) ) ).
% SUP_Sup_eq2
tff(fact_2572_SUP__UN__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_fk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).
% SUP_UN_eq2
tff(fact_2573_Sup__nat__def,axiom,
! [X5: set(nat)] :
aa(set(nat),nat,complete_Sup_Sup(nat),X5) = $ite(X5 = bot_bot(set(nat)),zero_zero(nat),lattic643756798349783984er_Max(nat,X5)) ).
% Sup_nat_def
tff(fact_2574_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).
% Sup_SUP_eq2
tff(fact_2575_Union__empty,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).
% Union_empty
tff(fact_2576_Union__empty__conv,axiom,
! [A: $tType,Aa2: set(set(A))] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Aa2) = bot_bot(set(A)) )
<=> ! [X4: set(A)] :
( member(set(A),X4,Aa2)
=> ( X4 = bot_bot(set(A)) ) ) ) ).
% Union_empty_conv
tff(fact_2577_empty__Union__conv,axiom,
! [A: $tType,Aa2: set(set(A))] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Aa2) )
<=> ! [X4: set(A)] :
( member(set(A),X4,Aa2)
=> ( X4 = bot_bot(set(A)) ) ) ) ).
% empty_Union_conv
tff(fact_2578_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image2($o,A,Aa2),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,Aa2,$true)),aa($o,A,Aa2,$false)) ) ).
% SUP_UNIV_bool_expand
tff(fact_2579_less__eq__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A),U: A] :
( ! [V2: A] :
( member(A,V2,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2) )
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ).
% less_eq_Sup
tff(fact_2580_SUP__eq__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),F: fun(A,B),X: B] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> ( aa(A,B,F,I3) = X ) )
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),I4)) = X ) ) ) ) ).
% SUP_eq_const
tff(fact_2581_Sup__union__distrib,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A),Ba: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Ba)) ) ).
% Sup_union_distrib
tff(fact_2582_Union__disjoint,axiom,
! [A: $tType,Ca: set(set(A)),Aa2: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca)),Aa2) = bot_bot(set(A)) )
<=> ! [X4: set(A)] :
( member(set(A),X4,Ca)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),Aa2) = bot_bot(set(A)) ) ) ) ).
% Union_disjoint
tff(fact_2583_SUP__absorb,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [K: A,I4: set(A),Aa2: fun(A,B)] :
( member(A,K,I4)
=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Aa2,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,Aa2),I4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,Aa2),I4)) ) ) ) ).
% SUP_absorb
tff(fact_2584_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Aa2: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),Aa2))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),Aa2))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fl(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Aa2)) ) ).
% complete_lattice_class.SUP_sup_distrib
tff(fact_2585_UNION__empty__conv_I2_J,axiom,
! [B: $tType,A: $tType,Ba: fun(B,set(A)),Aa2: set(B)] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Aa2)) = bot_bot(set(A)) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,set(A),Ba,X4) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(2)
tff(fact_2586_UNION__empty__conv_I1_J,axiom,
! [B: $tType,A: $tType,Ba: fun(B,set(A)),Aa2: set(B)] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Aa2)) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,set(A),Ba,X4) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(1)
tff(fact_2587_UN__empty,axiom,
! [B: $tType,A: $tType,Ba: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),bot_bot(set(B)))) = bot_bot(set(A)) ).
% UN_empty
tff(fact_2588_UN__empty2,axiom,
! [B: $tType,A: $tType,Aa2: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fm(B,set(A))),Aa2)) = bot_bot(set(A)) ).
% UN_empty2
tff(fact_2589_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),C2: B,F: fun(A,B)] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F,I3)) )
=> ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),I4)) = C2 )
<=> ! [X4: A] :
( member(A,X4,I4)
=> ( aa(A,B,F,X4) = C2 ) ) ) ) ) ) ).
% SUP_eq_iff
tff(fact_2590_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).
% SUP_empty
tff(fact_2591_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [C2: A,Aa2: set(B)] :
aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_fn(A,fun(B,A),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(A),C2) ) ).
% SUP_constant
tff(fact_2592_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A3: B,Aa2: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),insert2(B,A3),Aa2))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),Aa2))) ) ).
% SUP_insert
tff(fact_2593_SUP__union,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [M4: fun(B,A),Aa2: set(B),Ba: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),Ba))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),Aa2))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M4),Ba))) ) ).
% SUP_union
tff(fact_2594_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,A3: A,Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(A),set(A),insert2(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca))) = $ite(Ca = bot_bot(set(B)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fg(A,fun(fun(B,set(A)),fun(B,set(A))),A3),Ba)),Ca))) ).
% UN_extend_simps(1)
tff(fact_2595_UN__extend__simps_I2_J,axiom,
! [A: $tType,B: $tType,Aa2: fun(B,set(A)),Ca: set(B),Ba: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba) = $ite(Ca = bot_bot(set(B)),Ba,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fe(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% UN_extend_simps(2)
tff(fact_2596_UN__extend__simps_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca))) = $ite(Ca = bot_bot(set(B)),Aa2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fd(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% UN_extend_simps(3)
tff(fact_2597_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fo(fun(B,A),fun(B,set(A)),F)),Aa2)) = aa(set(B),set(A),image2(B,A,F),Aa2) ).
% UNION_singleton_eq_range
tff(fact_2598_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% ccpo_Sup_singleton
tff(fact_2599_Union__image__empty,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F),bot_bot(set(B))))) = Aa2 ).
% Union_image_empty
tff(fact_2600_bit__cut__integer__code,axiom,
! [K: code_integer] :
code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_fp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).
% bit_cut_integer_code
tff(fact_2601_subset__mset_OcSUP__const,axiom,
! [A: $tType,B: $tType,Aa2: set(A),C2: multiset(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_fq(multiset(B),fun(A,multiset(B)),C2)),Aa2)) = C2 ) ) ).
% subset_mset.cSUP_const
tff(fact_2602_Sup__finite__insert,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ! [A3: A,Aa2: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,A3),Aa2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ).
% Sup_finite_insert
tff(fact_2603_top__finite__def,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( top_top(A) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).
% top_finite_def
tff(fact_2604_subset__mset_OcSup__singleton,axiom,
! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.cSup_singleton
tff(fact_2605_Sup__multiset__empty,axiom,
! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).
% Sup_multiset_empty
tff(fact_2606_UNIV__bool,axiom,
top_top(set($o)) = aa(set($o),set($o),insert2($o,$false),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o)))) ).
% UNIV_bool
tff(fact_2607_divmod__abs__code_I6_J,axiom,
! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).
% divmod_abs_code(6)
tff(fact_2608_divmod__abs__code_I5_J,axiom,
! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).
% divmod_abs_code(5)
tff(fact_2609_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).
% divmod_abs_def
tff(fact_2610_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_zero(code_integer),
aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
$ite(
aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
$ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_fr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
$ite(
L = zero_zero(code_integer),
aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
$ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_fs(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_2611_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Ba: set(A),A3: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),Ba)),A3) = bot_bot(A) )
<=> ! [X4: A] :
( member(A,X4,Ba)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).
% Sup_inf_eq_bot_iff
tff(fact_2612_mlex__eq,axiom,
! [A: $tType,F: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F,R) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_ft(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F),R))) ).
% mlex_eq
tff(fact_2613_Gcd__eq__Max,axiom,
! [M4: set(nat)] :
( aa(set(nat),$o,finite_finite(nat),M4)
=> ( ( M4 != bot_bot(set(nat)) )
=> ( ~ member(nat,zero_zero(nat),M4)
=> ( gcd_Gcd(nat,M4) = lattic643756798349783984er_Max(nat,aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),aTP_Lamp_fv(nat,set(nat))),M4))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_2614_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,Aa2: fun(B,set(A)),I: B,Ba: set(A),J3: set(B)] :
aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),Aa2,I,Ba)),J3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J3),aa(set(B),set(B),insert2(B,I),bot_bot(set(B))))))),
$ite(member(B,I,J3),Ba,bot_bot(set(A)))) ).
% UNION_fun_upd
tff(fact_2615_fold__union__pair,axiom,
! [B: $tType,A: $tType,Ba: set(A),X: B,Aa2: set(product_prod(B,A))] :
( aa(set(A),$o,finite_finite(A),Ba)
=> ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_fw(B,fun(A,set(product_prod(B,A))),X)),Ba))),Aa2) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_fx(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),Aa2,Ba) ) ) ).
% fold_union_pair
tff(fact_2616_Inf__top__conv_I2_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),Aa2) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ( X4 = top_top(A) ) ) ) ) ).
% Inf_top_conv(2)
tff(fact_2617_Inf__top__conv_I1_J,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,complete_Inf_Inf(A),Aa2) = top_top(A) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ( X4 = top_top(A) ) ) ) ) ).
% Inf_top_conv(1)
tff(fact_2618_fold__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,fun(A,A)),Z2: A] : finite_fold(B,A,F,Z2,bot_bot(set(B))) = Z2 ).
% fold_empty
tff(fact_2619_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F,Y)) ).
% apsnd_conv
tff(fact_2620_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,complete_Inf_Inf(A),Aa2) = bot_bot(A) )
<=> ! [X4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
=> ? [Xa2: A] :
( member(A,Xa2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X4) ) ) ) ) ).
% Inf_eq_bot_iff
tff(fact_2621_Inf__empty,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).
% Inf_empty
tff(fact_2622_Inf__UNIV,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) = bot_bot(A) ) ) ).
% Inf_UNIV
tff(fact_2623_cInf__singleton,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ).
% cInf_singleton
tff(fact_2624_Inf__atMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),set_ord_atMost(A,X)) = bot_bot(A) ) ).
% Inf_atMost
tff(fact_2625_INF__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_fj(B,fun(A,B),F)),Aa2)) = F ) ) ) ).
% INF_const
tff(fact_2626_cINF__const,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),C2: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ev(B,fun(A,B),C2)),Aa2)) = C2 ) ) ) ).
% cINF_const
tff(fact_2627_INF__top,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_fy(B,A)),Aa2)) = top_top(A) ) ).
% INF_top
tff(fact_2628_INF__top__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Ba: fun(B,A),Aa2: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,Ba),Aa2)) = top_top(A) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,A,Ba,X4) = top_top(A) ) ) ) ) ).
% INF_top_conv(1)
tff(fact_2629_INF__top__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Ba: fun(B,A),Aa2: set(B)] :
( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,Ba),Aa2)) )
<=> ! [X4: B] :
( member(B,X4,Aa2)
=> ( aa(B,A,Ba,X4) = top_top(A) ) ) ) ) ).
% INF_top_conv(2)
tff(fact_2630_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),Aa2: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),Aa2)) = bot_bot(A) )
<=> ! [X4: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
=> ? [Xa2: B] :
( member(B,Xa2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F,Xa2)),X4) ) ) ) ) ).
% INF_eq_bot_iff
tff(fact_2631_INT__constant,axiom,
! [B: $tType,A: $tType,C2: set(A),Aa2: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_fh(set(A),fun(B,set(A)),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),top_top(set(A)),C2) ).
% INT_constant
tff(fact_2632_Inf__atMostLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),X)
=> ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,X)) = bot_bot(A) ) ) ) ).
% Inf_atMostLessThan
tff(fact_2633_INT__simps_I2_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fz(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca)))) ).
% INT_simps(2)
tff(fact_2634_INT__simps_I1_J,axiom,
! [B: $tType,A: $tType,Aa2: fun(B,set(A)),Ba: set(A),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ga(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba)) ).
% INT_simps(1)
tff(fact_2635_INT__simps_I3_J,axiom,
! [B: $tType,A: $tType,Aa2: fun(B,set(A)),Ba: set(A),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_gb(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba)) ).
% INT_simps(3)
tff(fact_2636_INT__simps_I4_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gc(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca)) = $ite(Ca = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca)))) ).
% INT_simps(4)
tff(fact_2637_sup__Inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,Ba: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),Ba)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),Ba)) ) ).
% sup_Inf
tff(fact_2638_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Ba: set(A),A3: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),Ba)),A3) = top_top(A) )
<=> ! [X4: A] :
( member(A,X4,Ba)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).
% Inf_sup_eq_top_iff
tff(fact_2639_Inf__fold__inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,complete_Inf_Inf(A),Aa2) = finite_fold(A,A,inf_inf(A),top_top(A),Aa2) ) ) ) ).
% Inf_fold_inf
tff(fact_2640_INF__sup,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),Ba: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),Ba))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_gd(fun(B,A),fun(A,fun(B,A)),F),A3)),Ba)) ) ).
% INF_sup
tff(fact_2641_Inf__sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Ba: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),Ba)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_ge(A,fun(A,A),A3)),Ba)) ) ).
% Inf_sup
tff(fact_2642_sup__INF,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),Ba: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),Ba))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gf(A,fun(fun(B,A),fun(B,A)),A3),F)),Ba)) ) ).
% sup_INF
tff(fact_2643_INF__sup__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),Aa2: set(B),G: fun(C,A),Ba: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),Aa2))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),Ba))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_gh(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),Ba)),Aa2)) ) ).
% INF_sup_distrib2
tff(fact_2644_Inf__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A),U: A] :
( ! [V2: A] :
( member(A,V2,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U) )
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),U) ) ) ) ).
% Inf_less_eq
tff(fact_2645_cInf__greatest,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ).
% cInf_greatest
tff(fact_2646_cInf__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3) )
=> ( ! [Y2: A] :
( ! [X2: A] :
( member(A,X2,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),A3) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = A3 ) ) ) ) ) ).
% cInf_eq_non_empty
tff(fact_2647_cInf__lessD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Z2: A] :
( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z2)
=> ? [X3: A] :
( member(A,X3,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z2) ) ) ) ) ).
% cInf_lessD
tff(fact_2648_INF__eq__const,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),F: fun(A,B),X: B] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> ( aa(A,B,F,I3) = X ) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4)) = X ) ) ) ) ).
% INF_eq_const
tff(fact_2649_bot__finite__def,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( bot_bot(A) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).
% bot_finite_def
tff(fact_2650_Inter__subset,axiom,
! [A: $tType,Aa2: set(set(A)),Ba: set(A)] :
( ! [X6: set(A)] :
( member(set(A),X6,Aa2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),Ba) )
=> ( ( Aa2 != bot_bot(set(set(A))) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Aa2)),Ba) ) ) ).
% Inter_subset
tff(fact_2651_Inter__empty,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),bot_bot(set(set(A)))) = top_top(set(A)) ).
% Inter_empty
tff(fact_2652_Inter__UNIV,axiom,
! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),top_top(set(set(A)))) = bot_bot(set(A)) ).
% Inter_UNIV
tff(fact_2653_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),F: fun(A,B),C2: B] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),C2) )
=> ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4)) = C2 )
<=> ! [X4: A] :
( member(A,X4,I4)
=> ( aa(A,B,F,X4) = C2 ) ) ) ) ) ) ).
% INF_eq_iff
tff(fact_2654_cINF__greatest,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),M2: B,F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M2),aa(A,B,F,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M2),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ) ).
% cINF_greatest
tff(fact_2655_finite__less__Inf__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),A3: A] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5))
<=> ! [X4: A] :
( member(A,X4,X5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X4) ) ) ) ) ) ).
% finite_less_Inf_iff
tff(fact_2656_Inf__le__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ).
% Inf_le_Sup
tff(fact_2657_finite__Inf__in,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y2),Aa2) ) )
=> member(A,aa(set(A),A,complete_Inf_Inf(A),Aa2),Aa2) ) ) ) ) ).
% finite_Inf_in
tff(fact_2658_cInf__abs__ge,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3) ) ) ) ).
% cInf_abs_ge
tff(fact_2659_less__eq__Inf__inter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A),Ba: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),Ba))),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)),Aa2),Ba))) ) ).
% less_eq_Inf_inter
tff(fact_2660_Sup__finite__empty,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).
% Sup_finite_empty
tff(fact_2661_Inf__finite__empty,axiom,
! [A: $tType] :
( finite_lattice(A)
=> ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).
% Inf_finite_empty
tff(fact_2662_cInf__eq__Min,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = lattic643756798350308766er_Min(A,X5) ) ) ) ) ).
% cInf_eq_Min
tff(fact_2663_Min__Inf,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798350308766er_Min(A,Aa2) = aa(set(A),A,complete_Inf_Inf(A),Aa2) ) ) ) ) ).
% Min_Inf
tff(fact_2664_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A),Ba: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),Ba) = finite_fold(A,A,sup_sup(A),Ba,Aa2) ) ) ) ).
% sup_Sup_fold_sup
tff(fact_2665_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A)] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( ( X5 != bot_bot(set(A)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X5) = lattic7752659483105999362nf_fin(A,X5) ) ) ) ) ).
% cInf_eq_Inf_fin
tff(fact_2666_Inf__fin__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic7752659483105999362nf_fin(A,Aa2) = aa(set(A),A,complete_Inf_Inf(A),Aa2) ) ) ) ) ).
% Inf_fin_Inf
tff(fact_2667_INT__greaterThan__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).
% INT_greaterThan_UNIV
tff(fact_2668_INF__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).
% INF_empty
tff(fact_2669_INF__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [C2: A,Aa2: set(B)] :
aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_fn(A,fun(B,A),C2)),Aa2)) = $ite(Aa2 = bot_bot(set(B)),top_top(A),C2) ) ).
% INF_constant
tff(fact_2670_INF__inf__const2,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),F: fun(A,B),X: B] :
( ( I4 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_gi(fun(A,B),fun(B,fun(A,B)),F),X)),I4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4))),X) ) ) ) ).
% INF_inf_const2
tff(fact_2671_INF__inf__const1,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [I4: set(A),X: B,F: fun(A,B)] :
( ( I4 != bot_bot(set(A)) )
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gj(B,fun(fun(A,B),fun(A,B)),X),F)),I4)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),X),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),I4))) ) ) ) ).
% INF_inf_const1
tff(fact_2672_INT__empty,axiom,
! [B: $tType,A: $tType,Ba: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),bot_bot(set(B)))) = top_top(set(A)) ).
% INT_empty
tff(fact_2673_INT__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,Aa2: fun(B,set(A)),Ca: set(B),Ba: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba) = $ite(Ca = bot_bot(set(B)),Ba,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ga(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% INT_extend_simps(1)
tff(fact_2674_INT__extend__simps_I2_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca))) = $ite(Ca = bot_bot(set(B)),Aa2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fz(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% INT_extend_simps(2)
tff(fact_2675_Int__Inter__eq_I1_J,axiom,
! [A: $tType,Aa2: set(A),B10: set(set(A))] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B10)) = $ite(B10 = bot_bot(set(set(A))),Aa2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2)),B10))) ).
% Int_Inter_eq(1)
tff(fact_2676_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B10: set(set(A)),Aa2: 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)),B10)),Aa2) = $ite(B10 = bot_bot(set(set(A))),Aa2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_gk(set(A),fun(set(A),set(A)),Aa2)),B10))) ).
% Int_Inter_eq(2)
tff(fact_2677_INF__le__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ).
% INF_le_SUP
tff(fact_2678_cInf__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S: set(A),L: A,E3: A] :
( ( S != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S)),L))),E3) ) ) ) ).
% cInf_asclose
tff(fact_2679_Inf__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),insert2(A,X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S))) ) ) ) ).
% Inf_insert_finite
tff(fact_2680_Sup__fold__sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,complete_Sup_Sup(A),Aa2) = finite_fold(A,A,sup_sup(A),bot_bot(A),Aa2) ) ) ) ).
% Sup_fold_sup
tff(fact_2681_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),insert2(A,X),Aa2)) = finite_fold(A,A,sup_sup(A),X,Aa2) ) ) ) ).
% Sup_fin.eq_fold
tff(fact_2682_image__fold__insert,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),set(B),image2(A,B,F),Aa2) = finite_fold(A,set(B),aTP_Lamp_gl(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),Aa2) ) ) ).
% image_fold_insert
tff(fact_2683_INT__extend__simps_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: fun(B,set(A)),Ca: set(B),Ba: set(A)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),Ca))),Ba) = $ite(Ca = bot_bot(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),Ba),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_gb(fun(B,set(A)),fun(set(A),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% INT_extend_simps(3)
tff(fact_2684_INT__extend__simps_I4_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: fun(B,set(A)),Ca: set(B)] :
aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Ba),Ca))) = $ite(Ca = bot_bot(set(B)),Aa2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gc(set(A),fun(fun(B,set(A)),fun(B,set(A))),Aa2),Ba)),Ca))) ).
% INT_extend_simps(4)
tff(fact_2685_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
aa(set(A),A,semiring_gcd_Gcd_fin(A),Aa2) = $ite(aa(set(A),$o,finite_finite(A),Aa2),finite_fold(A,A,gcd_gcd(A),zero_zero(A),Aa2),one_one(A)) ) ).
% Gcd_fin.eq_fold
tff(fact_2686_fun__upd__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X: B,Y: A,Aa2: set(B)] :
aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X,Y)),Aa2) = $ite(member(B,X,Aa2),aa(set(A),set(A),insert2(A,Y),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F),Aa2)) ).
% fun_upd_image
tff(fact_2687_card__UNION,axiom,
! [A: $tType,Aa2: set(set(A))] :
( aa(set(set(A)),$o,finite_finite(set(A)),Aa2)
=> ( ! [X3: set(A)] :
( member(set(A),X3,Aa2)
=> aa(set(A),$o,finite_finite(A),X3) )
=> ( finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Aa2)) = nat2(aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_gm(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_gn(set(set(A)),fun(set(set(A)),$o),Aa2)))) ) ) ) ).
% card_UNION
tff(fact_2688_mlex__leq,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F,R)) ) ) ).
% mlex_leq
tff(fact_2689_mlex__less,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F,R)) ) ).
% mlex_less
tff(fact_2690_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat),R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),mlex_prod(A,F,R))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
| ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R) ) ) ) ).
% mlex_iff
tff(fact_2691_Set__filter__fold,axiom,
! [A: $tType,Aa2: set(A),P: fun(A,$o)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( filter3(A,P,Aa2) = finite_fold(A,set(A),aTP_Lamp_go(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),Aa2) ) ) ).
% Set_filter_fold
tff(fact_2692_subset__mset_OcInf__singleton,axiom,
! [A: $tType,X: multiset(A)] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.cInf_singleton
tff(fact_2693_subset__mset_OcINF__const,axiom,
! [A: $tType,B: $tType,Aa2: set(A),C2: multiset(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aTP_Lamp_fq(multiset(B),fun(A,multiset(B)),C2)),Aa2)) = C2 ) ) ).
% subset_mset.cINF_const
tff(fact_2694_Inf__nat__def1,axiom,
! [K5: set(nat)] :
( ( K5 != bot_bot(set(nat)) )
=> member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K5),K5) ) ).
% Inf_nat_def1
tff(fact_2695_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S)) ) ).
% INF_Int_eq2
tff(fact_2696_INF__INT__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_fk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).
% INF_INT_eq2
tff(fact_2697_Inf__multiset__empty,axiom,
! [A: $tType] : aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),bot_bot(set(multiset(A)))) = zero_zero(multiset(A)) ).
% Inf_multiset_empty
tff(fact_2698_INF__filter__not__bot,axiom,
! [A: $tType,B: $tType,Ba: set(A),F4: fun(A,filter(B))] :
( ! [X6: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),Ba)
=> ( aa(set(A),$o,finite_finite(A),X6)
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),X6)) != bot_bot(filter(B)) ) ) )
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),Ba)) != bot_bot(filter(B)) ) ) ).
% INF_filter_not_bot
tff(fact_2699_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).
% Inf_INT_eq2
tff(fact_2700_in__measure,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measure(A,F))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)) ) ).
% in_measure
tff(fact_2701_in__finite__psubset,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),Aa2),Ba),finite_psubset(A))
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),Ba)
& aa(set(A),$o,finite_finite(A),Ba) ) ) ).
% in_finite_psubset
tff(fact_2702_Id__on__fold,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( id_on(A,Aa2) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_gp(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),Aa2) ) ) ).
% Id_on_fold
tff(fact_2703_Id__on__def,axiom,
! [A: $tType,Aa2: set(A)] : id_on(A,Aa2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_gq(A,set(product_prod(A,A)))),Aa2)) ).
% Id_on_def
tff(fact_2704_Pow__fold,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( pow(A,Aa2) = finite_fold(A,set(set(A)),aTP_Lamp_gr(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),insert2(set(A),bot_bot(set(A))),bot_bot(set(set(A)))),Aa2) ) ) ).
% Pow_fold
tff(fact_2705_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),Aa2: set(A),Ba: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),S)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = finite_fold(A,B,F,finite_fold(A,B,F,Z2,Aa2),Ba) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
tff(fact_2706_Id__onI,axiom,
! [A: $tType,A3: A,Aa2: set(A)] :
( member(A,A3,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id_on(A,Aa2)) ) ).
% Id_onI
tff(fact_2707_Id__on__empty,axiom,
! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).
% Id_on_empty
tff(fact_2708_Pow__singleton__iff,axiom,
! [A: $tType,X5: set(A),Y5: set(A)] :
( ( pow(A,X5) = aa(set(set(A)),set(set(A)),insert2(set(A),Y5),bot_bot(set(set(A)))) )
<=> ( ( X5 = bot_bot(set(A)) )
& ( Y5 = bot_bot(set(A)) ) ) ) ).
% Pow_singleton_iff
tff(fact_2709_Pow__empty,axiom,
! [A: $tType] : pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),insert2(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_empty
tff(fact_2710_Inf__filter__not__bot,axiom,
! [A: $tType,Ba: set(filter(A))] :
( ! [X6: set(filter(A))] :
( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X6),Ba)
=> ( aa(set(filter(A)),$o,finite_finite(filter(A)),X6)
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X6) != bot_bot(filter(A)) ) ) )
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),Ba) != bot_bot(filter(A)) ) ) ).
% Inf_filter_not_bot
tff(fact_2711_INF__filter__bot__base,axiom,
! [A: $tType,B: $tType,I4: set(A),F4: fun(A,filter(B))] :
( ! [I3: A] :
( member(A,I3,I4)
=> ! [J2: A] :
( member(A,J2,I4)
=> ? [X2: A] :
( member(A,X2,I4)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),aa(A,filter(B),F4,J2))) ) ) )
=> ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I4)) = bot_bot(filter(B)) )
<=> ? [X4: A] :
( member(A,X4,I4)
& ( aa(A,filter(B),F4,X4) = bot_bot(filter(B)) ) ) ) ) ).
% INF_filter_bot_base
tff(fact_2712_Pow__bottom,axiom,
! [A: $tType,Ba: set(A)] : member(set(A),bot_bot(set(A)),pow(A,Ba)) ).
% Pow_bottom
tff(fact_2713_Pow__not__empty,axiom,
! [A: $tType,Aa2: set(A)] : pow(A,Aa2) != bot_bot(set(set(A))) ).
% Pow_not_empty
tff(fact_2714_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,Aa2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),id_on(A,Aa2))
<=> ( ( X = Y )
& member(A,X,Aa2) ) ) ).
% Id_on_iff
tff(fact_2715_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,Aa2: set(A)] :
( ( A3 = B2 )
=> ( member(A,A3,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id_on(A,Aa2)) ) ) ).
% Id_on_eqI
tff(fact_2716_Id__onE,axiom,
! [A: $tType,C2: product_prod(A,A),Aa2: set(A)] :
( member(product_prod(A,A),C2,id_on(A,Aa2))
=> ~ ! [X3: A] :
( member(A,X3,Aa2)
=> ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).
% Id_onE
tff(fact_2717_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),Aa2: set(A),X: A,Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( finite_fold(A,B,F,Z2,Aa2) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
tff(fact_2718_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,Aa2: set(A),Z2: B] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),Aa2)),S)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( finite_fold(A,B,F,Z2,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(B,B,aa(A,fun(B,B),F,X),finite_fold(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
tff(fact_2719_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),product_apfst(D,A,C,G,X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F,aa(product_prod(D,C),C,product_snd(D,C),X))) ).
% apsnd_apfst
tff(fact_2720_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : product_apfst(C,A,B,F,aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X))) ).
% apfst_apsnd
tff(fact_2721_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
code_divmod_integer(K,L) = $ite(
K = zero_zero(code_integer),
aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
$ite(
L = zero_zero(code_integer),
aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),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),
$ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_gs(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_2722_bijective__Empty,axiom,
! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).
% bijective_Empty
tff(fact_2723_total__on__singleton,axiom,
! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% total_on_singleton
tff(fact_2724_comp__fun__commute__product__fold,axiom,
! [B: $tType,A: $tType,Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Ba)
=> finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_gt(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Ba)) ) ).
% comp_fun_commute_product_fold
tff(fact_2725_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(C,A),X: C,Y: B] : product_apfst(C,A,B,F,aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X)),Y) ).
% apfst_conv
tff(fact_2726_total__onI,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> ( ( X3 != Y2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),R2) ) ) ) )
=> total_on(A,Aa2,R2) ) ).
% total_onI
tff(fact_2727_total__on__def,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( total_on(A,Aa2,R2)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ! [Xa2: A] :
( member(A,Xa2,Aa2)
=> ( ( X4 != Xa2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ) ).
% total_on_def
tff(fact_2728_total__on__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).
% total_on_empty
tff(fact_2729_bijective__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( bijective(A,B,R)
<=> ( ! [X4: A,Y3: B,Z4: B] :
( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R)
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R) )
=> ( Y3 = Z4 ) )
& ! [X4: A,Y3: A,Z4: B] :
( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R)
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z4),R) )
=> ( X4 = Y3 ) ) ) ) ).
% bijective_def
tff(fact_2730_comp__fun__commute__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S)
=> finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_gv(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).
% comp_fun_commute_relcomp_fold
tff(fact_2731_prod_Oreindex__nontrivial,axiom,
! [B: $tType,C: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [Aa2: set(A),H2: fun(A,B),G: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> ( ( X3 != Y2 )
=> ( ( aa(A,B,H2,X3) = aa(A,B,H2,Y2) )
=> ( aa(B,C,G,aa(A,B,H2,X3)) = one_one(C) ) ) ) ) )
=> ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image2(A,B,H2),Aa2)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),comp(B,C,A,G,H2)),Aa2) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_2732_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Aa2: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),Aa2) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),Aa2) ) ).
% prod.eq_fold
tff(fact_2733_sup__SUP__fold__sup,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),Ba: B,F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),Ba),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))) = finite_fold(A,B,comp(B,fun(B,B),A,sup_sup(B),F),Ba,Aa2) ) ) ) ).
% sup_SUP_fold_sup
tff(fact_2734_SUP__fold__sup,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2)) = finite_fold(A,B,comp(B,fun(B,B),A,sup_sup(B),F),bot_bot(B),Aa2) ) ) ) ).
% SUP_fold_sup
tff(fact_2735_INF__fold__inf,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2)) = finite_fold(A,B,comp(B,fun(B,B),A,inf_inf(B),F),top_top(B),Aa2) ) ) ) ).
% INF_fold_inf
tff(fact_2736_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F: fun(C,A),P5: product_prod(C,B)] :
( ( Q3 = product_apfst(C,A,B,F,P5) )
=> ~ ! [X3: C,Y2: B] :
( ( P5 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y2) )
=> ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X3)),Y2) ) ) ) ).
% apfst_convE
tff(fact_2737_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),X5: set(product_prod(C,B))] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S)
=> ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert2(product_prod(C,A),X),bot_bot(set(product_prod(C,A)))),S)),X5) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_gw(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X5,S) ) ) ).
% insert_relcomp_union_fold
tff(fact_2738_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Aa2: set(A),Z2: B,Y: B,A3: A] :
( finite4664212375090638736ute_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
=> ( finite_fold_graph(A,B,F,Z2,Aa2,Y)
=> ( member(A,A3,Aa2)
=> ? [Y6: B] :
( ( Y = aa(B,B,aa(A,fun(B,B),F,A3),Y6) )
& finite_fold_graph(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))),Y6) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_2739_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R2)) ).
% Field_insert
tff(fact_2740_max__ext_Omax__extI,axiom,
! [A: $tType,X5: set(A),Y5: set(A),R: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( aa(set(A),$o,finite_finite(A),Y5)
=> ( ( Y5 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,Y5)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R) ) )
=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Y5),max_ext(A,R)) ) ) ) ) ).
% max_ext.max_extI
tff(fact_2741_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(C,B))] : relcomp(A,C,B,bot_bot(set(product_prod(A,C))),R) = bot_bot(set(product_prod(A,B))) ).
% relcomp_empty1
tff(fact_2742_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,C))] : relcomp(A,C,B,R,bot_bot(set(product_prod(C,B)))) = bot_bot(set(product_prod(A,B))) ).
% relcomp_empty2
tff(fact_2743_Field__empty,axiom,
! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).
% Field_empty
tff(fact_2744_max__ext__compat,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
=> aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R)) ) ).
% max_ext_compat
tff(fact_2745_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C2),relcomp(A,C,B,R2,S2))
=> ~ ! [B3: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B3),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),C2),S2) ) ) ).
% relcompEpair
tff(fact_2746_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,S2))
=> ~ ! [X3: A,Y2: C,Z3: B] :
( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3) )
=> ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y2),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y2),Z3),S2) ) ) ) ).
% relcompE
tff(fact_2747_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),C2: C,S2: set(product_prod(B,C))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2),S2)
=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2),relcomp(A,B,C,R2,S2)) ) ) ).
% relcomp.relcompI
tff(fact_2748_relcomp_Osimps,axiom,
! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S2))
<=> ? [A5: A,B4: C,C4: B] :
( ( A1 = A5 )
& ( A22 = C4 )
& member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),B4),R2)
& member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C4),S2) ) ) ).
% relcomp.simps
tff(fact_2749_relcomp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S2))
=> ~ ! [B3: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B3),R2)
=> ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),A22),S2) ) ) ).
% relcomp.cases
tff(fact_2750_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : comp(C,set(B),A,aTP_Lamp_gx(C,set(B)),F) = comp(set(D),set(B),A,image2(D,B,G),aTP_Lamp_gy(A,set(D))) ).
% empty_natural
tff(fact_2751_fst__diag__fst,axiom,
! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_gz(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).
% fst_diag_fst
tff(fact_2752_snd__diag__snd,axiom,
! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ha(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).
% snd_diag_snd
tff(fact_2753_FieldI2,axiom,
! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R)
=> member(A,J,field2(A,R)) ) ).
% FieldI2
tff(fact_2754_FieldI1,axiom,
! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R)
=> member(A,I,field2(A,R)) ) ).
% FieldI1
tff(fact_2755_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,X: B] :
( finite_fold_graph(A,B,F,Z2,bot_bot(set(A)),X)
=> ( X = Z2 ) ) ).
% empty_fold_graphE
tff(fact_2756_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B] : finite_fold_graph(A,B,F,Z2,bot_bot(set(A)),Z2) ).
% fold_graph.emptyI
tff(fact_2757_snd__diag__fst,axiom,
! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_gz(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).
% snd_diag_fst
tff(fact_2758_fst__diag__snd,axiom,
! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ha(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).
% fst_diag_snd
tff(fact_2759_union__comp__emptyL,axiom,
! [A: $tType,Aa2: set(product_prod(A,A)),Ca: set(product_prod(A,A)),Ba: set(product_prod(A,A))] :
( ( relcomp(A,A,A,Aa2,Ca) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,Ba,Ca) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),Aa2),Ba),Ca) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyL
tff(fact_2760_union__comp__emptyR,axiom,
! [A: $tType,Aa2: set(product_prod(A,A)),Ba: set(product_prod(A,A)),Ca: set(product_prod(A,A))] :
( ( relcomp(A,A,A,Aa2,Ba) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,Aa2,Ca) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,A,A,Aa2,aa(set(product_prod(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))),Ba),Ca)) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyR
tff(fact_2761_fold__graph_Osimps,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F,Z2,A1,A22)
<=> ( ( ( A1 = bot_bot(set(A)) )
& ( A22 = Z2 ) )
| ? [X4: A,A8: set(A),Y3: B] :
( ( A1 = aa(set(A),set(A),insert2(A,X4),A8) )
& ( A22 = aa(B,B,aa(A,fun(B,B),F,X4),Y3) )
& ~ member(A,X4,A8)
& finite_fold_graph(A,B,F,Z2,A8,Y3) ) ) ) ).
% fold_graph.simps
tff(fact_2762_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,B)),Z2: B,A1: set(A),A22: B] :
( finite_fold_graph(A,B,F,Z2,A1,A22)
=> ( ( ( A1 = bot_bot(set(A)) )
=> ( A22 != Z2 ) )
=> ~ ! [X3: A,A7: set(A)] :
( ( A1 = aa(set(A),set(A),insert2(A,X3),A7) )
=> ! [Y2: B] :
( ( A22 = aa(B,B,aa(A,fun(B,B),F,X3),Y2) )
=> ( ~ member(A,X3,A7)
=> ~ finite_fold_graph(A,B,F,Z2,A7,Y2) ) ) ) ) ) ).
% fold_graph.cases
tff(fact_2763_max__ext__additive,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A),R: set(product_prod(A,A)),Ca: set(A),D4: set(A)] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),Aa2),Ba),max_ext(A,R))
=> ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),Ca),D4),max_ext(A,R))
=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ca)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Ba),D4)),max_ext(A,R)) ) ) ).
% max_ext_additive
tff(fact_2764_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),R)
=> ( aa(set(product_prod(B,C)),$o,finite_finite(product_prod(B,C)),S)
=> ( relcomp(A,B,C,R,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_hc(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R) ) ) ) ).
% relcomp_fold
tff(fact_2765_prod_OUnion__comp,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Ba: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,Ba)
=> aa(set(A),$o,finite_finite(A),X3) )
=> ( ! [A12: set(A)] :
( member(set(A),A12,Ba)
=> ! [A23: set(A)] :
( member(set(A),A23,Ba)
=> ( ( A12 != A23 )
=> ! [X3: A] :
( member(A,X3,A12)
=> ( member(A,X3,A23)
=> ( aa(A,B,G,X3) = one_one(B) ) ) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ba)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B),groups7121269368397514597t_prod(A,B)),G),Ba) ) ) ) ) ).
% prod.Union_comp
tff(fact_2766_fst__snd__flip,axiom,
! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_hd(A,fun(B,product_prod(B,A))))),Xy) ).
% fst_snd_flip
tff(fact_2767_snd__fst__flip,axiom,
! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))))),Xy) ).
% snd_fst_flip
tff(fact_2768_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),R: set(product_prod(C,A))] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S)
=> ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert2(product_prod(C,A),X),R),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_gw(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R,S),S) ) ) ).
% insert_relcomp_fold
tff(fact_2769_prod_OUnion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Ca: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> aa(set(A),$o,finite_finite(A),X3) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,Ca)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B),groups7121269368397514597t_prod(A,B)),G),Ca) ) ) ) ) ).
% prod.Union_disjoint
tff(fact_2770_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_2771_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_2772_sum_OUnion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Ca: set(set(A)),G: fun(A,B)] :
( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> aa(set(A),$o,finite_finite(A),X3) )
=> ( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> ! [Xa4: set(A)] :
( member(set(A),Xa4,Ca)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),Ca)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B),groups7311177749621191930dd_sum(A,B)),G),Ca) ) ) ) ) ).
% sum.Union_disjoint
tff(fact_2773_max__ext_Ocases,axiom,
! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
=> ~ ( aa(set(A),$o,finite_finite(A),A1)
=> ( aa(set(A),$o,finite_finite(A),A22)
=> ( ( A22 != bot_bot(set(A)) )
=> ~ ! [X2: A] :
( member(A,X2,A1)
=> ? [Xa4: A] :
( member(A,Xa4,A22)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa4),R) ) ) ) ) ) ) ).
% max_ext.cases
tff(fact_2774_max__ext_Osimps,axiom,
! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
<=> ( aa(set(A),$o,finite_finite(A),A1)
& aa(set(A),$o,finite_finite(A),A22)
& ( A22 != bot_bot(set(A)) )
& ! [X4: A] :
( member(A,X4,A1)
=> ? [Xa2: A] :
( member(A,Xa2,A22)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R) ) ) ) ) ).
% max_ext.simps
tff(fact_2775_min__ext__compat,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
=> aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R)) ) ).
% min_ext_compat
tff(fact_2776_max__extp__eq,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: set(A),Y: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R2),X),Y)
<=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y),max_ext(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% max_extp_eq
tff(fact_2777_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: set(A),Xa3: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X2),Xa3)
<=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X2),Xa3),max_ext(A,R)) ) ).
% max_extp_max_ext_eq
tff(fact_2778_Image__fold,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),R)
=> ( image(A,B,R,S) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_hg(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R) ) ) ).
% Image_fold
tff(fact_2779_pairself__image__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),P))) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_hh(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F),P)) ).
% pairself_image_eq
tff(fact_2780_chains__extend,axiom,
! [A: $tType,C2: set(set(A)),S: set(set(A)),Z2: set(A)] :
( member(set(set(A)),C2,chains2(A,S))
=> ( member(set(A),Z2,S)
=> ( ! [X3: set(A)] :
( member(set(A),X3,C2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Z2) )
=> member(set(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),insert2(set(A),Z2),bot_bot(set(set(A))))),C2),chains2(A,S)) ) ) ) ).
% chains_extend
tff(fact_2781_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),Aa2: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> ( member(A,A3,Aa2)
=> member(B,B2,image(A,B,R2,Aa2)) ) ) ).
% ImageI
tff(fact_2782_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(B,A))] : image(B,A,R,bot_bot(set(B))) = bot_bot(set(A)) ).
% Image_empty2
tff(fact_2783_ex__assn__proper,axiom,
! [A: $tType,P: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o))] :
( ! [X3: A] : aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),P,X3))
=> aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),$o,proper,aTP_Lamp_hi(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ) ).
% ex_assn_proper
tff(fact_2784_Image__empty1,axiom,
! [B: $tType,A: $tType,X5: set(B)] : image(B,A,bot_bot(set(product_prod(B,A))),X5) = bot_bot(set(A)) ).
% Image_empty1
tff(fact_2785_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A3: B] :
( member(A,B2,image(B,A,R2,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2),R2) ) ).
% Image_singleton_iff
tff(fact_2786_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),Aa2: set(B)] :
( member(A,B2,image(B,A,R2,Aa2))
=> ~ ! [X3: B] :
( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2),R2)
=> ~ member(B,X3,Aa2) ) ) ).
% ImageE
tff(fact_2787_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),Aa2: set(B)] :
( member(A,B2,image(B,A,R2,Aa2))
<=> ? [X4: B] :
( member(B,X4,Aa2)
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2),R2) ) ) ).
% Image_iff
tff(fact_2788_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,Aa2: set(A),B2: B,R2: set(product_prod(A,B))] :
( member(A,A3,Aa2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> member(B,B2,image(A,B,R2,Aa2)) ) ) ).
% rev_ImageI
tff(fact_2789_Image__singleton,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A3: B] : image(B,A,R2,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A3)) ).
% Image_singleton
tff(fact_2790_relcomp__unfold,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R2,S2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_hk(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S2))) ).
% relcomp_unfold
tff(fact_2791_ex__assn__def,axiom,
! [A: $tType,P: fun(A,assn)] : ex_assn(A,P) = aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),assn,abs_assn,aTP_Lamp_hl(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),P)) ).
% ex_assn_def
tff(fact_2792_Image__eq__UN,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Ba: set(B)] : image(B,A,R2,Ba) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_hm(set(product_prod(B,A)),fun(B,set(A)),R2)),Ba)) ).
% Image_eq_UN
tff(fact_2793_inf__Sup2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)),aa(set(A),A,lattic5882676163264333800up_fin(A),Ba)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_hn(set(A),fun(set(A),fun(A,$o)),Aa2),Ba))) ) ) ) ) ) ) ).
% inf_Sup2_distrib
tff(fact_2794_inf__Sup1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ho(set(A),fun(A,fun(A,$o)),Aa2),X))) ) ) ) ) ).
% inf_Sup1_distrib
tff(fact_2795_sup__Inf2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,Aa2)),lattic7752659483105999362nf_fin(A,Ba)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_hp(set(A),fun(set(A),fun(A,$o)),Aa2),Ba))) ) ) ) ) ) ) ).
% sup_Inf2_distrib
tff(fact_2796_sup__Inf1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),lattic7752659483105999362nf_fin(A,Aa2)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_hq(set(A),fun(A,fun(A,$o)),Aa2),X))) ) ) ) ) ).
% sup_Inf1_distrib
tff(fact_2797_max__ext__def,axiom,
! [A: $tType,X2: set(product_prod(A,A))] : max_ext(A,X2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),X2)))) ).
% max_ext_def
tff(fact_2798_brk__rel__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : brk_rel(A,B,R) = aa(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B))),aa(set(product_prod(product_prod($o,A),product_prod($o,B))),fun(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B)))),sup_sup(set(product_prod(product_prod($o,A),product_prod($o,B)))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_hr(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),R))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_hs(product_prod(product_prod($o,A),product_prod($o,B)),$o))) ).
% brk_rel_def
tff(fact_2799_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType,Aa2: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,Aa2,F,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_ht(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Aa2),F),G)) ).
% image2_def
tff(fact_2800_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( order_preorder_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R2) ) ) ) ) ).
% subset_Image1_Image1_iff
tff(fact_2801_Rep__unit__induct,axiom,
! [Y: $o,P: fun($o,$o)] :
( member($o,(Y),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> ( ! [X3: product_unit] : aa($o,$o,P,aa(product_unit,$o,product_Rep_unit,X3))
=> aa($o,$o,P,(Y)) ) ) ).
% Rep_unit_induct
tff(fact_2802_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( member($o,(X),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> ( member($o,(Y),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> ( ( aa($o,product_unit,product_Abs_unit,(X)) = aa($o,product_unit,product_Abs_unit,(Y)) )
<=> ( (X)
<=> (Y) ) ) ) ) ).
% Abs_unit_inject
tff(fact_2803_Abs__unit__induct,axiom,
! [P: fun(product_unit,$o),X: product_unit] :
( ! [Y2: $o] :
( member($o,(Y2),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> aa(product_unit,$o,P,aa($o,product_unit,product_Abs_unit,(Y2))) )
=> aa(product_unit,$o,P,X) ) ).
% Abs_unit_induct
tff(fact_2804_preorder__on__empty,axiom,
! [A: $tType] : order_preorder_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% preorder_on_empty
tff(fact_2805_Abs__unit__inverse,axiom,
! [Y: $o] :
( member($o,(Y),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> ( aa(product_unit,$o,product_Rep_unit,aa($o,product_unit,product_Abs_unit,(Y)))
<=> (Y) ) ) ).
% Abs_unit_inverse
tff(fact_2806_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: fun(B,A),X: B,C2: C,G: fun(B,C),Aa2: set(B)] :
( ( B2 = aa(B,A,F,X) )
=> ( ( C2 = aa(B,C,G,X) )
=> ( member(B,X,Aa2)
=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2),bNF_Greatest_image2(B,A,C,Aa2,F,G)) ) ) ) ).
% image2_eqI
tff(fact_2807_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),Ba: set(A)] :
( order_preorder_on(A,field2(A,R2),R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),field2(A,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,Aa2)),image(A,A,R2,Ba))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> ? [Xa2: A] :
( member(A,Xa2,Ba)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ) ) ).
% subset_Image_Image_iff
tff(fact_2808_Rep__unit,axiom,
! [X: product_unit] : member($o,aa(product_unit,$o,product_Rep_unit,X),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o)))) ).
% Rep_unit
tff(fact_2809_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y2: $o] :
( ( X = aa($o,product_unit,product_Abs_unit,(Y2)) )
=> ~ member($o,(Y2),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o)))) ) ).
% Abs_unit_cases
tff(fact_2810_Rep__unit__cases,axiom,
! [Y: $o] :
( member($o,(Y),aa(set($o),set($o),insert2($o,$true),bot_bot(set($o))))
=> ~ ! [X3: product_unit] :
( (Y)
<=> ~ aa(product_unit,$o,product_Rep_unit,X3) ) ) ).
% Rep_unit_cases
tff(fact_2811_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: B,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5)),lex_prod(A,B,R2,S2))
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A6),R2)
| ( ( A3 = A6 )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B5),S2) ) ) ) ).
% in_lex_prod
tff(fact_2812_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% linear_order_on_singleton
tff(fact_2813_Total__subset__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( total_on(A,field2(A,R2),R2)
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
=> ( ( R2 = bot_bot(set(product_prod(A,A))) )
| ? [A4: A] : R2 = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ).
% Total_subset_Id
tff(fact_2814_type__definition__unit,axiom,
type_definition(product_unit,$o,product_Rep_unit,product_Abs_unit,aa(set($o),set($o),insert2($o,$true),bot_bot(set($o)))) ).
% type_definition_unit
tff(fact_2815_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( order_7125193373082350890der_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( ( image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) )
<=> ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
tff(fact_2816_refl__on__singleton,axiom,
! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).
% refl_on_singleton
tff(fact_2817_IdI,axiom,
! [A: $tType,A3: A] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id2(A)) ).
% IdI
tff(fact_2818_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id2(A))
<=> ( A3 = B2 ) ) ).
% pair_in_Id_conv
tff(fact_2819_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ) ) ) ).
% Linear_order_in_diff_Id
tff(fact_2820_IdE,axiom,
! [A: $tType,P5: product_prod(A,A)] :
( member(product_prod(A,A),P5,id2(A))
=> ~ ! [X3: A] : P5 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).
% IdE
tff(fact_2821_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id2(A))
=> ( A3 = B2 ) ) ).
% BNF_Greatest_Fixpoint.IdD
tff(fact_2822_refl__onD,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A] :
( refl_on(A,Aa2,R2)
=> ( member(A,A3,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R2) ) ) ).
% refl_onD
tff(fact_2823_refl__onD1,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( refl_on(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,X,Aa2) ) ) ).
% refl_onD1
tff(fact_2824_refl__onD2,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( refl_on(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,Y,Aa2) ) ) ).
% refl_onD2
tff(fact_2825_refl__on__domain,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
( refl_on(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(A,A3,Aa2)
& member(A,B2,Aa2) ) ) ) ).
% refl_on_domain
tff(fact_2826_refl__on__empty,axiom,
! [A: $tType] : refl_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% refl_on_empty
tff(fact_2827_Id__def,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_hu(product_prod(A,A),$o)) ).
% Id_def
tff(fact_2828_partial__order__on__empty,axiom,
! [A: $tType] : order_7125193373082350890der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% partial_order_on_empty
tff(fact_2829_lnear__order__on__empty,axiom,
! [A: $tType] : order_679001287576687338der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% lnear_order_on_empty
tff(fact_2830_lex__prod__def,axiom,
! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_hw(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).
% lex_prod_def
tff(fact_2831_reflcl__set__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X2: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X2),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),id2(A))) ) ).
% reflcl_set_eq
tff(fact_2832_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( refl_on(A,field2(A,R2),R2)
=> ( antisym(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( ( image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) )
<=> ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
tff(fact_2833_Zorns__po__lemma,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_7125193373082350890der_on(A,field2(A,R2),R2)
=> ( ! [C6: set(A)] :
( member(set(A),C6,chains(A,R2))
=> ? [X2: A] :
( member(A,X2,field2(A,R2))
& ! [Xa4: A] :
( member(A,Xa4,C6)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X2),R2) ) ) )
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& ! [Xa3: A] :
( member(A,Xa3,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R2)
=> ( Xa3 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
tff(fact_2834_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R2),R2)
=> ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
<=> ! [A8: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),field2(A,R2))
=> ( ( A8 != bot_bot(set(A)) )
=> ? [X4: A] :
( member(A,X4,A8)
& ! [Xa2: A] :
( member(A,Xa2,A8)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
tff(fact_2835_eventually__INF__base,axiom,
! [B: $tType,A: $tType,Ba: set(A),F4: fun(A,filter(B)),P: fun(B,$o)] :
( ( Ba != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Ba)
=> ! [B3: A] :
( member(A,B3,Ba)
=> ? [X2: A] :
( member(A,X2,Ba)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A4)),aa(A,filter(B),F4,B3))) ) ) )
=> ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),Ba)))
<=> ? [X4: A] :
( member(A,X4,Ba)
& eventually(B,P,aa(A,filter(B),F4,X4)) ) ) ) ) ).
% eventually_INF_base
tff(fact_2836_pair__lessI2,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),T2)
=> member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_less) ) ) ).
% pair_lessI2
tff(fact_2837_wf__empty,axiom,
! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).
% wf_empty
tff(fact_2838_eventually__const,axiom,
! [A: $tType,F4: filter(A),P: $o] :
( ( F4 != bot_bot(filter(A)) )
=> ( eventually(A,aTP_Lamp_af($o,fun(A,$o),(P)),F4)
<=> (P) ) ) ).
% eventually_const
tff(fact_2839_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z2)),fun_pair_less)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z2) ) ).
% pair_less_iff1
tff(fact_2840_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,$o)))] :
( wf(A,R)
=> ( ! [F3: fun(A,B),G3: fun(A,B),X3: A,R3: B] :
( ! [Z5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),X3),R)
=> ( aa(A,B,F3,Z5) = aa(A,B,G3,Z5) ) )
=> ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X3),R3)
<=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,G3),X3),R3) ) )
=> ( ! [X3: A,F3: fun(A,B)] :
( ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R)
=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),Y4),aa(A,B,F3,Y4)) )
=> ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X3),X_13) )
=> ? [F3: fun(A,B)] :
! [X2: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P,F3),X2),aa(A,B,F3,X2)) ) ) ) ).
% dependent_wf_choice
tff(fact_2841_wf__induct__rule,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
( wf(A,R2)
=> ( ! [X3: A] :
( ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2)
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A3) ) ) ).
% wf_induct_rule
tff(fact_2842_wf__eq__minimal,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [Q8: set(A)] :
( ? [X4: A] : member(A,X4,Q8)
=> ? [X4: A] :
( member(A,X4,Q8)
& ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R2)
=> ~ member(A,Y3,Q8) ) ) ) ) ).
% wf_eq_minimal
tff(fact_2843_wf__not__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( wf(A,R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R2) ) ).
% wf_not_refl
tff(fact_2844_wf__not__sym,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,X: A] :
( wf(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3),R2) ) ) ).
% wf_not_sym
tff(fact_2845_wf__irrefl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( wf(A,R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R2) ) ).
% wf_irrefl
tff(fact_2846_wf__induct,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
( wf(A,R2)
=> ( ! [X3: A] :
( ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2)
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A3) ) ) ).
% wf_induct
tff(fact_2847_wf__asym,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,X: A] :
( wf(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3),R2) ) ) ).
% wf_asym
tff(fact_2848_wfUNIVI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [P3: fun(A,$o),X3: A] :
( ! [Xa3: A] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R2)
=> aa(A,$o,P3,Y2) )
=> aa(A,$o,P3,Xa3) )
=> aa(A,$o,P3,X3) )
=> wf(A,R2) ) ).
% wfUNIVI
tff(fact_2849_wfI__min,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X3: A,Q4: set(A)] :
( member(A,X3,Q4)
=> ? [Xa3: A] :
( member(A,Xa3,Q4)
& ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R)
=> ~ member(A,Y2,Q4) ) ) )
=> wf(A,R) ) ).
% wfI_min
tff(fact_2850_wfE__min,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,Q: set(A)] :
( wf(A,R)
=> ( member(A,X,Q)
=> ~ ! [Z3: A] :
( member(A,Z3,Q)
=> ~ ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
=> ~ member(A,Y4,Q) ) ) ) ) ).
% wfE_min
tff(fact_2851_wf__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [P4: fun(A,$o)] :
( ! [X4: A] :
( ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R2)
=> aa(A,$o,P4,Y3) )
=> aa(A,$o,P4,X4) )
=> ! [X_12: A] : aa(A,$o,P4,X_12) ) ) ).
% wf_def
tff(fact_2852_eventually__happens_H,axiom,
! [A: $tType,F4: filter(A),P: fun(A,$o)] :
( ( F4 != bot_bot(filter(A)) )
=> ( eventually(A,P,F4)
=> ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).
% eventually_happens'
tff(fact_2853_eventually__happens,axiom,
! [A: $tType,P: fun(A,$o),Net: filter(A)] :
( eventually(A,P,Net)
=> ( ( Net = bot_bot(filter(A)) )
| ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).
% eventually_happens
tff(fact_2854_eventually__bot,axiom,
! [A: $tType,P: fun(A,$o)] : eventually(A,P,bot_bot(filter(A))) ).
% eventually_bot
tff(fact_2855_antisym__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( antisym(A,R2)
<=> ! [X4: A,Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R2)
=> ( X4 = Y3 ) ) ) ) ).
% antisym_def
tff(fact_2856_antisymI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),R2)
=> ( X3 = Y2 ) ) )
=> antisym(A,R2) ) ).
% antisymI
tff(fact_2857_antisymD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( antisym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R2)
=> ( A3 = B2 ) ) ) ) ).
% antisymD
tff(fact_2858_trivial__limit__def,axiom,
! [A: $tType,F4: filter(A)] :
( ( F4 = bot_bot(filter(A)) )
<=> eventually(A,aTP_Lamp_al(A,$o),F4) ) ).
% trivial_limit_def
tff(fact_2859_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter(A)] :
( eventually(A,aTP_Lamp_af($o,fun(A,$o),(P)),F4)
<=> ( (P)
| ( F4 = bot_bot(filter(A)) ) ) ) ).
% eventually_const_iff
tff(fact_2860_False__imp__not__eventually,axiom,
! [A: $tType,P: fun(A,$o),Net: filter(A)] :
( ! [X3: A] : ~ aa(A,$o,P,X3)
=> ( ( Net != bot_bot(filter(A)) )
=> ~ eventually(A,P,Net) ) ) ).
% False_imp_not_eventually
tff(fact_2861_antisym__empty,axiom,
! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).
% antisym_empty
tff(fact_2862_wfE__min_H,axiom,
! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
( wf(A,R)
=> ( ( Q != bot_bot(set(A)) )
=> ~ ! [Z3: A] :
( member(A,Z3,Q)
=> ~ ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
=> ~ member(A,Y4,Q) ) ) ) ) ).
% wfE_min'
tff(fact_2863_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set(product_prod(A,A)),F: fun(nat,A)] :
( wf(A,R2)
=> ~ ! [K2: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F,aa(nat,nat,suc,K2))),aa(nat,A,F,K2)),R2) ) ).
% wf_no_infinite_down_chainE
tff(fact_2864_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ~ ? [F6: fun(nat,A)] :
! [I2: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,aa(nat,nat,suc,I2))),aa(nat,A,F6,I2)),R2) ) ).
% wf_iff_no_infinite_down_chain
tff(fact_2865_wf__no__loop,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( relcomp(A,A,A,R,R) = bot_bot(set(product_prod(A,A))) )
=> wf(A,R) ) ).
% wf_no_loop
tff(fact_2866_wf__bounded__measure,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B3)),aa(A,nat,Ub,A4))
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,B3)),aa(A,nat,Ub,A4))
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,A4)),aa(A,nat,F,B3)) ) )
=> wf(A,R2) ) ).
% wf_bounded_measure
tff(fact_2867_wfI__pf,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [A7: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),image(A,A,R,A7))
=> ( A7 = bot_bot(set(A)) ) )
=> wf(A,R) ) ).
% wfI_pf
tff(fact_2868_wfE__pf,axiom,
! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] :
( wf(A,R)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),image(A,A,R,Aa2))
=> ( Aa2 = bot_bot(set(A)) ) ) ) ).
% wfE_pf
tff(fact_2869_antisym__singleton,axiom,
! [A: $tType,X: product_prod(A,A)] : antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),X),bot_bot(set(product_prod(A,A))))) ).
% antisym_singleton
tff(fact_2870_eventually__Inf__base,axiom,
! [A: $tType,Ba: set(filter(A)),P: fun(A,$o)] :
( ( Ba != bot_bot(set(filter(A))) )
=> ( ! [F5: filter(A)] :
( member(filter(A),F5,Ba)
=> ! [G4: filter(A)] :
( member(filter(A),G4,Ba)
=> ? [X2: filter(A)] :
( member(filter(A),X2,Ba)
& aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X2),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G4)) ) ) )
=> ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),Ba))
<=> ? [X4: filter(A)] :
( member(filter(A),X4,Ba)
& eventually(A,P,X4) ) ) ) ) ).
% eventually_Inf_base
tff(fact_2871_wf__eq__minimal2,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wf(A,R2)
<=> ! [A8: set(A)] :
( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),field2(A,R2))
& ( A8 != bot_bot(set(A)) ) )
=> ? [X4: A] :
( member(A,X4,A8)
& ! [Xa2: A] :
( member(A,Xa2,A8)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ).
% wf_eq_minimal2
tff(fact_2872_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2)
=> ( aa(set(B),$o,finite_finite(B),aa(A,set(B),Ub,A4))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B3)),aa(A,set(B),Ub,A4))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F,B3)),aa(A,set(B),Ub,A4))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F,A4)),aa(A,set(B),F,B3)) ) )
=> wf(A,R2) ) ).
% wf_bounded_set
tff(fact_2873_pair__lessI1,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
=> member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_less) ) ).
% pair_lessI1
tff(fact_2874_reduction__pairI,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( wf(A,R)
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
=> fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),S)) ) ) ).
% reduction_pairI
tff(fact_2875_same__fst__def,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_hy(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R)))) ).
% same_fst_def
tff(fact_2876_smin__insertI,axiom,
! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( member(product_prod(nat,nat),X,XS)
=> ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_less)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_strict)
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),Y),YS)),fun_min_strict) ) ) ) ).
% smin_insertI
tff(fact_2877_Chains__subset_H,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R2)
=> aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).
% Chains_subset'
tff(fact_2878_bsqr__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_ia(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).
% bsqr_def
tff(fact_2879_pair__leqI2,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B2)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S2),T2)
=> member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_leq) ) ) ).
% pair_leqI2
tff(fact_2880_same__fstI,axiom,
! [B: $tType,A: $tType,P: fun(A,$o),X: A,Y7: B,Y: B,R: fun(A,set(product_prod(B,B)))] :
( aa(A,$o,P,X)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y),aa(A,set(product_prod(B,B)),R,X))
=> member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),same_fst(A,B,P,R)) ) ) ).
% same_fstI
tff(fact_2881_smin__emptyI,axiom,
! [X5: set(product_prod(nat,nat))] :
( ( X5 != bot_bot(set(product_prod(nat,nat))) )
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),bot_bot(set(product_prod(nat,nat)))),fun_min_strict) ) ).
% smin_emptyI
tff(fact_2882_pred__on_Ochain__empty,axiom,
! [A: $tType,Aa2: set(A),P: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,Aa2,P),bot_bot(set(A))) ).
% pred_on.chain_empty
tff(fact_2883_subset_Ochain__empty,axiom,
! [A: $tType,Aa2: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),bot_bot(set(set(A)))) ).
% subset.chain_empty
tff(fact_2884_subset__Zorn__nonempty,axiom,
! [A: $tType,A10: set(set(A))] :
( ( A10 != bot_bot(set(set(A))) )
=> ( ! [C7: set(set(A))] :
( ( C7 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A10,ord_less(set(A))),C7)
=> member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7),A10) ) )
=> ? [X3: set(A)] :
( member(set(A),X3,A10)
& ! [Xa3: set(A)] :
( member(set(A),Xa3,A10)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa3)
=> ( Xa3 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
tff(fact_2885_Union__in__chain,axiom,
! [A: $tType,B10: set(set(A)),A10: set(set(A))] :
( aa(set(set(A)),$o,finite_finite(set(A)),B10)
=> ( ( B10 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A10,ord_less(set(A))),B10)
=> member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B10),B10) ) ) ) ).
% Union_in_chain
tff(fact_2886_Inter__in__chain,axiom,
! [A: $tType,B10: set(set(A)),A10: set(set(A))] :
( aa(set(set(A)),$o,finite_finite(set(A)),B10)
=> ( ( B10 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A10,ord_less(set(A))),B10)
=> member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B10),B10) ) ) ) ).
% Inter_in_chain
tff(fact_2887_subset_Ochain__extend,axiom,
! [A: $tType,Aa2: set(set(A)),Ca: set(set(A)),Z2: set(A)] :
( aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),Ca)
=> ( member(set(A),Z2,Aa2)
=> ( ! [X3: set(A)] :
( member(set(A),X3,Ca)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Z2) )
=> aa(set(set(A)),$o,pred_chain(set(A),Aa2,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),insert2(set(A),Z2),bot_bot(set(set(A))))),Ca)) ) ) ) ).
% subset.chain_extend
tff(fact_2888_pred__on_Ochain__extend,axiom,
! [A: $tType,Aa2: set(A),P: fun(A,fun(A,$o)),Ca: set(A),Z2: A] :
( aa(set(A),$o,pred_chain(A,Aa2,P),Ca)
=> ( member(A,Z2,Aa2)
=> ( ! [X3: A] :
( member(A,X3,Ca)
=> aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X3),Z2) )
=> aa(set(A),$o,pred_chain(A,Aa2,P),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,Z2),bot_bot(set(A)))),Ca)) ) ) ) ).
% pred_on.chain_extend
tff(fact_2889_finite__subset__Union__chain,axiom,
! [A: $tType,Aa2: set(A),B10: set(set(A)),A10: set(set(A))] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B10))
=> ( ( B10 != bot_bot(set(set(A))) )
=> ( aa(set(set(A)),$o,pred_chain(set(A),A10,ord_less(set(A))),B10)
=> ~ ! [B8: set(A)] :
( member(set(A),B8,B10)
=> ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),B8) ) ) ) ) ) ).
% finite_subset_Union_chain
tff(fact_2890_Chains__alt__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R2)
=> ( chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))) ) ) ).
% Chains_alt_def
tff(fact_2891_Chains__subset,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).
% Chains_subset
tff(fact_2892_pair__leqI1,axiom,
! [A3: nat,B2: nat,S2: nat,T2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B2)
=> member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_leq) ) ).
% pair_leqI1
tff(fact_2893_wmax__insertI,axiom,
! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
( member(product_prod(nat,nat),Y,YS)
=> ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_leq)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_max_weak)
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),X),XS)),YS),fun_max_weak) ) ) ) ).
% wmax_insertI
tff(fact_2894_wmin__insertI,axiom,
! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( member(product_prod(nat,nat),X,XS)
=> ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_leq)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_weak)
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),Y),YS)),fun_min_weak) ) ) ) ).
% wmin_insertI
tff(fact_2895_smax__insertI,axiom,
! [Y: product_prod(nat,nat),Y5: set(product_prod(nat,nat)),X: product_prod(nat,nat),X5: set(product_prod(nat,nat))] :
( member(product_prod(nat,nat),Y,Y5)
=> ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_less)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),Y5),fun_max_strict)
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),X),X5)),Y5),fun_max_strict) ) ) ) ).
% smax_insertI
tff(fact_2896_max__weak__def,axiom,
fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% max_weak_def
tff(fact_2897_min__weak__def,axiom,
fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% min_weak_def
tff(fact_2898_max__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).
% max_rpair_set
tff(fact_2899_smax__emptyI,axiom,
! [Y5: set(product_prod(nat,nat))] :
( aa(set(product_prod(nat,nat)),$o,finite_finite(product_prod(nat,nat)),Y5)
=> ( ( Y5 != bot_bot(set(product_prod(nat,nat))) )
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y5),fun_max_strict) ) ) ).
% smax_emptyI
tff(fact_2900_wmin__emptyI,axiom,
! [X5: set(product_prod(nat,nat))] : member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X5),bot_bot(set(product_prod(nat,nat)))),fun_min_weak) ).
% wmin_emptyI
tff(fact_2901_wmax__emptyI,axiom,
! [X5: set(product_prod(nat,nat))] :
( aa(set(product_prod(nat,nat)),$o,finite_finite(product_prod(nat,nat)),X5)
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X5),fun_max_weak) ) ).
% wmax_emptyI
tff(fact_2902_min__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).
% min_rpair_set
tff(fact_2903_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R2))
=> ( ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A))) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
=> ( ( ( A3 = B2 )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).
% wo_rel.cases_Total3
tff(fact_2904_product__fold,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(B),$o,finite_finite(B),Ba)
=> ( product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_id(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Ba),bot_bot(set(product_prod(A,B))),Aa2) ) ) ) ).
% product_fold
tff(fact_2905_Sigma__def,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: fun(A,set(B))] : product_Sigma(A,B,Aa2,Ba) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_if(fun(A,set(B)),fun(A,set(product_prod(A,B))),Ba)),Aa2)) ).
% Sigma_def
tff(fact_2906_cSUP__UNION,axiom,
! [B: $tType,C: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Aa2: set(A),Ba: fun(A,set(B)),F: fun(B,C)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> ( aa(A,set(B),Ba,X3) != bot_bot(set(B)) ) )
=> ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ig(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Ba),F)),Aa2)))
=> ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Ba),Aa2)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ih(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Ba),F)),Aa2)) ) ) ) ) ) ).
% cSUP_UNION
tff(fact_2907_cINF__UNION,axiom,
! [B: $tType,C: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Aa2: set(A),Ba: fun(A,set(B)),F: fun(B,C)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> ( aa(A,set(B),Ba,X3) != bot_bot(set(B)) ) )
=> ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ig(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Ba),F)),Aa2)))
=> ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Ba),Aa2)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ii(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Ba),F)),Aa2)) ) ) ) ) ) ).
% cINF_UNION
tff(fact_2908_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,Aa2: set(A),B2: B,Ba: fun(A,set(B))] :
( member(A,A3,Aa2)
=> ( member(B,B2,aa(A,set(B),Ba,A3))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,Aa2,Ba)) ) ) ).
% SigmaI
tff(fact_2909_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,Aa2: set(A),Ba: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,Aa2,Ba))
<=> ( member(A,A3,Aa2)
& member(B,B2,aa(A,set(B),Ba,A3)) ) ) ).
% mem_Sigma_iff
tff(fact_2910_bdd__below__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit1013018076250108175_below(A,bot_bot(set(A))) ) ).
% bdd_below_empty
tff(fact_2911_bdd__above__empty,axiom,
! [A: $tType] :
( preorder(A)
=> condit941137186595557371_above(A,bot_bot(set(A))) ) ).
% bdd_above_empty
tff(fact_2912_Sigma__empty1,axiom,
! [B: $tType,A: $tType,Ba: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),Ba) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty1
tff(fact_2913_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [F: fun(B,A),G: fun(B,A),Aa2: set(B)] :
( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ij(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Aa2))
<=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),Aa2))
& condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),Aa2)) ) ) ) ).
% bdd_above_image_sup
tff(fact_2914_Times__empty,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)) = bot_bot(set(product_prod(A,B))) )
<=> ( ( Aa2 = bot_bot(set(A)) )
| ( Ba = bot_bot(set(B)) ) ) ) ).
% Times_empty
tff(fact_2915_Sigma__empty2,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] : product_Sigma(A,B,Aa2,aTP_Lamp_ik(A,set(B))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty2
tff(fact_2916_fst__image__times,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba))) = $ite(Ba = bot_bot(set(B)),bot_bot(set(A)),Aa2) ).
% fst_image_times
tff(fact_2917_snd__image__times,axiom,
! [B: $tType,A: $tType,Aa2: set(B),Ba: set(A)] :
aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,Aa2,aTP_Lamp_fh(set(A),fun(B,set(A)),Ba))) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),Ba) ).
% snd_image_times
tff(fact_2918_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,Aa2: set(A),X5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),X5))),product_Sigma(A,B,Aa2,aTP_Lamp_il(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_UNIV_cancel
tff(fact_2919_insert__Times__insert,axiom,
! [A: $tType,B: $tType,A3: A,Aa2: set(A),B2: B,Ba: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert2(A,A3),Aa2),aa(set(B),fun(A,set(B)),aTP_Lamp_im(B,fun(set(B),fun(A,set(B))),B2),Ba)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,Aa2,aa(set(B),fun(A,set(B)),aTP_Lamp_im(B,fun(set(B),fun(A,set(B))),B2),Ba))),product_Sigma(A,B,aa(set(A),set(A),insert2(A,A3),Aa2),aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))) ).
% insert_Times_insert
tff(fact_2920_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod(A,B),Aa2: set(A),Ba: fun(A,set(B))] :
( member(product_prod(A,B),C2,product_Sigma(A,B,Aa2,Ba))
=> ~ ! [X3: A] :
( member(A,X3,Aa2)
=> ! [Y2: B] :
( member(B,Y2,aa(A,set(B),Ba,X3))
=> ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) ) ) ) ) ).
% SigmaE
tff(fact_2921_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,Aa2: set(A),Ba: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,Aa2,Ba))
=> member(A,A3,Aa2) ) ).
% SigmaD1
tff(fact_2922_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,Aa2: set(A),Ba: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,Aa2,Ba))
=> member(B,B2,aa(A,set(B),Ba,A3)) ) ).
% SigmaD2
tff(fact_2923_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,Aa2: set(A),Ba: fun(A,set(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,Aa2,Ba))
=> ~ ( member(A,A3,Aa2)
=> ~ member(B,B2,aa(A,set(B),Ba,A3)) ) ) ).
% SigmaE2
tff(fact_2924_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( ! [X3: A] :
( ! [Y4: A] :
( ( ( Y4 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2) )
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,A3) ) ) ).
% wo_rel.well_order_induct
tff(fact_2925_times__eq__iff,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B),Ca: set(A),D4: set(B)] :
( ( product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)) = product_Sigma(A,B,Ca,aTP_Lamp_ib(set(B),fun(A,set(B)),D4)) )
<=> ( ( ( Aa2 = Ca )
& ( Ba = D4 ) )
| ( ( ( Aa2 = bot_bot(set(A)) )
| ( Ba = bot_bot(set(B)) ) )
& ( ( Ca = bot_bot(set(A)) )
| ( D4 = bot_bot(set(B)) ) ) ) ) ) ).
% times_eq_iff
tff(fact_2926_wo__rel_OTOTALS,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ! [X2: A] :
( member(A,X2,field2(A,R2))
=> ! [Xa3: A] :
( member(A,Xa3,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),R2)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X2),R2) ) ) ) ) ).
% wo_rel.TOTALS
tff(fact_2927_wo__rel_Omax2__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2),B2,A3) ) ) ).
% wo_rel.max2_def
tff(fact_2928_cInf__le__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,Aa2)
=> ( condit1013018076250108175_below(A,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ) ).
% cInf_le_cSup
tff(fact_2929_well__order__induct__imp,axiom,
! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( ! [X3: A] :
( ! [Y4: A] :
( ( ( Y4 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2) )
=> ( member(A,Y4,field2(A,R2))
=> aa(A,$o,P,Y4) ) )
=> ( member(A,X3,field2(A,R2))
=> aa(A,$o,P,X3) ) )
=> ( member(A,A3,field2(A,R2))
=> aa(A,$o,P,A3) ) ) ) ).
% well_order_induct_imp
tff(fact_2930_Sigma__empty__iff,axiom,
! [A: $tType,B: $tType,I4: set(A),X5: fun(A,set(B))] :
( ( product_Sigma(A,B,I4,X5) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] :
( member(A,X4,I4)
=> ( aa(A,set(B),X5,X4) = bot_bot(set(B)) ) ) ) ).
% Sigma_empty_iff
tff(fact_2931_wo__rel_Omax2__greater,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B2)),R2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A3,B2)),R2) ) ) ) ) ).
% wo_rel.max2_greater
tff(fact_2932_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = B2 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ) ) ).
% wo_rel.max2_equals2
tff(fact_2933_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( ( bNF_We1388413361240627857o_max2(A,R2,A3,B2) = A3 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R2) ) ) ) ) ).
% wo_rel.max2_equals1
tff(fact_2934_le__cInf__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S))
<=> ! [X4: A] :
( member(A,X4,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X4) ) ) ) ) ) ).
% le_cInf_iff
tff(fact_2935_cInf__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Ba: set(A),Aa2: set(A)] :
( ( Ba != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,Aa2)
=> ( ! [B3: A] :
( member(A,B3,Ba)
=> ? [X2: A] :
( member(A,X2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B3) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),Ba)) ) ) ) ) ).
% cInf_mono
tff(fact_2936_cInf__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X5)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y)
<=> ? [X4: A] :
( member(A,X4,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y) ) ) ) ) ) ).
% cInf_less_iff
tff(fact_2937_cSup__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Ba: set(A),Aa2: set(A)] :
( ( Ba != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,Aa2)
=> ( ! [B3: A] :
( member(A,B3,Ba)
=> ? [X2: A] :
( member(A,X2,Aa2)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),X2) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),Ba)),aa(set(A),A,complete_Sup_Sup(A),Aa2)) ) ) ) ) ).
% cSup_mono
tff(fact_2938_cSup__le__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A),A3: A] :
( ( S != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,S)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3)
<=> ! [X4: A] :
( member(A,X4,S)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A3) ) ) ) ) ) ).
% cSup_le_iff
tff(fact_2939_less__cSup__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X5: set(A),Y: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,X5)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
<=> ? [X4: A] :
( member(A,X4,X5)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X4) ) ) ) ) ) ).
% less_cSup_iff
tff(fact_2940_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,Aa2: set(B)] : finite_card(product_prod(A,B),product_Sigma(A,B,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aTP_Lamp_ib(set(B),fun(A,set(B)),Aa2))) = finite_card(B,Aa2) ).
% card_cartesian_product_singleton
tff(fact_2941_times__subset__iff,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ca: set(B),Ba: set(A),D4: set(B)] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ca))),product_Sigma(A,B,Ba,aTP_Lamp_ib(set(B),fun(A,set(B)),D4)))
<=> ( ( Aa2 = bot_bot(set(A)) )
| ( Ca = bot_bot(set(B)) )
| ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ca),D4) ) ) ) ).
% times_subset_iff
tff(fact_2942_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B2),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) ) ) ) ).
% wo_rel.max2_among
tff(fact_2943_wfI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),Ba: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Ba)))
=> ( ! [X3: A,P3: fun(A,$o)] :
( ! [Xa3: A] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa3),R2)
=> aa(A,$o,P3,Y2) )
=> aa(A,$o,P3,Xa3) )
=> ( member(A,X3,Aa2)
=> ( member(A,X3,Ba)
=> aa(A,$o,P3,X3) ) ) )
=> wf(A,R2) ) ) ).
% wfI
tff(fact_2944_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))
<=> ( ( Aa2 = bot_bot(set(A)) )
| ( Ba = bot_bot(set(B)) )
| ( aa(set(A),$o,finite_finite(A),Aa2)
& aa(set(B),$o,finite_finite(B),Ba) ) ) ) ).
% finite_cartesian_product_iff
tff(fact_2945_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(set(B),$o,finite_finite(B),Ba) ) ) ).
% finite_cartesian_productD2
tff(fact_2946_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))
=> ( ( Ba != bot_bot(set(B)) )
=> aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% finite_cartesian_productD1
tff(fact_2947_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: fun(A,set(B))] :
( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,set(B)),fun(A,$o)),Aa2),Ba)))
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(set(B),$o,finite_finite(B),aa(A,set(B),Ba,A4)) )
=> aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),product_Sigma(A,B,Aa2,Ba)) ) ) ).
% finite_SigmaI2
tff(fact_2948_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,Aa2,Ba)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,set(B)),fun(A,$o)),Aa2),Ba)) ).
% fst_image_Sigma
tff(fact_2949_refl__onI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2) )
=> refl_on(A,Aa2,R2) ) ) ).
% refl_onI
tff(fact_2950_refl__on__def,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,Aa2,R2)
<=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))
& ! [X4: A] :
( member(A,X4,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2) ) ) ) ).
% refl_on_def
tff(fact_2951_swap__product,axiom,
! [A: $tType,B: $tType,Aa2: set(B),Ba: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,Aa2,aTP_Lamp_fh(set(A),fun(B,set(A)),Ba))) = product_Sigma(A,B,Ba,aTP_Lamp_ib(set(B),fun(A,set(B)),Aa2)) ).
% swap_product
tff(fact_2952_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,$o))] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R2))
=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R2)
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),Phi,A3),B2) ) ) ) ) ).
% wo_rel.cases_Total
tff(fact_2953_le__cINF__iff,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),U: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2)))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F,X4)) ) ) ) ) ) ).
% le_cINF_iff
tff(fact_2954_cINF__mono,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Ba: set(A),F: fun(C,B),Aa2: set(C),G: fun(A,B)] :
( ( Ba != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F),Aa2))
=> ( ! [M3: A] :
( member(A,M3,Ba)
=> ? [X2: C] :
( member(C,X2,Aa2)
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F,X2)),aa(A,B,G,M3)) ) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),Ba))) ) ) ) ) ).
% cINF_mono
tff(fact_2955_cInf__superset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,Ba)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Ba)),aa(set(A),A,complete_Inf_Inf(A),Aa2)) ) ) ) ) ).
% cInf_superset_mono
tff(fact_2956_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),U: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),U)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),U) ) ) ) ) ) ).
% cSUP_le_iff
tff(fact_2957_cSUP__mono,axiom,
! [A: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),G: fun(C,B),Ba: set(C),F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),Ba))
=> ( ! [N4: A] :
( member(A,N4,Aa2)
=> ? [X2: C] :
( member(C,X2,Ba)
& aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,N4)),aa(C,B,G,X2)) ) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),Ba))) ) ) ) ) ).
% cSUP_mono
tff(fact_2958_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R2,A3,B2)),R2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A3,B2)),R2)
& member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B2),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) ) ) ) ) ).
% wo_rel.max2_greater_among
tff(fact_2959_cSup__subset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,Ba)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Ba)) ) ) ) ) ).
% cSup_subset_mono
tff(fact_2960_cInf__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,X5)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),insert2(A,A3),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).
% cInf_insert
tff(fact_2961_cInf__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( condit1013018076250108175_below(A,X5)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),insert2(A,A3),X5)) = $ite(X5 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X5))) ) ) ) ).
% cInf_insert_If
tff(fact_2962_cSup__insert,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( ( X5 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,X5)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,A3),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).
% cSup_insert
tff(fact_2963_cSup__insert__If,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X5: set(A),A3: A] :
( condit941137186595557371_above(A,X5)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,A3),X5)) = $ite(X5 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X5))) ) ) ) ).
% cSup_insert_If
tff(fact_2964_cInf__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,Ba)
=> ( 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)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),Ba)) ) ) ) ) ) ) ).
% cInf_union_distrib
tff(fact_2965_cSup__union__distrib,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,Ba)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Ba)) ) ) ) ) ) ) ).
% cSup_union_distrib
tff(fact_2966_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( condit6923001295902523014norder(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))),A3)
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),A3) ) ) ) ) ) ).
% cINF_less_iff
tff(fact_2967_less__cSUP__iff,axiom,
! [B: $tType,A: $tType] :
( condit6923001295902523014norder(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: B] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2)))
<=> ? [X4: A] :
( member(A,X4,Aa2)
& aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F,X4)) ) ) ) ) ) ).
% less_cSUP_iff
tff(fact_2968_cINF__inf__distrib,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),G: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),Aa2))
=> ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),Aa2))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ip(fun(A,B),fun(fun(A,B),fun(A,B)),F),G)),Aa2)) ) ) ) ) ) ).
% cINF_inf_distrib
tff(fact_2969_image__paired__Times,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,A),G: fun(D,B),Aa2: set(C),Ba: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_iq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),product_Sigma(C,D,Aa2,aTP_Lamp_ir(set(D),fun(C,set(D)),Ba))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),Aa2),aa(set(D),fun(A,set(B)),aTP_Lamp_is(fun(D,B),fun(set(D),fun(A,set(B))),G),Ba)) ).
% image_paired_Times
tff(fact_2970_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),G: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),Aa2))
=> ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),Aa2))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_it(fun(A,B),fun(fun(A,B),fun(A,B)),F),G)),Aa2)) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_2971_cINF__superset__mono,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),G: fun(A,B),Ba: set(A),F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),Ba))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ! [X3: A] :
( member(A,X3,Ba)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X3)),aa(A,B,F,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),Ba))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ) ) ) ).
% cINF_superset_mono
tff(fact_2972_cSUP__subset__mono,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),G: fun(A,B),Ba: set(A),F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),Ba))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),Ba))) ) ) ) ) ) ).
% cSUP_subset_mono
tff(fact_2973_less__eq__cInf__inter,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( condit1013018076250108175_below(A,Aa2)
=> ( condit1013018076250108175_below(A,Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),Aa2)),aa(set(A),A,complete_Inf_Inf(A),Ba))),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)),Aa2),Ba))) ) ) ) ) ).
% less_eq_cInf_inter
tff(fact_2974_cINF__insert,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),insert2(A,A3),Aa2))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,A3)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ) ) ).
% cINF_insert
tff(fact_2975_cSup__inter__less__eq,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( condit941137186595557371_above(A,Aa2)
=> ( condit941137186595557371_above(A,Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) != bot_bot(set(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)),aa(set(A),A,complete_Sup_Sup(A),Ba))) ) ) ) ) ).
% cSup_inter_less_eq
tff(fact_2976_cSUP__insert,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),A3: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),insert2(A,A3),Aa2))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,A3)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ) ) ).
% cSUP_insert
tff(fact_2977_cINF__union,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F),Ba))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Ba))) ) ) ) ) ) ) ).
% cINF_union
tff(fact_2978_cSUP__union,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Aa2: set(A),F: fun(A,B),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Aa2))
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F),Ba))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Ba))) ) ) ) ) ) ) ).
% cSUP_union
tff(fact_2979_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( Ba != bot_bot(set(A)) )
=> ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R2,Ba,X_1) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
tff(fact_2980_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( Ba != bot_bot(set(A)) )
=> member(A,bNF_We6954850376910717587_minim(A,R2,Ba),field2(A,R2)) ) ) ) ).
% wo_rel.minim_inField
tff(fact_2981_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( Ba != bot_bot(set(A)) )
=> member(A,bNF_We6954850376910717587_minim(A,R2,Ba),Ba) ) ) ) ).
% wo_rel.minim_in
tff(fact_2982_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A),A3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( member(A,A3,Ba)
=> ( ! [B3: A] :
( member(A,B3,Ba)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R2) )
=> ( A3 = bNF_We6954850376910717587_minim(A,R2,Ba) ) ) ) ) ) ).
% wo_rel.equals_minim
tff(fact_2983_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( member(A,B2,Ba)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R2,Ba)),B2),R2) ) ) ) ).
% wo_rel.minim_least
tff(fact_2984_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( Ba != bot_bot(set(A)) )
=> bNF_We4791949203932849705sMinim(A,R2,Ba,bNF_We6954850376910717587_minim(A,R2,Ba)) ) ) ) ).
% wo_rel.minim_isMinim
tff(fact_2985_wo__rel_OisMinim__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( bNF_We4791949203932849705sMinim(A,R2,Aa2,B2)
<=> ( member(A,B2,Aa2)
& ! [X4: A] :
( member(A,X4,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4),R2) ) ) ) ) ).
% wo_rel.isMinim_def
tff(fact_2986_rtrancl__last__visit__node,axiom,
! [A: $tType,S2: A,S3: A,R: set(product_prod(A,A)),Sh: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3),transitive_rtrancl(A,R))
=> ( ( ( S2 != Sh )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_iu(A,fun(A,set(A)),Sh))))) )
| ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),Sh),transitive_rtrancl(A,R))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_iu(A,fun(A,set(A)),Sh))))) ) ) ) ).
% rtrancl_last_visit_node
tff(fact_2987_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),Aa2: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_iv(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),Aa2) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),Aa2),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_iw(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),Aa2)) ).
% image_split_eq_Sigma
tff(fact_2988_relation__of__def,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),Aa2: set(A)] : order_relation_of(A,P,Aa2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_ix(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),P),Aa2))) ).
% relation_of_def
tff(fact_2989_finite__def,axiom,
! [A: $tType] : finite_finite(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_iy(fun(set(A),$o),fun(set(A),$o))) ).
% finite_def
tff(fact_2990_bsqr__max2,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22)),bNF_Wellorder_bsqr(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R2,A1,A22)),bNF_We1388413361240627857o_max2(A,R2,B1,B22)),R2) ) ) ).
% bsqr_max2
tff(fact_2991_vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : vimage(A,B,F,bot_bot(set(B))) = bot_bot(set(A)) ).
% vimage_empty
tff(fact_2992_rtrancl__empty,axiom,
! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).
% rtrancl_empty
tff(fact_2993_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,Aa2: set(B)] :
vimage(A,B,aTP_Lamp_as(B,fun(A,B),C2),Aa2) = $ite(member(B,C2,Aa2),top_top(set(A)),bot_bot(set(A))) ).
% vimage_const
tff(fact_2994_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E4: set(product_prod(B,A))] : vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U),E4) = image(B,A,E4,aa(set(B),set(B),insert2(B,U),bot_bot(set(B)))) ).
% pair_vimage_is_Image
tff(fact_2995_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
<=> ( wf(A,R2)
& ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2)) ) ) ).
% wf_insert
tff(fact_2996_vimage__if,axiom,
! [B: $tType,A: $tType,Ba: set(A),C2: B,D3: B,Aa2: set(B)] :
vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_iz(set(A),fun(B,fun(B,fun(A,B))),Ba),C2),D3),Aa2) = $ite(
member(B,C2,Aa2),
$ite(member(B,D3,Aa2),top_top(set(A)),Ba),
$ite(member(B,D3,Aa2),aa(set(A),set(A),uminus_uminus(set(A)),Ba),bot_bot(set(A))) ) ).
% vimage_if
tff(fact_2997_well__order__on__domain,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
( order_well_order_on(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(A,A3,Aa2)
& member(A,B2,Aa2) ) ) ) ).
% well_order_on_domain
tff(fact_2998_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_rtrancl(A,R2)) ) ) ).
% converse_rtrancl_into_rtrancl
tff(fact_2999_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,B2)
=> ( ! [Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,Z3)
=> aa(A,$o,P,Y2) ) ) )
=> aa(A,$o,P,A3) ) ) ) ).
% converse_rtrancl_induct
tff(fact_3000_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2))
=> ( ( X != Z2 )
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),transitive_rtrancl(A,R2)) ) ) ) ).
% converse_rtranclE
tff(fact_3001_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( aa(A,$o,P,A3)
=> ( ! [Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> ( aa(A,$o,P,Y2)
=> aa(A,$o,P,Z3) ) ) )
=> aa(A,$o,P,B2) ) ) ) ).
% rtrancl_induct
tff(fact_3002_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2)) ) ) ).
% rtrancl_trans
tff(fact_3003_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( ( A3 != B2 )
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2),R2) ) ) ) ).
% rtranclE
tff(fact_3004_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_rtrancl(A,R2)) ) ) ).
% rtrancl.rtrancl_into_rtrancl
tff(fact_3005_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),transitive_rtrancl(A,R2)) ).
% rtrancl.rtrancl_refl
tff(fact_3006_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
<=> ( ? [A5: A] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: A,B4: A,C4: A] :
( ( A1 = A5 )
& ( A22 = C4 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),transitive_rtrancl(A,R2))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),C4),R2) ) ) ) ).
% rtrancl.simps
tff(fact_3007_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
=> ( ( A22 != A1 )
=> ~ ! [B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22),R2) ) ) ) ).
% rtrancl.cases
tff(fact_3008_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,R))
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V),transitive_rtrancl(A,R)) ) ) ) ) ).
% converse_rtranclE'
tff(fact_3009_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F: fun(A,B),B2: B] :
( member(A,A3,vimage(A,B,F,aa(set(B),set(B),insert2(B,B2),bot_bot(set(B)))))
<=> ( aa(A,B,F,A3) = B2 ) ) ).
% vimage_singleton_eq
tff(fact_3010_well__order__on__empty,axiom,
! [A: $tType] : order_well_order_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).
% well_order_on_empty
tff(fact_3011_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q3: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
( member(A,Q3,image(A,A,transitive_rtrancl(A,R),Q0))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X),transitive_rtrancl(A,R))
=> member(A,X,image(A,A,transitive_rtrancl(A,R),Q0)) ) ) ).
% rtrancl_image_advance_rtrancl
tff(fact_3012_rtrancl__image__advance,axiom,
! [A: $tType,Q3: A,R: set(product_prod(A,A)),Q0: set(A),X: A] :
( member(A,Q3,image(A,A,transitive_rtrancl(A,R),Q0))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X),R)
=> member(A,X,image(A,A,transitive_rtrancl(A,R),Q0)) ) ) ).
% rtrancl_image_advance
tff(fact_3013_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),transitive_rtrancl(A,P))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),Q)
=> ( X3 = Y2 ) ) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P)) ) ) ).
% rtrancl_Un_separatorE
tff(fact_3014_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2),transitive_rtrancl(A,P))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),Q)
=> ( Y2 = X3 ) ) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P)) ) ) ).
% rtrancl_Un_separator_converseE
tff(fact_3015_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),transitive_rtrancl(product_prod(A,B),R2))
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2)),R2)
=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% rtrancl_induct2
tff(fact_3016_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R2))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
=> ~ ! [A4: A,B3: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),R2)
=> ~ member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R2)) ) ) ) ).
% converse_rtranclE2
tff(fact_3017_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2)),R2)
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2)
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).
% converse_rtrancl_induct2
tff(fact_3018_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,Aa2: set(B),F: fun(B,set(A))] :
vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,Aa2,F)) = $ite(member(B,X,Aa2),aa(B,set(A),F,X),bot_bot(set(A))) ).
% Pair_vimage_Sigma
tff(fact_3019_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Aa2: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( ( vimage(B,A,F,Aa2) = bot_bot(set(B)) )
<=> ( Aa2 = bot_bot(set(A)) ) ) ) ).
% surj_vimage_empty
tff(fact_3020_vimage__insert,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A3: B,Ba: set(B)] : vimage(A,B,F,aa(set(B),set(B),insert2(B,A3),Ba)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),vimage(A,B,F,Ba)) ).
% vimage_insert
tff(fact_3021_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R: set(product_prod(A,A)),V: A] :
( ( image(A,A,R,aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_rtrancl(A,R),aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,V),bot_bot(set(A))) ) ) ).
% Image_empty_rtrancl_Image_id
tff(fact_3022_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),Aa2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))
=> ( ( A3 = B2 )
| member(A,A3,Aa2) ) ) ) ).
% trancl_subset_Sigma_aux
tff(fact_3023_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W2: A,E4: set(product_prod(A,A)),U2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E4),product_Sigma(A,A,U2,aTP_Lamp_in(set(A),fun(A,set(A)),U2)))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_rtrancl(A,E4)) ) ).
% Restr_rtrancl_mono
tff(fact_3024_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,finite_finite(A),aa(set(B),set(A),image2(B,A,F),Aa2))
=> ( ~ aa(set(B),$o,finite_finite(B),Aa2)
=> ? [X3: A] :
( member(A,X3,aa(set(B),set(A),image2(B,A,F),Aa2))
& ~ aa(set(B),$o,finite_finite(B),vimage(B,A,F,aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_dom
tff(fact_3025_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,finite_finite(A),aa(set(B),set(A),image2(B,A,F),Aa2))
=> ( ~ aa(set(B),$o,finite_finite(B),Aa2)
=> ~ ! [Y2: A] :
( member(A,Y2,aa(set(B),set(A),image2(B,A,F),Aa2))
=> aa(set(B),$o,finite_finite(B),vimage(B,A,F,aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_domE
tff(fact_3026_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F4: set(A),H2: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,finite_finite(A),F4)
=> ( ! [Y2: A] :
( member(A,Y2,F4)
=> aa(set(B),$o,finite_finite(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H2,aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A))))),Aa2)) )
=> aa(set(B),$o,finite_finite(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H2,F4)),Aa2)) ) ) ).
% finite_finite_vimage_IntI
tff(fact_3027_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E4: set(product_prod(A,A)),F: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,E4))
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,A3)),aa(A,B,F,B2)),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F)),E4))) ) ).
% rtrancl_mapI
tff(fact_3028_rtrancl__apply__insert,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,S: set(A)] : image(A,A,transitive_rtrancl(A,R),aa(set(A),set(A),insert2(A,X),S)) = aa(set(A),set(A),insert2(A,X),image(A,A,transitive_rtrancl(A,R),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),image(A,A,R,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ).
% rtrancl_apply_insert
tff(fact_3029_vimage__eq__UN,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Ba: set(B)] : vimage(A,B,F,Ba) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ja(fun(A,B),fun(B,set(A)),F)),Ba)) ).
% vimage_eq_UN
tff(fact_3030_rtrancl__last__touch,axiom,
! [A: $tType,Q3: A,Q6: A,R: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q6),transitive_rtrancl(A,R))
=> ( member(A,Q3,S)
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_touch
tff(fact_3031_rtrancl__last__visit_H,axiom,
! [A: $tType,Q3: A,Q6: A,R: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q6),transitive_rtrancl(A,R))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q6),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),S)))))
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_visit'
tff(fact_3032_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,finite_finite(A),aa(set(B),set(A),image2(B,A,F),Aa2))
=> ( ~ aa(set(B),$o,finite_finite(B),Aa2)
=> ~ ! [Y2: A] :
( member(A,Y2,aa(set(B),set(A),image2(B,A,F),Aa2))
=> aa(set(B),$o,finite_finite(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F,aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A))))),Aa2)) ) ) ) ).
% inf_img_fin_domE'
tff(fact_3033_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,finite_finite(A),aa(set(B),set(A),image2(B,A,F),Aa2))
=> ( ~ aa(set(B),$o,finite_finite(B),Aa2)
=> ? [X3: A] :
( member(A,X3,aa(set(B),set(A),image2(B,A,F),Aa2))
& ~ aa(set(B),$o,finite_finite(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F,aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))))),Aa2)) ) ) ) ).
% inf_img_fin_dom'
tff(fact_3034_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R2)))) ).
% rtrancl_insert
tff(fact_3035_finite__reachable__advance,axiom,
! [A: $tType,E4: set(product_prod(A,A)),V0: A,V: A] :
( aa(set(A),$o,finite_finite(A),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V0),bot_bot(set(A)))))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V),transitive_rtrancl(A,E4))
=> aa(set(A),$o,finite_finite(A),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))) ) ) ).
% finite_reachable_advance
tff(fact_3036_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,U),bot_bot(set(A))))) ) ).
% rtrancl_Image_advance_ss
tff(fact_3037_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R2),R2)
=> ( order_well_order_on(A,field2(A,R2),R2)
<=> ! [A8: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),field2(A,R2))
=> ( ( A8 != bot_bot(set(A)) )
=> ? [X4: A] :
( member(A,X4,A8)
& ! [Xa2: A] :
( member(A,Xa2,A8)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
tff(fact_3038_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),E4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),R)))))
=> ( ~ member(A,U,R)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E4,R))) ) ) ).
% rtrancl_restrictI
tff(fact_3039_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),M2: A,X5: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),insert2(A,M2),bot_bot(set(A))),aTP_Lamp_in(set(A),fun(A,set(A)),X5)))))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M2),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2),transitive_rtrancl(A,R2)) ) ) ) ) ).
% trancl_multi_insert2
tff(fact_3040_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),X5: set(A),M2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),product_Sigma(A,A,X5,aTP_Lamp_jc(A,fun(A,set(A)),M2)))))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ~ ! [X3: A] :
( member(A,X3,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),transitive_rtrancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),B2),transitive_rtrancl(A,R2)) ) ) ) ) ).
% trancl_multi_insert
tff(fact_3041_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N),transitive_rtrancl(nat,pred_nat))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N) ) ).
% pred_nat_trancl_eq_le
tff(fact_3042_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_ec(product_prod(int,int),product_prod(int,int))) ).
% inverse_rat_def
tff(fact_3043_rel__restrict__empty,axiom,
! [A: $tType,R: set(product_prod(A,A))] : rel_restrict(A,R,bot_bot(set(A))) = R ).
% rel_restrict_empty
tff(fact_3044_trancl__empty,axiom,
! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).
% trancl_empty
tff(fact_3045_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A)))) ).
% trancl_single
tff(fact_3046_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W2: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,rel_restrict(A,E4,R)))
=> ~ member(A,W2,R) ) ).
% rel_restrict_trancl_notR(2)
tff(fact_3047_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W2: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,rel_restrict(A,E4,R)))
=> ~ member(A,V,R) ) ).
% rel_restrict_trancl_notR(1)
tff(fact_3048_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,Aa2: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,rel_restrict(A,Aa2,R)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),rel_restrict(A,transitive_trancl(A,Aa2),R)) ) ).
% rel_restrict_trancl_mem
tff(fact_3049_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),R2)
=> ~ ! [B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3),transitive_trancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22),R2) ) ) ) ).
% trancl.cases
tff(fact_3050_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
<=> ( ? [A5: A,B4: A] :
( ( A1 = A5 )
& ( A22 = B4 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),R2) )
| ? [A5: A,B4: A,C4: A] :
( ( A1 = A5 )
& ( A22 = C4 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),transitive_trancl(A,R2))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),C4),R2) ) ) ) ).
% trancl.simps
tff(fact_3051_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2)) ) ).
% trancl.r_into_trancl
tff(fact_3052_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ~ ! [C3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3),transitive_trancl(A,R2))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C3),B2),R2) ) ) ) ).
% tranclE
tff(fact_3053_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2)) ) ) ).
% trancl_trans
tff(fact_3054_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),R2)
=> aa(A,$o,P,Y2) )
=> ( ! [Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y2),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> ( aa(A,$o,P,Y2)
=> aa(A,$o,P,Z3) ) ) )
=> aa(A,$o,P,B2) ) ) ) ).
% trancl_induct
tff(fact_3055_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R)) ) ) ).
% r_r_into_trancl
tff(fact_3056_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2)
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),transitive_trancl(A,R2)) ) ) ) ).
% converse_tranclE
tff(fact_3057_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
tff(fact_3058_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).
% Transitive_Closure.trancl_into_trancl
tff(fact_3059_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).
% trancl_into_trancl2
tff(fact_3060_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),P: fun(A,fun(A,$o))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R2))
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> aa(A,$o,aa(A,fun(A,$o),P,X3),Y2) )
=> ( ! [X3: A,Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),transitive_trancl(A,R2))
=> ( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),transitive_trancl(A,R2))
=> ( aa(A,$o,aa(A,fun(A,$o),P,Y2),Z3)
=> aa(A,$o,aa(A,fun(A,$o),P,X3),Z3) ) ) ) )
=> aa(A,$o,aa(A,fun(A,$o),P,X),Y) ) ) ) ).
% trancl_trans_induct
tff(fact_3061_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2),R2)
=> aa(A,$o,P,Y2) )
=> ( ! [Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_trancl(A,R2))
=> ( aa(A,$o,P,Z3)
=> aa(A,$o,P,Y2) ) ) )
=> aa(A,$o,P,A3) ) ) ) ).
% converse_trancl_induct
tff(fact_3062_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,E4,R))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E4) ) ).
% rel_restrict_lift
tff(fact_3063_rel__restrictI,axiom,
! [A: $tType,X: A,R: set(A),Y: A,E4: set(product_prod(A,A))] :
( ~ member(A,X,R)
=> ( ~ member(A,Y,R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),E4)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,E4,R)) ) ) ) ).
% rel_restrictI
tff(fact_3064_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,Aa2: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,Aa2,R))
=> ~ member(A,X,R) ) ).
% rel_restrict_notR(1)
tff(fact_3065_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,Aa2: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),rel_restrict(A,Aa2,R))
=> ~ member(A,Y,R) ) ).
% rel_restrict_notR(2)
tff(fact_3066_less__eq,axiom,
! [M2: nat,N: nat] :
( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N),transitive_trancl(nat,pred_nat))
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N) ) ).
% less_eq
tff(fact_3067_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,E4))
=> ( ~ member(A,X,R)
=> ( ~ member(A,Y,R)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E4,R)),R)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,rel_restrict(A,E4,R))) ) ) ) ) ).
% rel_restrict_tranclI
tff(fact_3068_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).
% trancl_rtrancl_trancl
tff(fact_3069_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_trancl_trancl
tff(fact_3070_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_into_trancl2
tff(fact_3071_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R2)) ) ) ).
% rtrancl_into_trancl1
tff(fact_3072_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R))
<=> ( ( X = Y )
| ( ( X != Y )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R)) ) ) ) ).
% rtrancl_eq_or_trancl
tff(fact_3073_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R2)) ) ).
% trancl_into_rtrancl
tff(fact_3074_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R))
=> ? [Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3),transitive_rtrancl(A,R))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),R) ) ) ).
% tranclD2
tff(fact_3075_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R))
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)) ) ) ) ).
% rtranclD
tff(fact_3076_tranclD,axiom,
! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,R))
=> ? [Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3),R)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),transitive_rtrancl(A,R)) ) ) ).
% tranclD
tff(fact_3077_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R2))
=> ( ! [A4: A,B3: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),R2)
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) )
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),transitive_trancl(product_prod(A,B),R2))
=> ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2)),R2)
=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% trancl_induct2
tff(fact_3078_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R: set(product_prod(A,A)),V: A] :
( ( image(A,A,R,aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_trancl(A,R),aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).
% Image_empty_trancl_Image_empty
tff(fact_3079_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,$o),K: B,M2: fun(B,A)] :
( wf(A,R2)
=> ( ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),transitive_trancl(A,R2))
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X3),transitive_rtrancl(A,R2)) )
=> ( aa(B,$o,P,K)
=> ? [X3: B] :
( aa(B,$o,P,X3)
& ! [Y4: B] :
( aa(B,$o,P,Y4)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M2,X3)),aa(B,A,M2,Y4)),transitive_rtrancl(A,R2)) ) ) ) ) ) ).
% wf_linord_ex_has_least
tff(fact_3080_trancl__union__outside,axiom,
! [A: $tType,V: A,W2: A,E4: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U2)))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,E4))
=> ? [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),X3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U2)))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),U2)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),W2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U2))) ) ) ) ).
% trancl_union_outside
tff(fact_3081_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W2: A,V1: A,V22: A,E4: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V1),V22)),E4)))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2),transitive_trancl(A,E4))
=> ~ ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1),transitive_rtrancl(A,E4))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W2),transitive_rtrancl(A,E4)) ) ) ) ).
% trancl_over_edgeE
tff(fact_3082_rel__restrict__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),Aa2: set(A)] : rel_restrict(A,R,Aa2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_jd(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),R),Aa2))) ).
% rel_restrict_def
tff(fact_3083_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W2: A,E4: set(product_prod(A,A)),U2: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E4),product_Sigma(A,A,U2,aTP_Lamp_in(set(A),fun(A,set(A)),U2)))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W2),transitive_trancl(A,E4)) ) ).
% Restr_trancl_mono
tff(fact_3084_rel__restrict__Int__empty,axiom,
! [A: $tType,Aa2: 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)),Aa2),field2(A,R)) = bot_bot(set(A)) )
=> ( rel_restrict(A,R,Aa2) = R ) ) ).
% rel_restrict_Int_empty
tff(fact_3085_rel__restrict__compl,axiom,
! [A: $tType,R: set(product_prod(A,A)),Aa2: 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,Aa2)),rel_restrict(A,R,aa(set(A),set(A),uminus_uminus(set(A)),Aa2))) = bot_bot(set(product_prod(A,A))) ).
% rel_restrict_compl
tff(fact_3086_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_je(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A3),B2),R2)))) ).
% trancl_insert2
tff(fact_3087_uminus__rat__def,axiom,
uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_ef(product_prod(int,int),product_prod(int,int))) ).
% uminus_rat_def
tff(fact_3088_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E4)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))),image(A,A,transitive_trancl(A,E4),aa(set(A),set(A),insert2(A,U),bot_bot(set(A))))) ) ).
% trancl_Image_advance_ss
tff(fact_3089_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),R: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,E4))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E4,R)),R)
=> ( ~ member(A,V,R)
=> ~ ( ~ member(A,U,R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E4,R))) ) ) ) ) ).
% E_closed_restr_reach_cases
tff(fact_3090_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,E4))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,E4,S)),S)
=> ( member(A,U,S)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),E4),product_Sigma(A,A,S,aTP_Lamp_in(set(A),fun(A,set(A)),S))))) ) ) ) ).
% trancl_restrict_reachable
tff(fact_3091_rtrancl__last__visit,axiom,
! [A: $tType,Q3: A,Q6: A,R: set(product_prod(A,A)),S: set(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q6),transitive_rtrancl(A,R))
=> ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q6),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),S)))))
=> ~ ! [Qt: A] :
( member(A,Qt,S)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_trancl(A,R))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_in(set(A),fun(A,set(A)),S))))) ) ) ) ) ).
% rtrancl_last_visit
tff(fact_3092_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),X),R2)))) ).
% trancl_insert
tff(fact_3093_plus__rat__def,axiom,
plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_eg(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat_def
tff(fact_3094_times__rat__def,axiom,
times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_eh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat_def
tff(fact_3095_subset__singleton__iff__Uniq,axiom,
! [A: $tType,Aa2: set(A)] :
( ? [A5: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,A5),bot_bot(set(A))))
<=> uniq(A,aTP_Lamp_a(set(A),fun(A,$o),Aa2)) ) ).
% subset_singleton_iff_Uniq
tff(fact_3096_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_3097_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S: set(A),F: fun(A,B)] :
( aa(set(A),$o,finite_finite(A),S)
=> ( ( S != bot_bot(set(A)) )
=> ? [X_1: A] : lattic501386751177426532rg_min(A,B,F,aTP_Lamp_a(set(A),fun(A,$o),S),X_1) ) ) ) ).
% ex_is_arg_min_if_finite
tff(fact_3098_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,Aa2: set(A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),insert2(A,A3),Aa2)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
tff(fact_3099_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,Aa2: set(A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( member(A,A3,Aa2)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),Aa2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).
% bounded_quasi_semilattice_set.remove
tff(fact_3100_relImage__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,B)),F: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_jf(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R),F)) ).
% relImage_def
tff(fact_3101_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_jg(A,fun(A,$o)),aTP_Lamp_jh(A,fun(A,$o))) ) ).
% Sup_fin.semilattice_order_set_axioms
tff(fact_3102_less__than__iff,axiom,
! [X: nat,Y: nat] :
( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y),less_than)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ).
% less_than_iff
tff(fact_3103_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
=> ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),bot_bot(set(A))) = Top ) ) ).
% bounded_quasi_semilattice_set.empty
tff(fact_3104_mlex__prod__def,axiom,
! [A: $tType,F: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F,R) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R),aTP_Lamp_ji(fun(A,nat),fun(A,product_prod(nat,A)),F)) ).
% mlex_prod_def
tff(fact_3105_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),Aa2: set(A)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F),Aa2))) ) ) ) ) ).
% mono_cInf
tff(fact_3106_mono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( condit1219197933456340205attice(B)
& condit1219197933456340205attice(A) )
=> ! [F: fun(A,B),Aa2: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,Aa2),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,Aa2),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_jj(fun(A,B),fun(fun(C,A),fun(C,B)),F),Aa2)),I4))) ) ) ) ) ).
% mono_cINF
tff(fact_3107_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),Aa2: set(A)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),Aa2))) ) ) ) ) ).
% mono_cSup
tff(fact_3108_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),Aa2: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,Aa2),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_jj(fun(A,B),fun(fun(C,A),fun(C,B)),F),Aa2)),I4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,Aa2),I4)))) ) ) ) ) ).
% mono_cSUP
tff(fact_3109_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(B,B)),F: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),inv_image(B,A,R2,F))
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X)),aa(A,B,F,Y)),R2) ) ).
% in_inv_image
tff(fact_3110_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_3111_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).
% mono_strict_invE
tff(fact_3112_monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).
% monoD
tff(fact_3113_monoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).
% monoE
tff(fact_3114_monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> order_mono(A,B,F) ) ) ).
% monoI
tff(fact_3115_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_mono(A,B,F)
<=> ! [X4: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,F,Y3)) ) ) ) ).
% mono_def
tff(fact_3116_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,M2)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),M2),N)) ) ) ) ).
% min_of_mono
tff(fact_3117_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_mono(A,B,F)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% mono_invE
tff(fact_3118_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf(A)
& semilattice_inf(B) )
=> ! [F: fun(A,B),Aa2: A,Ba: A] :
( order_mono(A,B,F)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),Aa2),Ba))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,Aa2)),aa(A,B,F,Ba))) ) ) ).
% mono_inf
tff(fact_3119_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup(A)
& semilattice_sup(B) )
=> ! [F: fun(A,B),Aa2: A,Ba: A] :
( order_mono(A,B,F)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,Aa2)),aa(A,B,F,Ba))),aa(A,B,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba))) ) ) ).
% mono_sup
tff(fact_3120_Rings_Omono__mult,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_3121_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),Aa2: set(A)] :
( order_mono(A,B,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798349783984er_Max(A,Aa2)) = lattic643756798349783984er_Max(B,aa(set(A),set(B),image2(A,B,F),Aa2)) ) ) ) ) ) ).
% mono_Max_commute
tff(fact_3122_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),Aa2: set(A)] :
( order_mono(A,B,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798350308766er_Min(A,Aa2)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),image2(A,B,F),Aa2)) ) ) ) ) ) ).
% mono_Min_commute
tff(fact_3123_inv__image__def,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F: fun(A,B)] : inv_image(B,A,R2,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jk(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F))) ).
% inv_image_def
tff(fact_3124_finite_Omono,axiom,
! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_iy(fun(set(A),$o),fun(set(A),$o))) ).
% finite.mono
tff(fact_3125_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_jl(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).
% rp_inv_image_def
tff(fact_3126_cInf__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( condit1013018076250108175_below(A,S)
=> ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_jm(set(A),fun(A,$o),S))) ) ) ) ) ).
% cInf_cSup
tff(fact_3127_cSup__cInf,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( condit941137186595557371_above(A,S)
=> ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_jn(set(A),fun(A,$o),S))) ) ) ) ) ).
% cSup_cInf
tff(fact_3128_flip__pred,axiom,
! [A: $tType,B: $tType,Aa2: set(product_prod(A,B)),R: fun(B,fun(A,$o))] :
( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),Aa2),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),conversep(B,A,R))))
=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),set(product_prod(B,A)),image2(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_hd(A,fun(B,product_prod(B,A))))),Aa2)),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),R))) ) ).
% flip_pred
tff(fact_3129_prod__set__simps_I2_J,axiom,
! [B: $tType,A: $tType,X: B,Y: A] : basic_snds(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))) ).
% prod_set_simps(2)
tff(fact_3130_ball__empty,axiom,
! [A: $tType,P: fun(A,$o),X2: A] :
( member(A,X2,bot_bot(set(A)))
=> aa(A,$o,P,X2) ) ).
% ball_empty
tff(fact_3131_mono__compose,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Q: fun(A,fun(B,C)),F: fun(D,B)] :
( order_mono(A,fun(B,C),Q)
=> order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_jo(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F)) ) ) ).
% mono_compose
tff(fact_3132_Chains__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_jp(set(product_prod(A,A)),fun(set(A),$o),R2)) ).
% Chains_def
tff(fact_3133_refl__on__def_H,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( refl_on(A,Aa2,R2)
<=> ( ! [X4: product_prod(A,A)] :
( member(product_prod(A,A),X4,R2)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_jq(set(A),fun(A,fun(A,$o)),Aa2)),X4) )
& ! [X4: A] :
( member(A,X4,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2) ) ) ) ).
% refl_on_def'
tff(fact_3134_UnderS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : order_UnderS(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jr(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ).
% UnderS_def
tff(fact_3135_Under__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : order_Under(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_js(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ).
% Under_def
tff(fact_3136_prod__set__defs_I2_J,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B)] : basic_snds(A,B,X2) = aa(set(B),set(B),insert2(B,aa(product_prod(A,B),B,product_snd(A,B),X2)),bot_bot(set(B))) ).
% prod_set_defs(2)
tff(fact_3137_Above__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : order_Above(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jt(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ).
% Above_def
tff(fact_3138_min__ext__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_ju(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).
% min_ext_def
tff(fact_3139_bex__empty,axiom,
! [A: $tType,P: fun(A,$o)] :
~ ? [X2: A] :
( member(A,X2,bot_bot(set(A)))
& aa(A,$o,P,X2) ) ).
% bex_empty
tff(fact_3140_Image__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S2: set(B)] : image(B,A,R2,S2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_jv(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S2)) ).
% Image_def
tff(fact_3141_max__extp_Ocases,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
=> ~ ( aa(set(A),$o,finite_finite(A),A1)
=> ( aa(set(A),$o,finite_finite(A),A22)
=> ( ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
=> ~ ! [X2: A] :
( member(A,X2,A1)
=> ? [Xa4: A] :
( member(A,Xa4,A22)
& aa(A,$o,aa(A,fun(A,$o),R,X2),Xa4) ) ) ) ) ) ) ).
% max_extp.cases
tff(fact_3142_max__extp_Osimps,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
<=> ( aa(set(A),$o,finite_finite(A),A1)
& aa(set(A),$o,finite_finite(A),A22)
& ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
& ! [X4: A] :
( member(A,X4,A1)
=> ? [Xa2: A] :
( member(A,Xa2,A22)
& aa(A,$o,aa(A,fun(A,$o),R,X4),Xa2) ) ) ) ) ).
% max_extp.simps
tff(fact_3143_max__extp_Omax__extI,axiom,
! [A: $tType,X5: set(A),Y5: set(A),R: fun(A,fun(A,$o))] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( aa(set(A),$o,finite_finite(A),Y5)
=> ( ( Y5 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
=> ( ! [X3: A] :
( member(A,X3,X5)
=> ? [Xa3: A] :
( member(A,Xa3,Y5)
& aa(A,$o,aa(A,fun(A,$o),R,X3),Xa3) ) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),X5),Y5) ) ) ) ) ).
% max_extp.max_extI
tff(fact_3144_max__ext__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_jw(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).
% max_ext_eq
tff(fact_3145_AboveS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] : order_AboveS(A,R2,Aa2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_jx(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),Aa2)) ).
% AboveS_def
tff(fact_3146_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] : basic_fsts(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).
% prod_set_simps(1)
tff(fact_3147_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
<=> ( transitive_acyclic(A,R2)
& ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2)) ) ) ).
% acyclic_insert
tff(fact_3148_irrefl__tranclI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R2)),transitive_rtrancl(A,R2)) = bot_bot(set(product_prod(A,A))) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R2)) ) ).
% irrefl_tranclI
tff(fact_3149_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R2))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R2) ) ).
% converse_iff
tff(fact_3150_converse__empty,axiom,
! [B: $tType,A: $tType] : converse(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(product_prod(A,B))) ).
% converse_empty
tff(fact_3151_acyclic__empty,axiom,
! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).
% acyclic_empty
tff(fact_3152_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),converse(A,B,R2)) ) ).
% converseI
tff(fact_3153_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod(A,B),R2: set(product_prod(B,A))] :
( member(product_prod(A,B),Yx,converse(B,A,R2))
=> ~ ! [X3: B,Y2: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X3) )
=> ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y2),R2) ) ) ).
% converseE
tff(fact_3154_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R2))
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R2) ) ).
% converseD
tff(fact_3155_converse_Osimps,axiom,
! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
<=> ? [A5: B,B4: A] :
( ( A1 = B4 )
& ( A22 = A5 )
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B4),R2) ) ) ).
% converse.simps
tff(fact_3156_converse_Ocases,axiom,
! [A: $tType,B: $tType,A1: A,A22: B,R2: set(product_prod(B,A))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),converse(B,A,R2))
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A22),A1),R2) ) ).
% converse.cases
tff(fact_3157_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,R2))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,converse(A,A,R2))) ) ).
% rtrancl_converseI
tff(fact_3158_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,converse(A,A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,R2)) ) ).
% rtrancl_converseD
tff(fact_3159_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),converse(A,A,transitive_trancl(A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,converse(A,A,R2))) ) ).
% trancl_converseI
tff(fact_3160_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_trancl(A,converse(A,A,R2)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),converse(A,A,transitive_trancl(A,R2))) ) ).
% trancl_converseD
tff(fact_3161_converse__unfold,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : converse(B,A,R2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_jy(set(product_prod(B,A)),fun(A,fun(B,$o)),R2))) ).
% converse_unfold
tff(fact_3162_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),conversep(B,A,aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),converse(B,A,R2)) ) ).
% conversep_converse_eq
tff(fact_3163_acyclicI__order,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [R2: set(product_prod(A,A)),F: fun(A,B)] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,B3)),aa(A,B,F,A4)) )
=> transitive_acyclic(A,R2) ) ) ).
% acyclicI_order
tff(fact_3164_cyclicE,axiom,
! [A: $tType,G: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,G)
=> ~ ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,G)) ) ).
% cyclicE
tff(fact_3165_acyclic__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( transitive_acyclic(A,R2)
<=> ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),transitive_trancl(A,R2)) ) ).
% acyclic_def
tff(fact_3166_acyclicI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R2))
=> transitive_acyclic(A,R2) ) ).
% acyclicI
tff(fact_3167_converse__def,axiom,
! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : converse(A,B,X2) = aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),X2)))) ).
% converse_def
tff(fact_3168_AboveS__disjoint,axiom,
! [A: $tType,Aa2: 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)),Aa2),order_AboveS(A,R2,Aa2)) = bot_bot(set(A)) ).
% AboveS_disjoint
tff(fact_3169_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set(product_prod(A,A)),X: A,Y: A] :
( transitive_acyclic(A,G)
=> ( ~ transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),G))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),transitive_rtrancl(A,G)) ) ) ).
% acyclic_insert_cyclic
tff(fact_3170_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] : basic_fsts(A,B,X2) = aa(set(A),set(A),insert2(A,aa(product_prod(A,B),A,product_fst(A,B),X2)),bot_bot(set(A))) ).
% prod_set_defs(1)
tff(fact_3171_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( order_AboveS(A,R2,Ba) != bot_bot(set(A)) )
=> ( member(A,B2,Ba)
=> ( ( bNF_Wellorder_wo_suc(A,R2,Ba) != B2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R2,Ba)),R2) ) ) ) ) ) ).
% wo_rel.suc_greater
tff(fact_3172_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( order_AboveS(A,R2,Ba) != bot_bot(set(A)) )
=> member(A,bNF_Wellorder_wo_suc(A,R2,Ba),field2(A,R2)) ) ) ) ).
% wo_rel.suc_inField
tff(fact_3173_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( ( order_AboveS(A,R2,Ba) != bot_bot(set(A)) )
=> member(A,bNF_Wellorder_wo_suc(A,R2,Ba),order_AboveS(A,R2,Ba)) ) ) ) ).
% wo_rel.suc_AboveS
tff(fact_3174_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Ba: set(A),A3: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),field2(A,R2))
=> ( member(A,A3,order_AboveS(A,R2,Ba))
=> ( ! [A11: A] :
( member(A,A11,order_AboveS(A,R2,Ba))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A11),R2) )
=> ( A3 = bNF_Wellorder_wo_suc(A,R2,Ba) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
tff(fact_3175_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),insert2(A,A5),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_jz(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A5)))) ) ) ).
% trans_wf_iff
tff(fact_3176_trans__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
<=> ! [X4: A,Y3: A,Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z4),R2) ) ) ) ).
% trans_def
tff(fact_3177_transI,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3),R2) ) )
=> trans(A,R2) ) ).
% transI
tff(fact_3178_transE,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ).
% transE
tff(fact_3179_transD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ).
% transD
tff(fact_3180_trans__empty,axiom,
! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).
% trans_empty
tff(fact_3181_trans__singleton,axiom,
! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).
% trans_singleton
tff(fact_3182_trans__join,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( trans(A,R2)
<=> ! [X4: product_prod(A,A)] :
( member(product_prod(A,A),X4,R2)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_kb(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4) ) ) ).
% trans_join
tff(fact_3183_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,Ba: set(A)] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( member(A,A3,order_AboveS(A,R2,Ba))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R2,Ba)),A3),R2) ) ) ).
% wo_rel.suc_least_AboveS
tff(fact_3184_Image__INT__eq,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),Aa2: set(C),Ba: fun(C,set(B))] :
( single_valued(A,B,converse(B,A,R2))
=> ( ( Aa2 != bot_bot(set(C)) )
=> ( image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),Ba),Aa2))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_kc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),Ba)),Aa2)) ) ) ) ).
% Image_INT_eq
tff(fact_3185_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R2)
=> ( order_ofilter(A,R2,Aa2)
=> ( ( order_AboveS(A,R2,Aa2) != bot_bot(set(A)) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R2,Aa2)),R2)
=> ( ( B2 != bNF_Wellorder_wo_suc(A,R2,Aa2) )
=> member(A,B2,Aa2) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
tff(fact_3186_wf__finite__segments,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
=> ( trans(A,R2)
=> ( ! [X3: A] : aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_kd(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),X3)))
=> wf(A,R2) ) ) ) ).
% wf_finite_segments
tff(fact_3187_init__seg__of__def,axiom,
! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),$o),aTP_Lamp_ke(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ).
% init_seg_of_def
tff(fact_3188_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
( ( nat_prod_decode_aux(X,Xa) = Y )
=> ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa))
=> ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X)))) )
=> ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa)) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_3189_single__valued__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B))] :
( single_valued(A,B,R2)
<=> ! [X4: A,Y3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R2)
=> ! [Z4: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R2)
=> ( Y3 = Z4 ) ) ) ) ).
% single_valued_def
tff(fact_3190_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( ! [X3: A,Y2: B,Z3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3),R2)
=> ( Y2 = Z3 ) ) )
=> single_valued(A,B,R2) ) ).
% single_valuedI
tff(fact_3191_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,B)),X: A,Y: B,Z2: B] :
( single_valued(A,B,R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R2)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2),R2)
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
tff(fact_3192_irrefl__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( irrefl(A,R2)
<=> ! [A5: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R2) ) ).
% irrefl_def
tff(fact_3193_irreflI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [A4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R)
=> irrefl(A,R) ) ).
% irreflI
tff(fact_3194_single__valued__empty,axiom,
! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).
% single_valued_empty
tff(fact_3195_trans__init__seg__of,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A)),T2: set(product_prod(A,A))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S2),init_seg_of(A))
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S2),T2),init_seg_of(A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),T2),init_seg_of(A)) ) ) ).
% trans_init_seg_of
tff(fact_3196_antisym__init__seg__of,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S2),init_seg_of(A))
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S2),R2),init_seg_of(A))
=> ( R2 = S2 ) ) ) ).
% antisym_init_seg_of
tff(fact_3197_refl__on__init__seg__of,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),init_seg_of(A)) ).
% refl_on_init_seg_of
tff(fact_3198_single__valued__confluent,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( single_valued(A,A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),transitive_rtrancl(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R2))
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y),transitive_rtrancl(A,R2)) ) ) ) ) ).
% single_valued_confluent
tff(fact_3199_prod__decode__aux_Ocases,axiom,
! [X: product_prod(nat,nat)] :
~ ! [K2: nat,M3: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M3) ).
% prod_decode_aux.cases
tff(fact_3200_Chains__init__seg__of__Union,axiom,
! [A: $tType,R: set(set(product_prod(A,A))),R2: set(product_prod(A,A))] :
( member(set(set(product_prod(A,A))),R,chains(set(product_prod(A,A)),init_seg_of(A)))
=> ( member(set(product_prod(A,A)),R2,R)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)),init_seg_of(A)) ) ) ).
% Chains_init_seg_of_Union
tff(fact_3201_initial__segment__of__Diff,axiom,
! [A: $tType,P5: set(product_prod(A,A)),Q3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P5),Q3),init_seg_of(A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P5),S2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Q3),S2)),init_seg_of(A)) ) ).
% initial_segment_of_Diff
tff(fact_3202_prod__decode__aux_Osimps,axiom,
! [K: nat,M2: nat] :
nat_prod_decode_aux(K,M2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M2)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,K)))) ).
% prod_decode_aux.simps
tff(fact_3203_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
( ( nat_prod_decode_aux(X,Xa) = Y )
=> ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)),nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X)))) ) ) ).
% prod_decode_aux.elims
tff(fact_3204_ofilter__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,Aa2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),field2(A,R2))
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).
% ofilter_ordLess
tff(fact_3205_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),Ba: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,Aa2)
=> ( order_ofilter(A,R2,Ba)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Aa2),Ba)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))),aa(set(product_prod(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,Ba,aTP_Lamp_in(set(A),fun(A,set(A)),Ba)))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ) ).
% ofilter_subset_ordLess
tff(fact_3206_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),Ba: set(A)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_ofilter(A,R2,Aa2)
=> ( order_ofilter(A,R2,Ba)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))),aa(set(product_prod(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,Ba,aTP_Lamp_in(set(A),fun(A,set(A)),Ba)))),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).
% ofilter_subset_ordLeq
tff(fact_3207_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R2))
<=> single_valued(A,B,R2) ) ).
% single_valuedp_single_valued_eq
tff(fact_3208_set__encode__empty,axiom,
nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).
% set_encode_empty
tff(fact_3209_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordLeq(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_Wellorder_ordLeq(A,C)) ) ) ).
% ordLeq_transitive
tff(fact_3210_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ~ member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A)) ) ).
% not_ordLess_ordLeq
tff(fact_3211_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B)) ) ).
% ordLess_imp_ordLeq
tff(fact_3212_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_We4044943003108391690rdLess(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_We4044943003108391690rdLess(A,C)) ) ) ).
% ordLeq_ordLess_trans
tff(fact_3213_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordLeq(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_We4044943003108391690rdLess(A,C)) ) ) ).
% ordLess_ordLeq_trans
tff(fact_3214_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_We4044943003108391690rdLess(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_We4044943003108391690rdLess(A,C)) ) ) ).
% ordLess_transitive
tff(fact_3215_ordLess__irreflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),bNF_We4044943003108391690rdLess(A,A)) ).
% ordLess_irreflexive
tff(fact_3216_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( ~ member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_We4044943003108391690rdLess(B,A))
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).
% not_ordLess_iff_ordLeq
tff(fact_3217_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( ~ member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A))
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ).
% not_ordLeq_iff_ordLess
tff(fact_3218_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
| member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).
% ordLess_or_ordLeq
tff(fact_3219_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : single_valuedp(A,B,bot_bot(fun(A,fun(B,$o)))) ).
% single_valuedp_bot
tff(fact_3220_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> ( order_well_order_on(A,field2(A,R2),R2)
& order_well_order_on(B,field2(B,R4),R4) ) ) ).
% ordLeq_Well_order_simp
tff(fact_3221_ordLeq__reflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),bNF_Wellorder_ordLeq(A,A)) ) ).
% ordLeq_reflexive
tff(fact_3222_ordLeq__total,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
| member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).
% ordLeq_total
tff(fact_3223_exists__minim__Well__order,axiom,
! [A: $tType,R: set(set(product_prod(A,A)))] :
( ( R != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
=> order_well_order_on(A,field2(A,X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
& ! [Xa3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa3,R)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).
% exists_minim_Well_order
tff(fact_3224_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( ~ aa(set(B),$o,finite_finite(B),field2(B,R4))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).
% finite_ordLess_infinite
tff(fact_3225_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
<=> ! [X4: A] :
( member(A,X4,field2(A,R2))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_kf(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),R4),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
tff(fact_3226_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)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,A3),aa(A,fun(A,set(A)),aTP_Lamp_kf(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A3)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ).
% underS_Restr_ordLess
tff(fact_3227_aboveS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_aboveS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_kg(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).
% aboveS_def
tff(fact_3228_relInvImage__def,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R: set(product_prod(B,B)),F: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,Aa2,R,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_kh(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Aa2),R),F)) ).
% relInvImage_def
tff(fact_3229_normalize__stable,axiom,
! [Q3: int,P5: int] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3)
=> ( algebr8660921524188924756oprime(int,P5,Q3)
=> ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) ) ) ) ).
% normalize_stable
tff(fact_3230_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_3231_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_3232_coprime__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,A3)
<=> dvd_dvd(A,A3,one_one(A)) ) ) ).
% coprime_self
tff(fact_3233_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_3234_coprime__0__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,zero_zero(A),A3)
<=> dvd_dvd(A,A3,one_one(A)) ) ) ).
% coprime_0_left_iff
tff(fact_3235_coprime__0__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,zero_zero(A))
<=> dvd_dvd(A,A3,one_one(A)) ) ) ).
% coprime_0_right_iff
tff(fact_3236_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))
<=> ( dvd_dvd(A,C2,one_one(A))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_3237_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))
<=> ( dvd_dvd(A,C2,one_one(A))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_3238_is__unit__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( 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_3239_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).
% coprime_1_left
tff(fact_3240_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).
% coprime_1_right
tff(fact_3241_underS__I,axiom,
! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
( ( I != J )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R)
=> member(A,I,order_underS(A,R,J)) ) ) ).
% underS_I
tff(fact_3242_underS__E,axiom,
! [A: $tType,I: A,R: set(product_prod(A,A)),J: A] :
( member(A,I,order_underS(A,R,J))
=> ( ( I != J )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R) ) ) ).
% underS_E
tff(fact_3243_underS__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_underS(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ki(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).
% underS_def
tff(fact_3244_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_3245_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_3246_coprime__doff__one__right,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).
% coprime_doff_one_right
tff(fact_3247_coprime__diff__one__left,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A)),A3) ) ).
% coprime_diff_one_left
tff(fact_3248_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> dvd_dvd(A,A3,B2) ) ) ) ).
% coprime_dvd_mult_right_iff
tff(fact_3249_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> dvd_dvd(A,A3,B2) ) ) ) ).
% coprime_dvd_mult_left_iff
tff(fact_3250_divides__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( dvd_dvd(A,A3,C2)
=> ( dvd_dvd(A,B2,C2)
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2) ) ) ) ) ).
% divides_mult
tff(fact_3251_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( dvd_dvd(A,B2,one_one(A))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_right_imp_coprime
tff(fact_3252_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( dvd_dvd(A,A3,one_one(A))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_left_imp_coprime
tff(fact_3253_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( dvd_dvd(A,C2,A3)
=> ( dvd_dvd(A,C2,B2)
=> dvd_dvd(A,C2,one_one(A)) ) ) ) ) ).
% coprime_common_divisor
tff(fact_3254_coprime__absorb__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Y: A,X: A] :
( dvd_dvd(A,Y,X)
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> dvd_dvd(A,Y,one_one(A)) ) ) ) ).
% coprime_absorb_right
tff(fact_3255_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,D3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,D3)
=> ( ! [E2: A] :
( ~ dvd_dvd(A,E2,one_one(A))
=> ( dvd_dvd(A,E2,A3)
=> ( dvd_dvd(A,E2,B2)
=> dvd_dvd(A,E2,C2) ) ) )
=> ( ! [E2: A] :
( ~ dvd_dvd(A,E2,one_one(A))
=> ( dvd_dvd(A,E2,A3)
=> ( dvd_dvd(A,E2,B2)
=> dvd_dvd(A,E2,D3) ) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% coprime_imp_coprime
tff(fact_3256_coprime__absorb__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,Y: A] :
( dvd_dvd(A,X,Y)
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> dvd_dvd(A,X,one_one(A)) ) ) ) ).
% coprime_absorb_left
tff(fact_3257_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( dvd_dvd(A,C2,A3)
=> ( dvd_dvd(A,C2,B2)
=> ( ~ dvd_dvd(A,C2,one_one(A))
=> ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% not_coprimeI
tff(fact_3258_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ~ algebr8660921524188924756oprime(A,A3,B2)
=> ~ ! [C3: A] :
( dvd_dvd(A,C3,A3)
=> ( dvd_dvd(A,C3,B2)
=> dvd_dvd(A,C3,one_one(A)) ) ) ) ) ).
% not_coprimeE
tff(fact_3259_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ! [C4: A] :
( dvd_dvd(A,C4,A3)
=> ( dvd_dvd(A,C4,B2)
=> dvd_dvd(A,C4,one_one(A)) ) ) ) ) ).
% coprime_def
tff(fact_3260_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ! [C3: A] :
( dvd_dvd(A,C3,A3)
=> ( dvd_dvd(A,C3,B2)
=> dvd_dvd(A,C3,one_one(A)) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprimeI
tff(fact_3261_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_3262_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_3263_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_3264_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_3265_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_3266_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_3267_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_3268_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_3269_quotient__of__coprime,axiom,
! [R2: rat,P5: int,Q3: int] :
( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> algebr8660921524188924756oprime(int,P5,Q3) ) ).
% quotient_of_coprime
tff(fact_3270_normalize__coprime,axiom,
! [R2: product_prod(int,int),P5: int,Q3: int] :
( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> algebr8660921524188924756oprime(int,P5,Q3) ) ).
% normalize_coprime
tff(fact_3271_underS__empty,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
( ~ member(A,A3,field2(A,R2))
=> ( order_underS(A,R2,A3) = bot_bot(set(A)) ) ) ).
% underS_empty
tff(fact_3272_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_3273_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) )
=> ? [A11: A,B6: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A11),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,A11,B6) ) ) ) ).
% gcd_coprime_exists
tff(fact_3274_gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,A6: A,B5: 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),A6),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),B5),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> algebr8660921524188924756oprime(A,A6,B5) ) ) ) ) ).
% gcd_coprime
tff(fact_3275_underS__Field3,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( ( field2(A,R2) != bot_bot(set(A)) )
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),field2(A,R2)) ) ).
% underS_Field3
tff(fact_3276_underS__incl__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> ( member(A,B2,field2(A,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B2))
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ) ) ).
% underS_incl_iff
tff(fact_3277_underS__incr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R2)
=> ( antisym(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B2)) ) ) ) ).
% underS_incr
tff(fact_3278_ordLess__iff__ordIso__Restr,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( order_well_order_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_We4044943003108391690rdLess(B,A))
<=> ? [X4: A] :
( member(A,X4,field2(A,R2))
& member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),aa(set(product_prod(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_kf(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),X4)))),bNF_Wellorder_ordIso(B,A)) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
tff(fact_3279_Refl__under__underS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( refl_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> ( order_under(A,R2,A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R2,A3)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ) ) ).
% Refl_under_underS
tff(fact_3280_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),Ba: set(A)] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F),Ba)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)) ) ) ) ) ).
% semilattice_order_set.subset_imp
tff(fact_3281_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( antisymp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> antisym(A,R2) ) ).
% antisymp_antisym_eq
tff(fact_3282_Max_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( lattic643756798349783984er_Max(A,Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ) ).
% Max.remove
tff(fact_3283_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_3284_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_3285_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_3286_max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
& aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).
% max.bounded_iff
tff(fact_3287_max_Oabsorb2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb2
tff(fact_3288_max_Oabsorb1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb1
tff(fact_3289_max__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2)
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
& aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z2) ) ) ) ).
% max_less_iff_conj
tff(fact_3290_max_Oabsorb4,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb4
tff(fact_3291_max_Oabsorb3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb3
tff(fact_3292_max__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).
% max_bot
tff(fact_3293_max__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).
% max_bot2
tff(fact_3294_max__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).
% max_top
tff(fact_3295_max__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).
% max_top2
tff(fact_3296_max__min__same_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).
% max_min_same(1)
tff(fact_3297_max__min__same_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).
% max_min_same(2)
tff(fact_3298_max__min__same_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).
% max_min_same(3)
tff(fact_3299_max__min__same_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).
% max_min_same(4)
tff(fact_3300_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_3301_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_3302_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_3303_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_3304_Max__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,Aa2)) ) ) ) ) ).
% Max_insert
tff(fact_3305_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_3306_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_3307_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_3308_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_3309_sup__max,axiom,
! [A: $tType] :
( ( semilattice_sup(A)
& linorder(A) )
=> ( sup_sup(A) = ord_max(A) ) ) ).
% sup_max
tff(fact_3310_complete__linorder__sup__max,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ( sup_sup(A) = ord_max(A) ) ) ).
% complete_linorder_sup_max
tff(fact_3311_max__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,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),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% max_diff_distrib_left
tff(fact_3312_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_3313_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_3314_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_3315_less__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
| aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).
% less_max_iff_disj
tff(fact_3316_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3) ) ) ) ).
% max.strict_boundedE
tff(fact_3317_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_3318_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).
% max.strict_coboundedI1
tff(fact_3319_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).
% max.strict_coboundedI2
tff(fact_3320_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_3321_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_3322_max_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,D3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
=> aa(A,$o,aa(A,fun(A,$o),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_3323_max_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).
% max.orderE
tff(fact_3324_max_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3) ) ) ).
% max.orderI
tff(fact_3325_max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)
=> ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3) ) ) ) ).
% max.boundedE
tff(fact_3326_max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) ) ) ) ).
% max.boundedI
tff(fact_3327_max_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_3328_max_Ocobounded1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).
% max.cobounded1
tff(fact_3329_max_Ocobounded2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).
% max.cobounded2
tff(fact_3330_le__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
<=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X)
| aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y) ) ) ) ).
% le_max_iff_disj
tff(fact_3331_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),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_3332_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),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_3333_max_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).
% max.coboundedI1
tff(fact_3334_max_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ) ).
% max.coboundedI2
tff(fact_3335_max__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] :
aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B2),B2,A3) ) ).
% max_def
tff(fact_3336_max__absorb1,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).
% max_absorb1
tff(fact_3337_max__absorb2,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).
% max_absorb2
tff(fact_3338_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordIso(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_Wellorder_ordIso(A,C)) ) ) ).
% ordIso_transitive
tff(fact_3339_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordIso(B,A)) ) ).
% ordIso_symmetric
tff(fact_3340_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M2: A,N: A] :
( order_mono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,M2)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),M2),N)) ) ) ) ).
% max_of_mono
tff(fact_3341_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_max(A)) ) ).
% max.abel_semigroup_axioms
tff(fact_3342_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_max(A)) ) ).
% max.semilattice_axioms
tff(fact_3343_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_max(A)) ) ).
% max.semigroup_axioms
tff(fact_3344_under__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_under(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_kd(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).
% under_def
tff(fact_3345_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_3346_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_3347_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
<=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
& member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).
% ordIso_iff_ordLeq
tff(fact_3348_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B)) ) ).
% ordIso_imp_ordLeq
tff(fact_3349_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordLeq(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_Wellorder_ordLeq(A,C)) ) ) ).
% ordIso_ordLeq_trans
tff(fact_3350_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordIso(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_Wellorder_ordLeq(A,C)) ) ) ).
% ordLeq_ordIso_trans
tff(fact_3351_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ~ member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B)) ) ).
% not_ordLess_ordIso
tff(fact_3352_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_We4044943003108391690rdLess(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_We4044943003108391690rdLess(A,C)) ) ) ).
% ordIso_ordLess_trans
tff(fact_3353_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),R6: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R4),R6),bNF_Wellorder_ordIso(B,C))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R2),R6),bNF_We4044943003108391690rdLess(A,C)) ) ) ).
% ordLess_ordIso_trans
tff(fact_3354_Sup__fin__def,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ( lattic5882676163264333800up_fin(A) = lattic1715443433743089157tice_F(A,sup_sup(A)) ) ) ).
% Sup_fin_def
tff(fact_3355_antisym__bot,axiom,
! [A: $tType] : antisymp(A,bot_bot(fun(A,fun(A,$o)))) ).
% antisym_bot
tff(fact_3356_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).
% max_nat.semilattice_neutr_axioms
tff(fact_3357_finite__well__order__on__ordIso,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( order_well_order_on(A,Aa2,R2)
=> ( order_well_order_on(A,Aa2,R4)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R4),bNF_Wellorder_ordIso(A,A)) ) ) ) ).
% finite_well_order_on_ordIso
tff(fact_3358_ordIso__reflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),bNF_Wellorder_ordIso(A,A)) ) ).
% ordIso_reflexive
tff(fact_3359_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
<=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
| member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B)) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
tff(fact_3360_min__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A,P5: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P5) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P5),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P5)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P5)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P5)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P5))) ) ).
% min_mult_distrib_right
tff(fact_3361_max__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A,P5: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P5) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P5),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P5)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P5)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P5)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P5))) ) ).
% max_mult_distrib_right
tff(fact_3362_min__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P5: A,X: A,Y: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),P5),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P5),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P5),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P5),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P5),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P5),Y))) ) ).
% min_mult_distrib_left
tff(fact_3363_max__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P5: A,X: A,Y: A] :
aa(A,A,aa(A,fun(A,A),times_times(A),P5),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P5),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P5),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P5),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P5),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P5),Y))) ) ).
% max_mult_distrib_left
tff(fact_3364_Sup__insert__finite,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),S)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),insert2(A,X),S)) = $ite(S = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).
% Sup_insert_finite
tff(fact_3365_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2))
=> ! [A9: A] :
( member(A,A9,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A9) ) ) ) ) ) ).
% semilattice_order_set.boundedE
tff(fact_3366_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),A4) )
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)) ) ) ) ) ).
% semilattice_order_set.boundedI
tff(fact_3367_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Aa2: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,X),X4) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
tff(fact_3368_under__incr,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A3)),order_under(A,R2,B2)) ) ) ).
% under_incr
tff(fact_3369_hom__Max__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H2: fun(A,A),N3: set(A)] :
( ! [X3: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H2,X3)),aa(A,A,H2,Y2))
=> ( aa(set(A),$o,finite_finite(A),N3)
=> ( ( N3 != bot_bot(set(A)) )
=> ( aa(A,A,H2,lattic643756798349783984er_Max(A,N3)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),image2(A,A,H2),N3)) ) ) ) ) ) ).
% hom_Max_commute
tff(fact_3370_Max_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,Ba)),lattic643756798349783984er_Max(A,Aa2)) = lattic643756798349783984er_Max(A,Aa2) ) ) ) ) ) ).
% Max.subset
tff(fact_3371_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,Aa2)) ) ) ) ) ) ).
% Max.insert_not_elem
tff(fact_3372_Max_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,lattic643756798349783984er_Max(A,Aa2),Aa2) ) ) ) ) ).
% Max.closed
tff(fact_3373_Max_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),ord_max(A),lattic643756798349783984er_Max(A,Aa2)),lattic643756798349783984er_Max(A,Ba)) ) ) ) ) ) ) ).
% Max.union
tff(fact_3374_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R4: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),R2),bNF_Wellorder_ordLeq(A,B))
<=> ? [P6: set(product_prod(B,B))] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),field2(B,P6)),field2(B,R2))
& member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),P6),bNF_Wellorder_ordIso(A,B))
& member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).
% internalize_ordLeq
tff(fact_3375_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R4: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),R2),bNF_We4044943003108391690rdLess(A,B))
<=> ? [P6: set(product_prod(B,B))] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),field2(B,P6)),field2(B,R2))
& member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R4),P6),bNF_Wellorder_ordIso(A,B))
& member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R2),bNF_We4044943003108391690rdLess(B,B)) ) ) ).
% internalize_ordLess
tff(fact_3376_Max_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),ord_max(A),X),lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ).
% Max.insert_remove
tff(fact_3377_semilattice__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ) ).
% semilattice_set.remove
tff(fact_3378_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),insert2(A,X),Aa2)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = bot_bot(set(A)),X,aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))))) ) ) ) ).
% semilattice_set.insert_remove
tff(fact_3379_semilattice__set_Ounion,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),Ba: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Ba)) ) ) ) ) ) ) ).
% semilattice_set.union
tff(fact_3380_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ~ member(A,X,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
tff(fact_3381_semilattice__set_Oinsert,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)) ) ) ) ) ).
% semilattice_set.insert
tff(fact_3382_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> lattic149705377957585745ce_set(A,sup_sup(A)) ) ).
% Sup_fin.semilattice_set_axioms
tff(fact_3383_semilattice__set_Osingleton,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = X ) ) ).
% semilattice_set.singleton
tff(fact_3384_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),H2: fun(A,A),N3: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ! [X3: A,Y2: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),F,X3),Y2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,H2,X3)),aa(A,A,H2,Y2))
=> ( aa(set(A),$o,finite_finite(A),N3)
=> ( ( N3 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic1715443433743089157tice_F(A,F),N3)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),image2(A,A,H2),N3)) ) ) ) ) ) ).
% semilattice_set.hom_commute
tff(fact_3385_semilattice__set_Osubset,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A),Ba: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(A)) )
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),Aa2)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),Ba)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2) ) ) ) ) ) ).
% semilattice_set.subset
tff(fact_3386_semilattice__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Aa2: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),F,X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,aa(set(A),A,lattic1715443433743089157tice_F(A,F),Aa2),Aa2) ) ) ) ) ).
% semilattice_set.closed
tff(fact_3387_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,Aa2: set(B),F: fun(B,A)] : bind2(B,A,Aa2,aTP_Lamp_fo(fun(B,A),fun(B,set(A)),F)) = aa(set(B),set(A),image2(B,A,F),Aa2) ).
% bind_singleton_conv_image
tff(fact_3388_set__decode__zero,axiom,
nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).
% set_decode_zero
tff(fact_3389_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_3390_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_3391_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,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).
% power_int_minus_mult
tff(fact_3392_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_3393_empty__bind,axiom,
! [B: $tType,A: $tType,F: fun(B,set(A))] : bind2(B,A,bot_bot(set(B)),F) = bot_bot(set(A)) ).
% empty_bind
tff(fact_3394_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,W2: num,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W2)),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),W2),M2)) ) ).
% power_int_mult_distrib_numeral2
tff(fact_3395_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( field(A)
=> ! [W2: 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),W2)),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),M2)),power_int(A,Y,M2)) ) ).
% power_int_mult_distrib_numeral1
tff(fact_3396_power__int__0__right,axiom,
! [A: $tType] :
( ( inverse(A)
& power(A) )
=> ! [X: A] : power_int(A,X,zero_zero(int)) = one_one(A) ) ).
% power_int_0_right
tff(fact_3397_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_3398_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_3399_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_3400_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_3401_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_3402_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_3403_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_3404_bind__const,axiom,
! [B: $tType,A: $tType,Aa2: set(B),Ba: set(A)] :
bind2(B,A,Aa2,aTP_Lamp_fh(set(A),fun(B,set(A)),Ba)) = $ite(Aa2 = bot_bot(set(B)),bot_bot(set(A)),Ba) ).
% bind_const
tff(fact_3405_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( bind2(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)) = Ba ) ) ).
% nonempty_bind_const
tff(fact_3406_power__int__0__left__If,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M2: int] :
power_int(A,zero_zero(A),M2) = $ite(M2 = zero_zero(int),one_one(A),zero_zero(A)) ) ).
% power_int_0_left_If
tff(fact_3407_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N3: int,A3: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N3)) ) ) ) ).
% power_int_increasing
tff(fact_3408_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N3: int,A3: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N3)) ) ) ) ).
% power_int_strict_increasing
tff(fact_3409_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_3410_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)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A3),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A3)) ) ).
% power_int_minus_one_diff_commute
tff(fact_3411_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N3: int,A3: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N3)),power_int(A,A3,N)) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_3412_one__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,X,N)) ) ) ) ).
% one_le_power_int
tff(fact_3413_one__less__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),N)
=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A3,N)) ) ) ) ).
% one_less_power_int
tff(fact_3414_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_3415_power__int__minus__left__distrib,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( division_ring(C)
& one(A)
& uminus(A) )
=> ! [X: B,A3: C,N: int] :
( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),X)
=> ( power_int(C,aa(C,C,uminus_uminus(C),A3),N) = aa(C,C,aa(C,fun(C,C),times_times(C),power_int(C,aa(C,C,uminus_uminus(C),one_one(C)),N)),power_int(C,A3,N)) ) ) ) ).
% power_int_minus_left_distrib
tff(fact_3416_power__int__le__one,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,N)),one_one(A)) ) ) ) ) ).
% power_int_le_one
tff(fact_3417_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N3: int,A3: A] :
( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),N),N3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
=> ( ( ( A3 != zero_zero(A) )
| ( N3 != zero_zero(int) )
| ( N = zero_zero(int) ) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N3)),power_int(A,A3,N)) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_3418_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M2: int,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,X,M2)),power_int(A,X,N))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M2),N) ) ) ) ) ).
% power_int_le_imp_le_exp
tff(fact_3419_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M2: int,N: int] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,X,M2)),power_int(A,X,N))
=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),M2),N) ) ) ) ) ).
% power_int_le_imp_less_exp
tff(fact_3420_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_3421_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Aa2: set(A),Ba: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))
<=> ( inj_on(A,B,F,Aa2)
& inj_on(A,B,F,Ba)
& ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),Ba))),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ba),Aa2))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_3422_quotient__def,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,Aa2,R2) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_kj(set(product_prod(A,A)),fun(A,set(set(A))),R2)),Aa2)) ).
% quotient_def
tff(fact_3423_Fract_Otransfer,axiom,
aa(fun(int,fun(int,rat)),$o,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),$o),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_el(int,fun(int,product_prod(int,int)))),fract) ).
% Fract.transfer
tff(fact_3424_mset__set__Union,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Ba)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,Aa2)),mset_set(A,Ba)) ) ) ) ) ).
% mset_set_Union
tff(fact_3425_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).
% inj_on_empty
tff(fact_3426_mset__set_Oempty,axiom,
! [A: $tType] : mset_set(A,bot_bot(set(A))) = zero_zero(multiset(A)) ).
% mset_set.empty
tff(fact_3427_quotient__is__empty2,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( ( bot_bot(set(set(A))) = equiv_quotient(A,Aa2,R2) )
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% quotient_is_empty2
tff(fact_3428_quotient__is__empty,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( ( equiv_quotient(A,Aa2,R2) = bot_bot(set(set(A))) )
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% quotient_is_empty
tff(fact_3429_quotient__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R2) = bot_bot(set(set(A))) ).
% quotient_empty
tff(fact_3430_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_3431_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,Aa2: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),insert2(A,A3),Aa2))
<=> ( inj_on(A,B,F,Aa2)
& ~ member(B,aa(A,B,F,A3),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ).
% inj_on_insert
tff(fact_3432_quotient__diff1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Aa2: set(A),A3: A] :
( inj_on(A,set(set(A)),aTP_Lamp_kk(set(product_prod(A,A)),fun(A,set(set(A))),R2),Aa2)
=> ( member(A,A3,Aa2)
=> ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))),R2) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),minus_minus(set(set(A))),equiv_quotient(A,Aa2,R2)),equiv_quotient(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))),R2)) ) ) ) ).
% quotient_diff1
tff(fact_3433_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_kl(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).
% inj_Pair(1)
tff(fact_3434_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_km(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).
% inj_Pair(2)
tff(fact_3435_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_kl(fun(A,B),fun(A,product_prod(A,B)),F),X5) ).
% inj_on_convol_ident
tff(fact_3436_inj__on__mult,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,Aa2: set(A)] :
( ( A3 != zero_zero(A) )
=> inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),Aa2) ) ) ).
% inj_on_mult
tff(fact_3437_funpow__times__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [F: fun(A,nat),X: A] : compow(fun(A,A),aa(A,nat,F,X),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F,X))) ) ).
% funpow_times_power
tff(fact_3438_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set(set(A)),F: fun(A,B)] :
( ( S != bot_bot(set(set(A))) )
=> ( ! [A7: set(A)] :
( member(set(A),A7,S)
=> inj_on(A,B,F,A7) )
=> inj_on(A,B,F,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ) ).
% inj_on_Inter
tff(fact_3439_inj__singleton,axiom,
! [A: $tType,Aa2: set(A)] : inj_on(A,set(A),aTP_Lamp_ff(A,set(A)),Aa2) ).
% inj_singleton
tff(fact_3440_quotient__of__eq,axiom,
! [A3: int,B2: int,P5: int,Q3: int] :
( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,P5),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).
% quotient_of_eq
tff(fact_3441_quotientE,axiom,
! [A: $tType,X5: set(A),Aa2: set(A),R2: set(product_prod(A,A))] :
( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ~ ! [X3: A] :
( ( X5 = image(A,A,R2,aa(set(A),set(A),insert2(A,X3),bot_bot(set(A)))) )
=> ~ member(A,X3,Aa2) ) ) ).
% quotientE
tff(fact_3442_quotientI,axiom,
! [A: $tType,X: A,Aa2: set(A),R2: set(product_prod(A,A))] :
( member(A,X,Aa2)
=> member(set(A),image(A,A,R2,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))),equiv_quotient(A,Aa2,R2)) ) ).
% quotientI
tff(fact_3443_swap__inj__on,axiom,
! [B: $tType,A: $tType,Aa2: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_hd(A,fun(B,product_prod(B,A)))),Aa2) ).
% swap_inj_on
tff(fact_3444_normalize__eq,axiom,
! [A3: int,B2: int,P5: int,Q3: int] :
( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P5),Q3) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,P5),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).
% normalize_eq
tff(fact_3445_mono__funpow,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Q: fun(A,A)] :
( order_mono(A,A,Q)
=> order_mono(nat,A,aTP_Lamp_kn(fun(A,A),fun(nat,A),Q)) ) ) ).
% mono_funpow
tff(fact_3446_inj__on__iff__surj,axiom,
! [B: $tType,A: $tType,Aa2: set(A),A13: set(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ? [F6: fun(A,B)] :
( inj_on(A,B,F6,Aa2)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),Aa2)),A13) )
<=> ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),A13) = Aa2 ) ) ).
% inj_on_iff_surj
tff(fact_3447_card__quotient__disjoint,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( inj_on(A,set(set(A)),aTP_Lamp_kk(set(product_prod(A,A)),fun(A,set(set(A))),R2),Aa2)
=> ( finite_card(set(A),equiv_quotient(A,Aa2,R2)) = finite_card(A,Aa2) ) ) ) ).
% card_quotient_disjoint
tff(fact_3448_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [F: fun(A,A),P5: A,K: nat] :
( order_mono(A,A,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F,P5)),P5)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),K,F),bot_bot(A))),P5) ) ) ) ).
% Kleene_iter_lpfp
tff(fact_3449_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( order_top(A)
=> ! [F: fun(A,A),P5: A,K: nat] :
( order_mono(A,A,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P5),aa(A,A,F,P5))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P5),aa(A,A,compow(fun(A,A),K,F),top_top(A))) ) ) ) ).
% Kleene_iter_gpfp
tff(fact_3450_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> ( compow(fun(A,fun(A,$o)),N,bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).
% relpowp_bot
tff(fact_3451_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),F: fun(B,C),Aa2: fun(A,set(B))] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> inj_on(B,C,F,aa(A,set(B),Aa2,I3)) )
=> inj_on(B,C,F,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),Aa2),I4))) ) ) ).
% inj_on_INTER
tff(fact_3452_mset__set__empty__iff,axiom,
! [A: $tType,Aa2: set(A)] :
( ( mset_set(A,Aa2) = zero_zero(multiset(A)) )
<=> ( ( Aa2 = bot_bot(set(A)) )
| ~ aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% mset_set_empty_iff
tff(fact_3453_quotient__of__Fract,axiom,
! [A3: int,B2: int] : quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) ).
% quotient_of_Fract
tff(fact_3454_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_3455_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_3456_funpow__decreasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [M2: nat,N: nat,F: fun(A,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( order_mono(A,A,F)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),M2,F),bot_bot(A))),aa(A,A,compow(fun(A,A),N,F),bot_bot(A))) ) ) ) ).
% funpow_decreasing
tff(fact_3457_funpow__increasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [M2: nat,N: nat,F: fun(A,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( order_mono(A,A,F)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,compow(fun(A,A),N,F),top_top(A))),aa(A,A,compow(fun(A,A),M2,F),top_top(A))) ) ) ) ).
% funpow_increasing
tff(fact_3458_lfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),K: nat] :
( order_mono(A,A,F)
=> ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F),bot_bot(A)) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) )
=> ( complete_lattice_lfp(A,F) = aa(A,A,compow(fun(A,A),K,F),bot_bot(A)) ) ) ) ) ).
% lfp_Kleene_iter
tff(fact_3459_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Aa2: set(A),G: fun(A,B),Ba: set(A)] :
( inj_on(A,B,F,Aa2)
=> ( inj_on(A,B,G,Ba)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F),Aa2)),aa(set(A),set(B),image2(A,B,G),Ba)) = 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_ko(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),Aa2),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba)) ) ) ) ).
% inj_on_disjoint_Un
tff(fact_3460_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,
$ite(X = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),X))) ).
% Fract.abs_eq
tff(fact_3461_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),insert2(set(A),image(A,A,R2,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),bot_bot(set(set(A)))) ).
% singleton_quotient
tff(fact_3462_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Aa2: set(A),A3: B] :
( inj_on(A,B,F,Aa2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),Aa2)),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_kp(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F),Aa2),A3))),bot_bot(set(A)))) ) ).
% inj_on_vimage_singleton
tff(fact_3463_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: B] :
( inj_on(A,B,F,top_top(set(A)))
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,$o),aTP_Lamp_kq(fun(A,B),fun(B,fun(A,$o)),F),A3))),bot_bot(set(A)))) ) ).
% inj_vimage_singleton
tff(fact_3464_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
( order_well_order_on(A,field2(A,R2),R2)
=> ( inj_on(A,B,F,field2(A,R2))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_We2720479622203943262_image(A,B,R2,F)),bNF_Wellorder_ordIso(A,B)) ) ) ).
% dir_image_ordIso
tff(fact_3465_antimono__funpow,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [Q: fun(A,A)] :
( order_mono(A,A,Q)
=> order_antimono(nat,A,aTP_Lamp_kr(fun(A,A),fun(nat,A),Q)) ) ) ).
% antimono_funpow
tff(fact_3466_proj__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B] : equiv_proj(B,A,R2,X) = image(B,A,R2,aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))) ).
% proj_def
tff(fact_3467_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ks(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
% The_split_eq
tff(fact_3468_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_antimono(A,B,F)
<=> ! [X4: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y3)),aa(A,B,F,X4)) ) ) ) ).
% antimono_def
tff(fact_3469_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y2)),aa(A,B,F,X3)) )
=> order_antimono(A,B,F) ) ) ).
% antimonoI
tff(fact_3470_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X)) ) ) ) ).
% antimonoE
tff(fact_3471_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X)) ) ) ) ).
% antimonoD
tff(fact_3472_dir__image__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F: fun(B,A)] : bNF_We2720479622203943262_image(B,A,R2,F) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_jf(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),R2),F)) ).
% dir_image_def
tff(fact_3473_Least__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_kt(fun(A,$o),fun(A,$o),P)) ) ).
% Least_def
tff(fact_3474_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,$o))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),complete_lattice_lfp(set(product_prod(A,B)),F))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
=> ( ! [A4: A,B3: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))))
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) )
=> aa(B,$o,aa(A,fun(B,$o),P,A3),B2) ) ) ) ).
% lfp_induct2
tff(fact_3475_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).
% min_of_antimono
tff(fact_3476_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).
% max_of_antimono
tff(fact_3477_the__elem__def,axiom,
! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_ku(set(A),fun(A,$o),X5)) ).
% the_elem_def
tff(fact_3478_Greatest__def,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_kv(fun(A,$o),fun(A,$o),P)) ) ).
% Greatest_def
tff(fact_3479_ord_OLeast__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_kw(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).
% ord.Least_def
tff(fact_3480_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [F: fun(A,C),G: fun(B,C),Aa2: set(A),Ba: set(B)] :
( inj_on(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_kx(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),F),G)),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))
=> ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F),Aa2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),Ba)) = aa(set(C),C,aa(fun(C,C),fun(set(C),C),groups7311177749621191930dd_sum(C,C),id(C)),aa(fun(C,$o),set(C),collect(C),aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_ky(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),F),G),Aa2),Ba))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_3481_flat__lub__def,axiom,
! [A: $tType,B2: A,Aa2: set(A)] :
aa(set(A),A,partial_flat_lub(A,B2),Aa2) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),B2,the(A,aa(set(A),fun(A,$o),aTP_Lamp_kz(A,fun(set(A),fun(A,$o)),B2),Aa2))) ).
% flat_lub_def
tff(fact_3482_proj__iff,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),Aa2)
=> ( ( equiv_proj(A,A,R2,X) = equiv_proj(A,A,R2,Y) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% proj_iff
tff(fact_3483_case__prod__Pair,axiom,
! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).
% case_prod_Pair
tff(fact_3484_ord_OLeast_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).
% ord.Least.cong
tff(fact_3485_in__quotient__imp__closed,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( member(A,X,X5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(A,Y,X5) ) ) ) ) ).
% in_quotient_imp_closed
tff(fact_3486_quotient__eq__iff,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
=> ( member(A,X,X5)
=> ( member(A,Y,Y5)
=> ( ( X5 = Y5 )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ) ) ).
% quotient_eq_iff
tff(fact_3487_quotient__eqI,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
=> ( member(A,X,X5)
=> ( member(A,Y,Y5)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( X5 = Y5 ) ) ) ) ) ) ) ).
% quotient_eqI
tff(fact_3488_in__quotient__imp__non__empty,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( X5 != bot_bot(set(A)) ) ) ) ).
% in_quotient_imp_non_empty
tff(fact_3489_equiv__class__self,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(A,A3,Aa2)
=> member(A,A3,image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) ) ) ).
% equiv_class_self
tff(fact_3490_quotient__disj,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A)] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
=> ( ( X5 = Y5 )
| ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),Y5) = bot_bot(set(A)) ) ) ) ) ) ).
% quotient_disj
tff(fact_3491_fst__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,comp(product_prod(A,A),A,A,product_fst(A,A),aTP_Lamp_gz(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% fst_diag_id
tff(fact_3492_snd__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,comp(product_prod(A,A),A,A,product_snd(A,A),aTP_Lamp_gz(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% snd_diag_id
tff(fact_3493_GreatestI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> ( ! [Y4: A] :
( aa(A,$o,P,Y4)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X3) )
=> aa(A,$o,Q,X3) ) )
=> aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).
% GreatestI2_order
tff(fact_3494_Greatest__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,$o),X: A] :
( aa(A,$o,P,X)
=> ( ! [Y2: A] :
( aa(A,$o,P,Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
=> ( order_Greatest(A,P) = X ) ) ) ) ).
% Greatest_equality
tff(fact_3495_eq__equiv__class,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A,B2: A,Aa2: set(A)] :
( ( image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) )
=> ( equiv_equiv(A,Aa2,R2)
=> ( member(A,B2,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ) ).
% eq_equiv_class
tff(fact_3496_equiv__class__eq,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> ( image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ) ) ).
% equiv_class_eq
tff(fact_3497_eq__equiv__class__iff,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(A,X,Aa2)
=> ( member(A,Y,Aa2)
=> ( ( image(A,A,R2,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ).
% eq_equiv_class_iff
tff(fact_3498_equiv__class__eq__iff,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
<=> ( ( image(A,A,R2,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) )
& member(A,X,Aa2)
& member(A,Y,Aa2) ) ) ) ).
% equiv_class_eq_iff
tff(fact_3499_eq__equiv__class__iff2,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(A,X,Aa2)
=> ( member(A,Y,Aa2)
=> ( ( equiv_quotient(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))),R2) )
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ) ).
% eq_equiv_class_iff2
tff(fact_3500_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),Aa2: set(A),A3: A] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
=> ( equiv_equiv(A,Aa2,R)
=> ( equiv_equiv(A,Aa2,S)
=> ( image(A,A,S,image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq2
tff(fact_3501_refines__equiv__class__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),Aa2: set(A),A3: A] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
=> ( equiv_equiv(A,Aa2,R)
=> ( equiv_equiv(A,Aa2,S)
=> ( image(A,A,R,image(A,A,S,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) = image(A,A,S,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ) ) ) ).
% refines_equiv_class_eq
tff(fact_3502_equiv__class__subset,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) ) ) ).
% equiv_class_subset
tff(fact_3503_subset__equiv__class,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),B2: A,A3: A] :
( equiv_equiv(A,Aa2,R2)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))
=> ( member(A,B2,Aa2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ) ).
% subset_equiv_class
tff(fact_3504_equiv__class__nondisjoint,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X: A,A3: A,B2: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ).
% equiv_class_nondisjoint
tff(fact_3505_in__quotient__imp__in__rel,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),X5: set(A),X: A,Y: A] :
( equiv_equiv(A,Aa2,R2)
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),X5)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% in_quotient_imp_in_rel
tff(fact_3506_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A1: A,A22: B] :
( equiv_equiv(A,A14,R1)
=> ( equiv_equiv(B,A24,R22)
=> ( equiv_congruent2(A,B,set(C),R1,R22,F)
=> ( member(A,A1,A14)
=> ( member(B,A22,A24)
=> ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_la(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F),A22)),image(A,A,R1,aa(set(A),set(A),insert2(A,A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F,A1),A22) ) ) ) ) ) ) ).
% UN_equiv_class2
tff(fact_3507_UN__equiv__class,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F: fun(A,set(B)),A3: A] :
( equiv_equiv(A,Aa2,R2)
=> ( equiv_congruent(A,set(B),R2,F)
=> ( member(A,A3,Aa2)
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) = aa(A,set(B),F,A3) ) ) ) ) ).
% UN_equiv_class
tff(fact_3508_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F: fun(A,set(B)),X5: set(A),Y5: set(A)] :
( equiv_equiv(A,Aa2,R2)
=> ( equiv_congruent(A,set(B),R2,F)
=> ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X5)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),Y5)) )
=> ( member(set(A),X5,equiv_quotient(A,Aa2,R2))
=> ( member(set(A),Y5,equiv_quotient(A,Aa2,R2))
=> ( ! [X3: A,Y2: A] :
( member(A,X3,Aa2)
=> ( member(A,Y2,Aa2)
=> ( ( aa(A,set(B),F,X3) = aa(A,set(B),F,Y2) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2) ) ) )
=> ( X5 = Y5 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
tff(fact_3509_disjnt__equiv__class,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,Aa2,R2)
=> ( disjnt(A,image(A,A,R2,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))),image(A,A,R2,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))
<=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2) ) ) ).
% disjnt_equiv_class
tff(fact_3510_congruent2__implies__congruent__UN,axiom,
! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A3: B] :
( equiv_equiv(A,A14,R1)
=> ( equiv_equiv(B,A24,R22)
=> ( equiv_congruent2(A,B,set(C),R1,R22,F)
=> ( member(B,A3,A24)
=> equiv_congruent(A,set(C),R1,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_la(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F),A3)) ) ) ) ) ).
% congruent2_implies_congruent_UN
tff(fact_3511_disjnt__self__iff__empty,axiom,
! [A: $tType,S: set(A)] :
( disjnt(A,S,S)
<=> ( S = bot_bot(set(A)) ) ) ).
% disjnt_self_iff_empty
tff(fact_3512_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,Ca: set(A),Aa2: set(B),Ba: set(B)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,Ca,aTP_Lamp_ib(set(B),fun(A,set(B)),Aa2)),product_Sigma(A,B,Ca,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))
<=> ( ( Ca = bot_bot(set(A)) )
| disjnt(B,Aa2,Ba) ) ) ).
% disjnt_Times1_iff
tff(fact_3513_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ca: set(B),Ba: set(A)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ca)),product_Sigma(A,B,Ba,aTP_Lamp_ib(set(B),fun(A,set(B)),Ca)))
<=> ( ( Ca = bot_bot(set(B)) )
| disjnt(A,Aa2,Ba) ) ) ).
% disjnt_Times2_iff
tff(fact_3514_disjnt__empty1,axiom,
! [A: $tType,Aa2: set(A)] : disjnt(A,bot_bot(set(A)),Aa2) ).
% disjnt_empty1
tff(fact_3515_disjnt__empty2,axiom,
! [A: $tType,Aa2: set(A)] : disjnt(A,Aa2,bot_bot(set(A))) ).
% disjnt_empty2
tff(fact_3516_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ca: fun(A,set(B)),Ba: set(A)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,Aa2,Ca),product_Sigma(A,B,Ba,Ca))
<=> ( ! [X4: A] :
( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba))
=> ( aa(A,set(B),Ca,X4) = bot_bot(set(B)) ) )
| disjnt(A,Aa2,Ba) ) ) ).
% disjnt_Sigma_iff
tff(fact_3517_disjnt__def,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( disjnt(A,Aa2,Ba)
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) ) ) ).
% disjnt_def
tff(fact_3518_congruentD,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B),Y: A,Z2: A] :
( equiv_congruent(A,B,R2,F)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> ( aa(A,B,F,Y) = aa(A,B,F,Z2) ) ) ) ).
% congruentD
tff(fact_3519_congruentI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,B)] :
( ! [Y2: A,Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> ( aa(A,B,F,Y2) = aa(A,B,F,Z3) ) )
=> equiv_congruent(A,B,R2,F) ) ).
% congruentI
tff(fact_3520_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C)),Y1: A,Z1: A,Y22: B,Z22: B] :
( equiv_congruent2(A,B,C,R1,R22,F)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1),R1)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z22),R22)
=> ( aa(B,C,aa(A,fun(B,C),F,Y1),Y22) = aa(B,C,aa(A,fun(B,C),F,Z1),Z22) ) ) ) ) ).
% congruent2D
tff(fact_3521_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( ! [Y12: A,Z12: A,Y23: B,Z23: B] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z12),R1)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z23),R22)
=> ( aa(B,C,aa(A,fun(B,C),F,Y12),Y23) = aa(B,C,aa(A,fun(B,C),F,Z12),Z23) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F) ) ).
% congruent2I'
tff(fact_3522_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F: fun(A,fun(A,B))] :
( equiv_equiv(A,Aa2,R2)
=> ( ! [Y2: A,Z3: A] :
( member(A,Y2,Aa2)
=> ( member(A,Z3,Aa2)
=> ( aa(A,B,aa(A,fun(A,B),F,Y2),Z3) = aa(A,B,aa(A,fun(A,B),F,Z3),Y2) ) ) )
=> ( ! [Y2: A,Z3: A,W: A] :
( member(A,W,Aa2)
=> ( member(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)
=> ( aa(A,B,aa(A,fun(A,B),F,W),Y2) = aa(A,B,aa(A,fun(A,B),F,W),Z3) ) ) )
=> equiv_congruent2(A,A,B,R2,R2,F) ) ) ) ).
% congruent2_commuteI
tff(fact_3523_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A14: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( equiv_equiv(A,A14,R1)
=> ( equiv_equiv(B,A24,R22)
=> ( ! [Y2: A,Z3: A,W: B] :
( member(B,W,A24)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R1)
=> ( aa(B,C,aa(A,fun(B,C),F,Y2),W) = aa(B,C,aa(A,fun(B,C),F,Z3),W) ) ) )
=> ( ! [Y2: B,Z3: B,W: A] :
( member(A,W,A14)
=> ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z3),R22)
=> ( aa(B,C,aa(A,fun(B,C),F,W),Y2) = aa(B,C,aa(A,fun(B,C),F,W),Z3) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F) ) ) ) ) ).
% congruent2I
tff(fact_3524_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A14: set(B),B12: set(A),F22: fun(C,D),B23: set(C),A24: set(D)] :
( ( aa(set(B),set(A),image2(B,A,F1),A14) = B12 )
=> ( inj_on(C,D,F22,B23)
=> ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B23)),A24)
=> ( ( ( B23 = bot_bot(set(C)) )
=> ( A24 = bot_bot(set(D)) ) )
=> ( bNF_Wellorder_Func(C,A,B23,B12) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A24,A14)) ) ) ) ) ) ).
% Func_map_surj
tff(fact_3525_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: fun(A,B),Ca: set(A),Ba: set(A),X: A] :
( inj_on(A,B,G,Ca)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ca),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Ba),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))
=> member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_lb(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),Ca),X),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Ba),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ).
% If_the_inv_into_in_Func
tff(fact_3526_mset__set_Oinsert__remove,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( mset_set(A,aa(set(A),set(A),insert2(A,X),Aa2)) = add_mset(A,X,mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ).
% mset_set.insert_remove
tff(fact_3527_mset__set_Oremove,axiom,
! [A: $tType,Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( mset_set(A,Aa2) = add_mset(A,X,mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% mset_set.remove
tff(fact_3528_gfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),K: nat] :
( order_mono(A,A,F)
=> ( ( aa(A,A,compow(fun(A,A),aa(nat,nat,suc,K),F),top_top(A)) = aa(A,A,compow(fun(A,A),K,F),top_top(A)) )
=> ( complete_lattice_gfp(A,F) = aa(A,A,compow(fun(A,A),K,F),top_top(A)) ) ) ) ) ).
% gfp_Kleene_iter
tff(fact_3529_Func__non__emp,axiom,
! [A: $tType,B: $tType,Ba: set(A),Aa2: set(B)] :
( ( Ba != bot_bot(set(A)) )
=> ( bNF_Wellorder_Func(B,A,Aa2,Ba) != bot_bot(set(fun(B,A))) ) ) ).
% Func_non_emp
tff(fact_3530_Func__is__emp,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( bNF_Wellorder_Func(A,B,Aa2,Ba) = bot_bot(set(fun(A,B))) )
<=> ( ( Aa2 != bot_bot(set(A)) )
& ( Ba = bot_bot(set(B)) ) ) ) ).
% Func_is_emp
tff(fact_3531_coinduct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),X5: A] :
( order_mono(A,A,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),complete_lattice_gfp(A,F)) ) ) ) ).
% coinduct
tff(fact_3532_def__coinduct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: A,F: fun(A,A),X5: A] :
( ( Aa2 = complete_lattice_gfp(A,F) )
=> ( order_mono(A,A,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),Aa2)))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Aa2) ) ) ) ) ).
% def_coinduct
tff(fact_3533_coinduct__lemma,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X5: A,F: fun(A,A)] :
( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))))
=> ( order_mono(A,A,F)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F))),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),complete_lattice_gfp(A,F)))) ) ) ) ).
% coinduct_lemma
tff(fact_3534_su__rel__fun_Of__def,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A] :
( su_rel_fun(A,B,F4,F)
=> ( aa(A,B,F,Aa2) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),Aa2)) ) ) ).
% su_rel_fun.f_def
tff(fact_3535_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B)] :
( ! [A7: A,B8: B,B11: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B8),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B11),F4)
=> ( B8 = B11 ) ) )
=> ( ! [A7: A,P3: $o] :
( ! [B13: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B13),F4)
=> (P3) )
=> (P3) )
=> ( ! [A7: A] : aa(A,B,F,A7) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A7))
=> su_rel_fun(A,B,F4,F) ) ) ) ).
% su_rel_fun.intro
tff(fact_3536_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A,Ba: B] :
( su_rel_fun(A,B,F4,F)
=> ( ( aa(A,B,F,Aa2) = Ba )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),F4) ) ) ).
% su_rel_fun.repr
tff(fact_3537_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A] :
( su_rel_fun(A,B,F4,F)
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),aa(A,B,F,Aa2)),F4) ) ).
% su_rel_fun.repr1
tff(fact_3538_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A,Ba: B] :
( su_rel_fun(A,B,F4,F)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),F4)
=> ( Ba = aa(A,B,F,Aa2) ) ) ) ).
% su_rel_fun.repr2
tff(fact_3539_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A,Ba: B,B14: B] :
( su_rel_fun(A,B,F4,F)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B14),F4)
=> ( Ba = B14 ) ) ) ) ).
% su_rel_fun.unique
tff(fact_3540_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B),Aa2: A] :
( su_rel_fun(A,B,F4,F)
=> ~ ! [B8: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),B8),F4) ) ).
% su_rel_fun.surjective
tff(fact_3541_su__rel__fun__def,axiom,
! [B: $tType,A: $tType,F4: set(product_prod(A,B)),F: fun(A,B)] :
( su_rel_fun(A,B,F4,F)
<=> ( ! [A8: A,B7: B,B15: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B7),F4)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B15),F4)
=> ( B7 = B15 ) ) )
& ! [A8: A,P4: $o] :
( ! [B7: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B7),F4)
=> (P4) )
=> (P4) )
& ! [A8: A] : aa(A,B,F,A8) = the(B,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),F4),A8)) ) ) ).
% su_rel_fun_def
tff(fact_3542_mult__cancel,axiom,
! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Z6: multiset(A),Y5: multiset(A)] :
( trans(A,S2)
=> ( irrefl(A,S2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X5),Z6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y5),Z6)),mult(A,S2))
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y5),mult(A,S2)) ) ) ) ).
% mult_cancel
tff(fact_3543_mult__cancel__add__mset,axiom,
! [A: $tType,S2: set(product_prod(A,A)),Uu: A,X5: multiset(A),Y5: multiset(A)] :
( trans(A,S2)
=> ( irrefl(A,S2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),add_mset(A,Uu,X5)),add_mset(A,Uu,Y5)),mult(A,S2))
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y5),mult(A,S2)) ) ) ) ).
% mult_cancel_add_mset
tff(fact_3544_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A3: A,M4: multiset(A)] :
( ~ member(A,A3,aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),add_mset(A,A3,zero_zero(multiset(A))))))
=> ( aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M4),add_mset(A,A3,zero_zero(multiset(A))))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(multiset(A),set(A),set_mset(A),M4)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ) ).
% at_most_one_mset_mset_diff
tff(fact_3545_Func__empty,axiom,
! [B: $tType,A: $tType,Ba: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),Ba) = aa(set(fun(A,B)),set(fun(A,B)),insert2(fun(A,B),aTP_Lamp_lc(A,B)),bot_bot(set(fun(A,B)))) ).
% Func_empty
tff(fact_3546_prod__filter__INF2,axiom,
! [C: $tType,B: $tType,A: $tType,J3: set(A),Aa2: filter(B),Ba: fun(A,filter(C))] :
( ( J3 != bot_bot(set(A)) )
=> ( prod_filter(B,C,Aa2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),Ba),J3))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_ld(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Aa2),Ba)),J3)) ) ) ).
% prod_filter_INF2
tff(fact_3547_set__mset__empty,axiom,
! [A: $tType] : aa(multiset(A),set(A),set_mset(A),zero_zero(multiset(A))) = bot_bot(set(A)) ).
% set_mset_empty
tff(fact_3548_set__mset__eq__empty__iff,axiom,
! [A: $tType,M4: multiset(A)] :
( ( aa(multiset(A),set(A),set_mset(A),M4) = bot_bot(set(A)) )
<=> ( M4 = zero_zero(multiset(A)) ) ) ).
% set_mset_eq_empty_iff
tff(fact_3549_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext(product_unit),P: assn] :
( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H2)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
<=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,P),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_normalize
tff(fact_3550_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,Aa2: filter(A),Ba: filter(B)] :
( ( prod_filter(A,B,Aa2,Ba) = bot_bot(filter(product_prod(A,B))) )
<=> ( ( Aa2 = bot_bot(filter(A)) )
| ( Ba = bot_bot(filter(B)) ) ) ) ).
% prod_filter_eq_bot
tff(fact_3551_one__step__implies__mult,axiom,
! [A: $tType,J3: multiset(A),K5: multiset(A),R2: set(product_prod(A,A)),I4: multiset(A)] :
( ( J3 != zero_zero(multiset(A)) )
=> ( ! [X3: A] :
( member(A,X3,aa(multiset(A),set(A),set_mset(A),K5))
=> ? [Xa3: A] :
( member(A,Xa3,aa(multiset(A),set(A),set_mset(A),J3))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R2) ) )
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),K5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),J3)),mult(A,R2)) ) ) ).
% one_step_implies_mult
tff(fact_3552_in__Inf__multiset__iff,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: A] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( member(A,X,aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)))
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,Aa2)
=> member(A,X,aa(multiset(A),set(A),set_mset(A),X4)) ) ) ) ).
% in_Inf_multiset_iff
tff(fact_3553_mult__implies__one__step,axiom,
! [A: $tType,R2: set(product_prod(A,A)),M4: multiset(A),N3: multiset(A)] :
( trans(A,R2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),N3),mult(A,R2))
=> ? [I5: multiset(A),J4: multiset(A)] :
( ( N3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),J4) )
& ? [K6: multiset(A)] :
( ( M4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),K6) )
& ( J4 != zero_zero(multiset(A)) )
& ! [X2: A] :
( member(A,X2,aa(multiset(A),set(A),set_mset(A),K6))
=> ? [Xa4: A] :
( member(A,Xa4,aa(multiset(A),set(A),set_mset(A),J4))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa4),R2) ) ) ) ) ) ) ).
% mult_implies_one_step
tff(fact_3554_eventually__prod__same,axiom,
! [A: $tType,P: fun(product_prod(A,A),$o),F4: filter(A)] :
( eventually(product_prod(A,A),P,prod_filter(A,A,F4,F4))
<=> ? [Q8: fun(A,$o)] :
( eventually(A,Q8,F4)
& ! [X4: A,Y3: A] :
( aa(A,$o,Q8,X4)
=> ( aa(A,$o,Q8,Y3)
=> aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3)) ) ) ) ) ).
% eventually_prod_same
tff(fact_3555_eventually__prod__filter,axiom,
! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F4: filter(A),G6: filter(B)] :
( eventually(product_prod(A,B),P,prod_filter(A,B,F4,G6))
<=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
( eventually(A,Pf,F4)
& eventually(B,Pg,G6)
& ! [X4: A,Y3: B] :
( aa(A,$o,Pf,X4)
=> ( aa(B,$o,Pg,Y3)
=> aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3)) ) ) ) ) ).
% eventually_prod_filter
tff(fact_3556_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,Aa2: filter(A),Ba: filter(B),Ca: filter(A),D4: filter(B)] :
( ( Aa2 != bot_bot(filter(A)) )
=> ( ( Ba != bot_bot(filter(B)) )
=> ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,Aa2,Ba)),prod_filter(A,B,Ca,D4))
<=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Aa2),Ca)
& aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),Ba),D4) ) ) ) ) ).
% prod_filter_mono_iff
tff(fact_3557_infinite__set__mset__mset__set,axiom,
! [A: $tType,Aa2: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(multiset(A),set(A),set_mset(A),mset_set(A,Aa2)) = bot_bot(set(A)) ) ) ).
% infinite_set_mset_mset_set
tff(fact_3558_mult__cancel__max,axiom,
! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Y5: multiset(A)] :
( trans(A,S2)
=> ( irrefl(A,S2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y5),mult(A,S2))
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X5),Y5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y5),X5)),mult(A,S2)) ) ) ) ).
% mult_cancel_max
tff(fact_3559_set__mset__Inf,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),Aa2)) ) ) ).
% set_mset_Inf
tff(fact_3560_set__mset__single,axiom,
! [A: $tType,B2: A] : aa(multiset(A),set(A),set_mset(A),add_mset(A,B2,zero_zero(multiset(A)))) = aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))) ).
% set_mset_single
tff(fact_3561_eventually__prod2,axiom,
! [A: $tType,B: $tType,Aa2: filter(A),P: fun(B,$o),Ba: filter(B)] :
( ( Aa2 != bot_bot(filter(A)) )
=> ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_le(fun(B,$o),fun(A,fun(B,$o)),P)),prod_filter(A,B,Aa2,Ba))
<=> eventually(B,P,Ba) ) ) ).
% eventually_prod2
tff(fact_3562_eventually__prod1,axiom,
! [A: $tType,B: $tType,Ba: filter(A),P: fun(B,$o),Aa2: filter(B)] :
( ( Ba != bot_bot(filter(A)) )
=> ( eventually(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_lf(fun(B,$o),fun(B,fun(A,$o)),P)),prod_filter(B,A,Aa2,Ba))
<=> eventually(B,P,Aa2) ) ) ).
% eventually_prod1
tff(fact_3563_prod__filter__INF,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,I4: set(A),J3: set(B),Aa2: fun(A,filter(C)),Ba: fun(B,filter(D))] :
( ( I4 != bot_bot(set(A)) )
=> ( ( J3 != bot_bot(set(B)) )
=> ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),Aa2),I4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),Ba),J3))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_lh(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J3),Aa2),Ba)),I4)) ) ) ) ).
% prod_filter_INF
tff(fact_3564_prod__filter__INF1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),Aa2: fun(A,filter(B)),Ba: filter(C)] :
( ( I4 != bot_bot(set(A)) )
=> ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),Aa2),I4)),Ba) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_li(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Aa2),Ba)),I4)) ) ) ).
% prod_filter_INF1
tff(fact_3565_multp__code__iff__mult,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N3: multiset(A),M4: multiset(A)] :
( irrefl(A,R)
=> ( trans(A,R)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R) )
=> ( multp_code(A,P,N3,M4)
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),M4),mult(A,R)) ) ) ) ) ).
% multp_code_iff_mult
tff(fact_3566_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,fun(A,$o)),N3: multiset(A),M4: multiset(A)] :
( irrefl(A,R)
=> ( trans(A,R)
=> ( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),P,X3),Y2)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R) )
=> ( multeqp_code(A,P,N3,M4)
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),M4),aa(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A))),aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A)))),sup_sup(set(product_prod(multiset(A),multiset(A)))),mult(A,R)),id2(multiset(A)))) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
tff(fact_3567_mult1__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : mult1(A,R2) = aa(fun(product_prod(multiset(A),multiset(A)),$o),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),$o)),fun(product_prod(multiset(A),multiset(A)),$o),product_case_prod(multiset(A),multiset(A),$o),aTP_Lamp_lj(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),R2))) ).
% mult1_def
tff(fact_3568_mult1E,axiom,
! [A: $tType,N3: multiset(A),M4: multiset(A),R2: set(product_prod(A,A))] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),M4),mult1(A,R2))
=> ~ ! [A4: A,M0: multiset(A)] :
( ( M4 = add_mset(A,A4,M0) )
=> ! [K6: multiset(A)] :
( ( N3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K6) )
=> ~ ! [B9: A] :
( member(A,B9,aa(multiset(A),set(A),set_mset(A),K6))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A4),R2) ) ) ) ) ).
% mult1E
tff(fact_3569_mult1I,axiom,
! [A: $tType,M4: multiset(A),A3: A,M02: multiset(A),N3: multiset(A),K5: multiset(A),R2: set(product_prod(A,A))] :
( ( M4 = add_mset(A,A3,M02) )
=> ( ( N3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K5) )
=> ( ! [B3: A] :
( member(A,B3,aa(multiset(A),set(A),set_mset(A),K5))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R2) )
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),M4),mult1(A,R2)) ) ) ) ).
% mult1I
tff(fact_3570_not__less__empty,axiom,
! [A: $tType,M4: multiset(A),R2: set(product_prod(A,A))] : ~ member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),zero_zero(multiset(A))),mult1(A,R2)) ).
% not_less_empty
tff(fact_3571_mult1__union,axiom,
! [A: $tType,Ba: multiset(A),D4: multiset(A),R2: set(product_prod(A,A)),Ca: multiset(A)] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),Ba),D4),mult1(A,R2))
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ca),Ba)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ca),D4)),mult1(A,R2)) ) ).
% mult1_union
tff(fact_3572_less__add,axiom,
! [A: $tType,N3: multiset(A),A3: A,M02: multiset(A),R2: set(product_prod(A,A))] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),add_mset(A,A3,M02)),mult1(A,R2))
=> ( ? [M5: multiset(A)] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M5),M02),mult1(A,R2))
& ( N3 = add_mset(A,A3,M5) ) )
| ? [K6: multiset(A)] :
( ! [B9: A] :
( member(A,B9,aa(multiset(A),set(A),set_mset(A),K6))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A3),R2) )
& ( N3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K6) ) ) ) ) ).
% less_add
tff(fact_3573_mult1__lessE,axiom,
! [A: $tType,N3: multiset(A),M4: multiset(A),R2: fun(A,fun(A,$o))] :
( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N3),M4),mult1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))))
=> ( asymp(A,R2)
=> ~ ! [A4: A,M0: multiset(A)] :
( ( M4 = add_mset(A,A4,M0) )
=> ! [K6: multiset(A)] :
( ( N3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K6) )
=> ( ~ member(A,A4,aa(multiset(A),set(A),set_mset(A),K6))
=> ~ ! [B9: A] :
( member(A,B9,aa(multiset(A),set(A),set_mset(A),K6))
=> aa(A,$o,aa(A,fun(A,$o),R2,B9),A4) ) ) ) ) ) ) ).
% mult1_lessE
tff(fact_3574_smsI,axiom,
! [Aa2: multiset(product_prod(nat,nat)),Ba: multiset(product_prod(nat,nat)),Z6: multiset(product_prod(nat,nat))] :
( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Ba)),fun_max_strict)
=> member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Aa2)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Ba)),ms_strict) ) ).
% smsI
tff(fact_3575_wmsI,axiom,
! [Aa2: multiset(product_prod(nat,nat)),Ba: multiset(product_prod(nat,nat)),Z6: multiset(product_prod(nat,nat))] :
( ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Ba)),fun_max_strict)
| ( ( Aa2 = zero_zero(multiset(product_prod(nat,nat))) )
& ( Ba = zero_zero(multiset(product_prod(nat,nat))) ) ) )
=> member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Aa2)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Ba)),ms_weak) ) ).
% wmsI
tff(fact_3576_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X: A] :
aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% set_mset_replicate_mset_subset
tff(fact_3577_mult__cancel__max0,axiom,
! [A: $tType,S2: set(product_prod(A,A)),X5: multiset(A),Y5: multiset(A)] :
( trans(A,S2)
=> ( irrefl(A,S2)
=> ( member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X5),Y5),mult(A,S2))
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X5),Y5))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X5),Y5))),mult(A,S2)) ) ) ) ).
% mult_cancel_max0
tff(fact_3578_subset__mset_Oinf_Osemigroup__axioms,axiom,
! [A: $tType] : semigroup(multiset(A),inter_mset(A)) ).
% subset_mset.inf.semigroup_axioms
tff(fact_3579_ms__reduction__pair,axiom,
fun_reduction_pair(multiset(product_prod(nat,nat)),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))))),product_Pair(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),ms_weak)) ).
% ms_reduction_pair
tff(fact_3580_ms__weakI1,axiom,
! [Z6: multiset(product_prod(nat,nat)),Z7: multiset(product_prod(nat,nat)),Aa2: multiset(product_prod(nat,nat)),Ba: multiset(product_prod(nat,nat))] :
( pw_leq(Z6,Z7)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Ba)),fun_max_strict)
=> member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Aa2)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),Ba)),ms_weak) ) ) ).
% ms_weakI1
tff(fact_3581_ms__strictI,axiom,
! [Z6: multiset(product_prod(nat,nat)),Z7: multiset(product_prod(nat,nat)),Aa2: multiset(product_prod(nat,nat)),Ba: multiset(product_prod(nat,nat))] :
( pw_leq(Z6,Z7)
=> ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Aa2)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),Ba)),fun_max_strict)
=> member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),Aa2)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),Ba)),ms_strict) ) ) ).
% ms_strictI
tff(fact_3582_pw__leq__split,axiom,
! [X5: multiset(product_prod(nat,nat)),Y5: multiset(product_prod(nat,nat))] :
( pw_leq(X5,Y5)
=> ? [A7: multiset(product_prod(nat,nat)),B8: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat))] :
( ( X5 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),A7),Z8) )
& ( Y5 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),B8),Z8) )
& ( member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A7)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B8)),fun_max_strict)
| ( ( B8 = zero_zero(multiset(product_prod(nat,nat))) )
& ( A7 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).
% pw_leq_split
tff(fact_3583_ms__weakI2,axiom,
! [Z6: multiset(product_prod(nat,nat)),Z7: multiset(product_prod(nat,nat))] :
( pw_leq(Z6,Z7)
=> member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z6),zero_zero(multiset(product_prod(nat,nat))))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),zero_zero(multiset(product_prod(nat,nat))))),ms_weak) ) ).
% ms_weakI2
tff(fact_3584_pw__leq_Ocases,axiom,
! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
( pw_leq(A1,A22)
=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
=> ( A22 != zero_zero(multiset(product_prod(nat,nat))) ) )
=> ~ ! [X3: product_prod(nat,nat),Y2: product_prod(nat,nat),X6: multiset(product_prod(nat,nat))] :
( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),X3,zero_zero(multiset(product_prod(nat,nat))))),X6) )
=> ! [Y8: multiset(product_prod(nat,nat))] :
( ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),Y2,zero_zero(multiset(product_prod(nat,nat))))),Y8) )
=> ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X3),Y2),fun_pair_leq)
=> ~ pw_leq(X6,Y8) ) ) ) ) ) ).
% pw_leq.cases
tff(fact_3585_pw__leq__lstep,axiom,
! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_leq)
=> pw_leq(add_mset(product_prod(nat,nat),X,zero_zero(multiset(product_prod(nat,nat)))),add_mset(product_prod(nat,nat),Y,zero_zero(multiset(product_prod(nat,nat))))) ) ).
% pw_leq_lstep
tff(fact_3586_pw__leq__step,axiom,
! [X: product_prod(nat,nat),Y: product_prod(nat,nat),X5: multiset(product_prod(nat,nat)),Y5: multiset(product_prod(nat,nat))] :
( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y),fun_pair_leq)
=> ( pw_leq(X5,Y5)
=> pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),X,zero_zero(multiset(product_prod(nat,nat))))),X5),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),Y,zero_zero(multiset(product_prod(nat,nat))))),Y5)) ) ) ).
% pw_leq_step
tff(fact_3587_pw__leq_Osimps,axiom,
! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
( pw_leq(A1,A22)
<=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
& ( A22 = zero_zero(multiset(product_prod(nat,nat))) ) )
| ? [X4: product_prod(nat,nat),Y3: product_prod(nat,nat),X7: multiset(product_prod(nat,nat)),Y9: multiset(product_prod(nat,nat))] :
( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),X4,zero_zero(multiset(product_prod(nat,nat))))),X7) )
& ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat),Y3,zero_zero(multiset(product_prod(nat,nat))))),Y9) )
& member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X4),Y3),fun_pair_leq)
& pw_leq(X7,Y9) ) ) ) ).
% pw_leq.simps
tff(fact_3588_multp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),M4: multiset(A),N3: multiset(A)] :
( multp(A,R2,M4,N3)
<=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M4),N3),mult(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% multp_def
tff(fact_3589_prod__mset_Oremove,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,Aa2: multiset(A)] :
( member(A,X,aa(multiset(A),set(A),set_mset(A),Aa2))
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Aa2),add_mset(A,X,zero_zero(multiset(A)))))) ) ) ) ).
% prod_mset.remove
tff(fact_3590_subset__mset_OInf__fin_Osingleton,axiom,
! [A: $tType,X: multiset(A)] : lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.Inf_fin.singleton
tff(fact_3591_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) ) ) ) ).
% subset_mset.Inf_fin.insert
tff(fact_3592_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_3593_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_3594_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [X: A,N3: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),add_mset(A,X,N3)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N3)) ) ).
% prod_mset.add_mset
tff(fact_3595_prod__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M4: multiset(A),N3: 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)),M4),N3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M4)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N3)) ) ).
% prod_mset.union
tff(fact_3596_prod__mset__Un,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Aa2: multiset(A),Ba: 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)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Ba)) ) ).
% prod_mset_Un
tff(fact_3597_prod__mset_Oneutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Aa2: multiset(A)] :
( ! [X3: A] :
( member(A,X3,aa(multiset(A),set(A),set_mset(A),Aa2))
=> ( X3 = one_one(A) ) )
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2) = one_one(A) ) ) ) ).
% prod_mset.neutral
tff(fact_3598_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,H2: fun(multiset(A),multiset(A)),N3: set(multiset(A))] :
( ! [X3: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),H2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),H2,X3)),aa(multiset(A),multiset(A),H2,Y2))
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),N3)
=> ( ( N3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),H2,lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),N3)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H2),N3)) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
tff(fact_3599_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Ba),Aa2)
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Ba)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2) ) ) ) ) ).
% subset_mset.Inf_fin.subset
tff(fact_3600_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),Aa2),Ba)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Ba)) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
tff(fact_3601_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Aa2: multiset(A)] :
( dvd_dvd(A,aa(multiset(A),A,comm_m9189036328036947845d_mset(A),Aa2),one_one(A))
<=> ! [X4: A] :
( member(A,X4,aa(multiset(A),set(A),set_mset(A),Aa2))
=> dvd_dvd(A,X4,one_one(A)) ) ) ) ).
% is_unit_prod_mset_iff
tff(fact_3602_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),X5)
=> ( ( X5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),X5) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
tff(fact_3603_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A),Y2: multiset(A)] : member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X3),Y2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X3),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),Y2),bot_bot(set(multiset(A))))))
=> member(multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2),Aa2) ) ) ) ).
% subset_mset.Inf_fin.closed
tff(fact_3604_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ~ member(multiset(A),X,Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
tff(fact_3605_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( member(multiset(A),X,Aa2)
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A)))))))) ) ) ) ).
% subset_mset.Inf_fin.remove
tff(fact_3606_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A)))))))) ) ) ).
% subset_mset.Inf_fin.insert_remove
tff(fact_3607_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Ba)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_lk(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Aa2),Ba))) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
tff(fact_3608_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) = lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_ll(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Aa2),X))) ) ) ) ).
% subset_mset.sup_Inf1_distrib
tff(fact_3609_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M4: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M4) = fold_mset(A,A,times_times(A),one_one(A),M4) ) ).
% prod_mset.eq_fold
tff(fact_3610_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2))
=> ! [A9: multiset(A)] :
( member(multiset(A),A9,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A9) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
tff(fact_3611_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( ! [A4: multiset(A)] :
( member(multiset(A),A4,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),A4) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) ) ) ) ).
% subset_mset.Inf_fin.boundedI
tff(fact_3612_subset__mset_Osemilattice__neutr__axioms,axiom,
! [A: $tType] : semilattice_neutr(multiset(A),union_mset(A),zero_zero(multiset(A))) ).
% subset_mset.semilattice_neutr_axioms
tff(fact_3613_subset__mset_Osemigroup__axioms,axiom,
! [A: $tType] : semigroup(multiset(A),union_mset(A)) ).
% subset_mset.semigroup_axioms
tff(fact_3614_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),X3) )
=> ( ! [Y2: multiset(A)] :
( ! [X2: multiset(A)] :
( member(multiset(A),X2,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),X2) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Y2),A3) )
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5) = A3 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
tff(fact_3615_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),X3) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Z2),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ).
% subset_mset.cInf_greatest
tff(fact_3616_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),A3) )
=> ( ! [Y2: multiset(A)] :
( ! [X2: multiset(A)] :
( member(multiset(A),X2,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),Y2) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),Y2) )
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = A3 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
tff(fact_3617_subset__mset_OcSup__least,axiom,
! [A: $tType,X5: set(multiset(A)),Z2: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A)] :
( member(multiset(A),X3,X5)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),Z2) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)),Z2) ) ) ).
% subset_mset.cSup_least
tff(fact_3618_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ? [X3: multiset(A)] :
( member(multiset(A),X3,Aa2)
& ! [Xa3: multiset(A)] :
( member(multiset(A),Xa3,Aa2)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X3),Xa3)
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
tff(fact_3619_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ? [X3: multiset(A)] :
( member(multiset(A),X3,Aa2)
& ! [Xa3: multiset(A)] :
( member(multiset(A),Xa3,Aa2)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa3),X3)
=> ( X3 = Xa3 ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
tff(fact_3620_subset__mset_OcINF__greatest,axiom,
! [B: $tType,A: $tType,Aa2: set(A),M2: multiset(B),F: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M2),aa(A,multiset(B),F,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M2),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))) ) ) ).
% subset_mset.cINF_greatest
tff(fact_3621_subset__mset_OcSUP__least,axiom,
! [A: $tType,B: $tType,Aa2: set(A),F: fun(A,multiset(B)),M4: multiset(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X3)),M4) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),M4) ) ) ).
% subset_mset.cSUP_least
tff(fact_3622_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Ba)),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
tff(fact_3623_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2))
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X),X4) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
tff(fact_3624_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A),Aa2)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
tff(fact_3625_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A)))))))) ) ) ).
% subset_mset.Sup_fin.insert_remove
tff(fact_3626_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( member(multiset(A),X,Aa2)
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2) = $ite(aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = bot_bot(set(multiset(A))),X,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),Aa2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A)))))))) ) ) ) ).
% subset_mset.Sup_fin.remove
tff(fact_3627_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_lm(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Aa2),X))) ) ) ) ).
% subset_mset.inf_Sup1_distrib
tff(fact_3628_subset__mset_OSup__fin_Osingleton,axiom,
! [A: $tType,X: multiset(A)] : lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = X ).
% subset_mset.Sup_fin.singleton
tff(fact_3629_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)) ) ) ) ).
% subset_mset.Sup_fin.insert
tff(fact_3630_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),X)
=> ! [A9: multiset(A)] :
( member(multiset(A),A9,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A9),X) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
tff(fact_3631_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( ! [A4: multiset(A)] :
( member(multiset(A),A4,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A4),X) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),X) ) ) ) ).
% subset_mset.Sup_fin.boundedI
tff(fact_3632_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),X)
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),X) ) ) ) ) ).
% subset_mset.Sup_fin.bounded_iff
tff(fact_3633_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X5: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),X5)
=> ( ( X5 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),X5) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
tff(fact_3634_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( ! [X3: multiset(A),Y2: multiset(A)] : member(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y2),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X3),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),Y2),bot_bot(set(multiset(A))))))
=> member(multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2),Aa2) ) ) ) ).
% subset_mset.Sup_fin.closed
tff(fact_3635_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: multiset(A)] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ~ member(multiset(A),X,Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),Aa2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)) ) ) ) ) ).
% subset_mset.Sup_fin.insert_not_elem
tff(fact_3636_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Ba),Aa2)
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Ba)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2) ) ) ) ) ).
% subset_mset.Sup_fin.subset
tff(fact_3637_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),Aa2),Ba)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Ba)) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
tff(fact_3638_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,H2: fun(multiset(A),multiset(A)),N3: set(multiset(A))] :
( ! [X3: multiset(A),Y2: multiset(A)] : aa(multiset(A),multiset(A),H2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X3),Y2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),H2,X3)),aa(multiset(A),multiset(A),H2,Y2))
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),N3)
=> ( ( N3 != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),H2,lattic4630905495605216202up_fin(multiset(A),union_mset(A),N3)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(set(multiset(A)),set(multiset(A)),image2(multiset(A),multiset(A),H2),N3)) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
tff(fact_3639_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Aa2),Ba)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Ba)) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
tff(fact_3640_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(set(multiset(A)),$o,finite_finite(multiset(A)),Ba)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Aa2)),lattic4630905495605216202up_fin(multiset(A),union_mset(A),Ba)) = lattic4630905495605216202up_fin(multiset(A),union_mset(A),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_ln(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Aa2),Ba))) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
tff(fact_3641_subset__mset_Omono__sup,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(multiset(A),B),Aa2: multiset(A),Ba: multiset(A)] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(multiset(A),B,F,Aa2)),aa(multiset(A),B,F,Ba))),aa(multiset(A),B,F,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Aa2),Ba))) ) ) ).
% subset_mset.mono_sup
tff(fact_3642_subset__mset_OcINF__union,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Ba))
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Ba))) ) ) ) ) ) ).
% subset_mset.cINF_union
tff(fact_3643_subset__mset_OcINF__insert,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),A3: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),insert2(A,A3),Aa2))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),F,A3)),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))) ) ) ) ).
% subset_mset.cINF_insert
tff(fact_3644_subset__mset_OcSUP__union,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),Ba: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( ( Ba != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Ba))
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Ba))) ) ) ) ) ) ).
% subset_mset.cSUP_union
tff(fact_3645_subset__mset_Obdd__above__empty,axiom,
! [A: $tType] : condit8047198070973881523_above(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).
% subset_mset.bdd_above_empty
tff(fact_3646_subset__mset_Obdd__below__empty,axiom,
! [A: $tType] : condit8119078960628432327_below(multiset(A),subseteq_mset(A),bot_bot(set(multiset(A)))) ).
% subset_mset.bdd_below_empty
tff(fact_3647_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [Less_eq: fun(A,fun(A,$o))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).
% order.mono.cong
tff(fact_3648_subset__mset_OcInf__le__cSup,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Aa2)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)) ) ) ) ).
% subset_mset.cInf_le_cSup
tff(fact_3649_subset__mset_OcSup__mono,axiom,
! [A: $tType,Ba: set(multiset(A)),Aa2: set(multiset(A))] :
( ( Ba != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( ! [B3: multiset(A)] :
( member(multiset(A),B3,Ba)
=> ? [X2: multiset(A)] :
( member(multiset(A),X2,Aa2)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B3),X2) ) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Ba)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)) ) ) ) ).
% subset_mset.cSup_mono
tff(fact_3650_subset__mset_OcSup__le__iff,axiom,
! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S)),A3)
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,S)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),A3) ) ) ) ) ).
% subset_mset.cSup_le_iff
tff(fact_3651_subset__mset_Ole__cInf__iff,axiom,
! [A: $tType,S: set(multiset(A)),A3: multiset(A)] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S))
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,S)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),X4) ) ) ) ) ).
% subset_mset.le_cInf_iff
tff(fact_3652_subset__mset_OcInf__mono,axiom,
! [A: $tType,Ba: set(multiset(A)),Aa2: set(multiset(A))] :
( ( Ba != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Aa2)
=> ( ! [B3: multiset(A)] :
( member(multiset(A),B3,Ba)
=> ? [X2: multiset(A)] :
( member(multiset(A),X2,Aa2)
& aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X2),B3) ) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Ba)) ) ) ) ).
% subset_mset.cInf_mono
tff(fact_3653_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),Aa2: set(multiset(A))] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F),Aa2))),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2))) ) ) ) ) ).
% subset_mset.mono_cSup
tff(fact_3654_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),Aa2: set(multiset(A))] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Aa2)
=> ( ( Aa2 != bot_bot(set(multiset(A))) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F),Aa2))) ) ) ) ) ).
% subset_mset.mono_cInf
tff(fact_3655_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,Aa2: set(A),F: fun(A,multiset(B)),U: multiset(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),U)
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X4)),U) ) ) ) ) ).
% subset_mset.cSUP_le_iff
tff(fact_3656_subset__mset_OcSUP__mono,axiom,
! [A: $tType,B: $tType,C: $tType,Aa2: set(A),G: fun(C,multiset(B)),Ba: set(C),F: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),Ba))
=> ( ! [N4: A] :
( member(A,N4,Aa2)
=> ? [X2: C] :
( member(C,X2,Ba)
& aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,N4)),aa(C,multiset(B),G,X2)) ) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),G),Ba))) ) ) ) ).
% subset_mset.cSUP_mono
tff(fact_3657_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Ba)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),Aa2),Ba) != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),Aa2),Ba))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Ba))) ) ) ) ).
% subset_mset.cSup_inter_less_eq
tff(fact_3658_subset__mset_Ole__cINF__iff,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),U: multiset(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2)))
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),U),aa(A,multiset(B),F,X4)) ) ) ) ) ).
% subset_mset.le_cINF_iff
tff(fact_3659_subset__mset_OcINF__mono,axiom,
! [C: $tType,B: $tType,A: $tType,Ba: set(A),F: fun(C,multiset(B)),Aa2: set(C),G: fun(A,multiset(B))] :
( ( Ba != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),Aa2))
=> ( ! [M3: A] :
( member(A,M3,Ba)
=> ? [X2: C] :
( member(C,X2,Aa2)
& aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(C,multiset(B),F,X2)),aa(A,multiset(B),G,M3)) ) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(C),set(multiset(B)),image2(C,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Ba))) ) ) ) ).
% subset_mset.cINF_mono
tff(fact_3660_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Aa2)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Ba)
=> ( ( aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),Aa2),Ba) != bot_bot(set(multiset(A))) )
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Ba))),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))),Aa2),Ba))) ) ) ) ).
% subset_mset.less_eq_cInf_inter
tff(fact_3661_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),Aa2: fun(C,multiset(A)),I4: set(C)] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),Aa2),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_lo(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),Aa2)),I4))),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),Aa2),I4)))) ) ) ) ) ).
% subset_mset.mono_cSUP
tff(fact_3662_subset__mset_Omono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(multiset(A),B),Aa2: fun(C,multiset(A)),I4: set(C)] :
( aa(fun(multiset(A),B),$o,mono(multiset(A),B,subseteq_mset(A)),F)
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),aa(set(C),set(multiset(A)),image2(C,multiset(A),Aa2),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(multiset(A),B,F,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),Aa2),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_lo(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F),Aa2)),I4))) ) ) ) ) ).
% subset_mset.mono_cINF
tff(fact_3663_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Ba)
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Aa2),Ba)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Ba)) ) ) ) ).
% subset_mset.cSup_subset_mono
tff(fact_3664_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Ba)
=> ( aa(set(multiset(A)),$o,aa(set(multiset(A)),fun(set(multiset(A)),$o),ord_less_eq(set(multiset(A))),Aa2),Ba)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Ba)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)) ) ) ) ).
% subset_mset.cInf_superset_mono
tff(fact_3665_subset__mset_OcSup__cInf,axiom,
! [A: $tType,S: set(multiset(A))] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),S)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_lp(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).
% subset_mset.cSup_cInf
tff(fact_3666_subset__mset_OcInf__cSup,axiom,
! [A: $tType,S: set(multiset(A))] :
( ( S != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),S)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(fun(multiset(A),$o),set(multiset(A)),collect(multiset(A)),aTP_Lamp_lq(set(multiset(A)),fun(multiset(A),$o),S))) ) ) ) ).
% subset_mset.cInf_cSup
tff(fact_3667_subset__mset_OcSUP__subset__mono,axiom,
! [B: $tType,A: $tType,Aa2: set(A),G: fun(A,multiset(B)),Ba: set(A),F: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Ba))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F,X3)),aa(A,multiset(B),G,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Ba))) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
tff(fact_3668_subset__mset_OcINF__superset__mono,axiom,
! [B: $tType,A: $tType,Aa2: set(A),G: fun(A,multiset(B)),Ba: set(A),F: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Ba))
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> ( ! [X3: A] :
( member(A,X3,Ba)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),G,X3)),aa(A,multiset(B),F,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Ba))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))) ) ) ) ) ).
% subset_mset.cINF_superset_mono
tff(fact_3669_subset__mset_OcSup__insert,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5)) ) ) ) ).
% subset_mset.cSup_insert
tff(fact_3670_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( condit8047198070973881523_above(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),X5))) ) ) ).
% subset_mset.cSup_insert_If
tff(fact_3671_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),X5)) = $ite(X5 = bot_bot(set(multiset(A))),A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5))) ) ) ).
% subset_mset.cInf_insert_If
tff(fact_3672_subset__mset_OcInf__insert,axiom,
! [A: $tType,X5: set(multiset(A)),A3: multiset(A)] :
( ( X5 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),X5)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),X5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A3),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X5)) ) ) ) ).
% subset_mset.cInf_insert
tff(fact_3673_subset__mset_OSUP__sup__distrib,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))
=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))) = aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_lr(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F),G)),Aa2)) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
tff(fact_3674_subset__mset_OcINF__inf__distrib,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( condit8119078960628432327_below(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))
=> ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),G),Aa2))) = aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_ls(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),F),G)),Aa2)) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
tff(fact_3675_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Ba)
=> ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),Aa2),Ba)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Ba)) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
tff(fact_3676_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,Aa2: set(multiset(A)),Ba: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Aa2)
=> ( ( Ba != bot_bot(set(multiset(A))) )
=> ( condit8119078960628432327_below(multiset(A),subseteq_mset(A),Ba)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),Aa2),Ba)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Ba)) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
tff(fact_3677_subset__mset_OcSUP__insert,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),A3: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( condit8047198070973881523_above(multiset(B),subseteq_mset(B),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))
=> ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),aa(set(A),set(A),insert2(A,A3),Aa2))) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),F,A3)),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image2(A,multiset(B),F),Aa2))) ) ) ) ).
% subset_mset.cSUP_insert
tff(fact_3678_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> aa(set(A),$o,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),aTP_Lamp_lu(multiset(A),set(A))),Aa2))) ) ) ).
% bdd_above_multiset_imp_finite_support
tff(fact_3679_Sup__multiset__in__multiset,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lw(set(multiset(A)),fun(A,$o),Aa2))) ) ) ).
% Sup_multiset_in_multiset
tff(fact_3680_count__Sup__multiset__nonempty,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: A] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2)
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_lv(A,fun(multiset(A),nat),X)),Aa2)) ) ) ) ).
% count_Sup_multiset_nonempty
tff(fact_3681_subset__mset_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),C2),bot_bot(set(multiset(A)))) )
<=> ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
tff(fact_3682_subset__mset_OatLeastatMost__empty__iff2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B2) ) ).
% subset_mset.atLeastatMost_empty_iff2
tff(fact_3683_subset__mset_OatLeastatMost__empty__iff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A3),B2) ) ).
% subset_mset.atLeastatMost_empty_iff
tff(fact_3684_subset__mset_OatLeastAtMost__singleton,axiom,
! [A: $tType,A3: multiset(A)] : set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,A3) = aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),bot_bot(set(multiset(A)))) ).
% subset_mset.atLeastAtMost_singleton
tff(fact_3685_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set(multiset(A)),X2: A] :
aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X2) = $ite(aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_lx(A,fun(fun(A,nat),nat),X2)),aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X)))) ).
% Inf_multiset.rep_eq
tff(fact_3686_count__Inf__multiset__nonempty,axiom,
! [A: $tType,Aa2: set(multiset(A)),X: A] :
( ( Aa2 != bot_bot(set(multiset(A))) )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),Aa2)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_lv(A,fun(multiset(A),nat),X)),Aa2)) ) ) ).
% count_Inf_multiset_nonempty
tff(fact_3687_subset__mset_OatLeastAtMost__singleton_H,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( ( A3 = B2 )
=> ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),bot_bot(set(multiset(A)))) ) ) ).
% subset_mset.atLeastAtMost_singleton'
tff(fact_3688_Sup__multiset__def,axiom,
! [A: $tType,Aa2: set(multiset(A))] :
aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),Aa2) = $ite(
( ( Aa2 != bot_bot(set(multiset(A))) )
& condit8047198070973881523_above(multiset(A),subseteq_mset(A),Aa2) ),
aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_ly(set(multiset(A)),fun(A,nat),Aa2)),
zero_zero(multiset(A)) ) ).
% Sup_multiset_def
tff(fact_3689_count__image__mset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Aa2: multiset(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),image_mset(B,A,F,Aa2)),X) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),aa(multiset(B),fun(B,nat),count(B),Aa2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),Aa2))) ).
% count_image_mset
tff(fact_3690_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,Aa2: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_lz(B,fun(A,fun(B,A)),Y),C2),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Aa2),Y))) ) ).
% sum_mset_delta'
tff(fact_3691_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Y: B,C2: A,Aa2: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ma(B,fun(A,fun(B,A)),Y),C2),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Aa2),Y))) ) ).
% sum_mset_delta
tff(fact_3692_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Aa2: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aTP_Lamp_bp(B,A),Aa2)) = one_one(A) ) ).
% prod_mset.neutral_const
tff(fact_3693_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),X: B,Aa2: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,add_mset(B,X,Aa2))) = 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,Aa2))) ) ).
% prod_mset.insert
tff(fact_3694_sum__mset__product,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add(B)
& times(B)
& semiring_0(A) )
=> ! [F: fun(B,A),Aa2: multiset(B),G: fun(C,A),Ba: multiset(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F,Aa2))),comm_m7189776963980413722m_mset(A,image_mset(C,A,G,Ba))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_mc(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),F),G),Ba),Aa2)) ) ).
% sum_mset_product
tff(fact_3695_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),M4: multiset(B),C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,image_mset(B,A,F,M4))),C2) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_cj(fun(B,A),fun(A,fun(B,A)),F),C2),M4)) ) ).
% sum_mset_distrib_right
tff(fact_3696_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F: fun(B,A),M4: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),comm_m7189776963980413722m_mset(A,image_mset(B,A,F,M4))) = comm_m7189776963980413722m_mset(A,image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ci(A,fun(fun(B,A),fun(B,A)),C2),F),M4)) ) ).
% sum_mset_distrib_left
tff(fact_3697_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H2: fun(B,A),Aa2: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bq(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,G,Aa2))),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,H2,Aa2))) ) ).
% prod_mset.distrib
tff(fact_3698_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M4: multiset(product_prod(A,B)),F: fun(A,fun(B,C)),G: fun(A,fun(B,C))] :
( ! [X3: A,Y2: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M4))
=> ( aa(B,C,aa(A,fun(B,C),F,X3),Y2) = aa(B,C,aa(A,fun(B,C),G,X3),Y2) ) )
=> ( image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),M4) = image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),M4) ) ) ).
% image_mset_cong_pair
tff(fact_3699_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,Aa2: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_md(B,fun(A,fun(B,A)),Y),C2),Aa2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Aa2),Y)) ) ).
% prod_mset_delta
tff(fact_3700_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,Aa2: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_me(B,fun(A,fun(B,A)),Y),C2),Aa2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),Aa2),Y)) ) ).
% prod_mset_delta'
tff(fact_3701_prod__mset_Ounion__disjoint,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Aa2: multiset(A),Ba: multiset(A),G: fun(A,B)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Aa2),Ba) = zero_zero(multiset(A)) )
=> ( aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Aa2),Ba))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,Aa2))),aa(multiset(B),B,comm_m9189036328036947845d_mset(B),image_mset(A,B,G,Ba))) ) ) ) ).
% prod_mset.union_disjoint
tff(fact_3702_Inf__multiset__def,axiom,
! [A: $tType] : complete_Inf_Inf(multiset(A)) = aa(fun(set(fun(A,nat)),fun(A,nat)),fun(set(multiset(A)),multiset(A)),map_fun(set(multiset(A)),set(fun(A,nat)),fun(A,nat),multiset(A),image2(multiset(A),fun(A,nat),count(A)),abs_multiset(A)),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat))) ).
% Inf_multiset_def
tff(fact_3703_sum__mset__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,Aa2: multiset(B)] : comm_m7189776963980413722m_mset(A,image_mset(B,A,aTP_Lamp_cd(A,fun(B,A),Y),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(multiset(B),nat,size_size(multiset(B)),Aa2))),Y) ) ).
% sum_mset_constant
tff(fact_3704_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),compow(set(product_prod(A,A)),N,R))
<=> ? [F6: fun(nat,A)] :
( ( aa(nat,A,F6,zero_zero(nat)) = A3 )
& ( aa(nat,A,F6,N) = B2 )
& ! [I2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),N)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,I2)),aa(nat,A,F6,aa(nat,nat,suc,I2))),R) ) ) ) ).
% relpow_fun_conv
tff(fact_3705_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
~ ! [F3: fun(code_integer,option(product_prod($o,list(code_term)))),D2: code_integer,I3: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F3),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),D2),I3)) ).
% exhaustive_integer'.cases
tff(fact_3706_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set(product_prod(A,A)),X2: A,Y4: A,Z5: A] :
( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),compow(set(product_prod(A,A)),N,R))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z5),R) )
=> ? [W: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),W),R)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z5),compow(set(product_prod(A,A)),N,R)) ) ) ).
% relpow_Suc_D2'
tff(fact_3707_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),compow(set(product_prod(A,A)),zero_zero(nat),R))
=> ( X = Y ) ) ).
% relpow_0_E
tff(fact_3708_relpow__0__I,axiom,
! [A: $tType,X: A,R: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),compow(set(product_prod(A,A)),zero_zero(nat),R)) ).
% relpow_0_I
tff(fact_3709_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),compow(set(product_prod(A,A)),N,R))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),R) ) ) ).
% relpow_Suc_E
tff(fact_3710_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R: set(product_prod(A,A)),Z2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),compow(set(product_prod(A,A)),N,R))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).
% relpow_Suc_I
tff(fact_3711_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
=> ? [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),N,R)) ) ) ).
% relpow_Suc_D2
tff(fact_3712_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R))
=> ~ ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),N,R)) ) ) ).
% relpow_Suc_E2
tff(fact_3713_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z2: A,N: nat] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),compow(set(product_prod(A,A)),N,R))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R)) ) ) ).
% relpow_Suc_I2
tff(fact_3714_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set(product_prod(A,A)),X2: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),compow(fun(A,fun(A,$o)),N,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X2),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),compow(set(product_prod(A,A)),N,R)) ) ).
% relpowp_relpow_eq
tff(fact_3715_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),N,R))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N = aa(nat,nat,suc,M3) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),compow(set(product_prod(A,A)),M3,R))
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),R) ) ) ) ) ).
% relpow_E
tff(fact_3716_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),compow(set(product_prod(A,A)),N,R))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y2: A,M3: nat] :
( ( N = aa(nat,nat,suc,M3) )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2),compow(set(product_prod(A,A)),M3,R)) ) ) ) ) ).
% relpow_E2
tff(fact_3717_relpow__empty,axiom,
! [A: $tType,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
=> ( compow(set(product_prod(A,A)),N,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).
% relpow_empty
tff(fact_3718_exhaustive__int_H_Ocases,axiom,
! [X: product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))] :
~ ! [F3: fun(int,option(product_prod($o,list(code_term)))),D2: int,I3: int] : X != aa(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3)) ).
% exhaustive_int'.cases
tff(fact_3719_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))] :
~ ! [F3: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: int,I3: int] : X != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3)) ).
% full_exhaustive_int'.cases
tff(fact_3720_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
~ ! [F3: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: code_integer,I3: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F3),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),D2),I3)) ).
% full_exhaustive_integer'.cases
tff(fact_3721_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X: set(fun(A,nat))] :
( aa(set(fun(A,nat)),$o,aa(set(fun(A,nat)),fun(set(fun(A,nat)),$o),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_mh(fun(A,nat),$o))),X),X)
=> ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(fun(A,nat)),set(multiset(A)),image2(fun(A,nat),multiset(A),abs_multiset(A)),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat)),X)) ) ) ).
% Inf_multiset.abs_eq
tff(fact_3722_irreflp__irrefl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irreflp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
<=> irrefl(A,R) ) ).
% irreflp_irrefl_eq
tff(fact_3723_equivp__equiv,axiom,
! [A: $tType,Aa2: set(product_prod(A,A))] :
( equiv_equiv(A,top_top(set(A)),Aa2)
<=> equiv_equivp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),Aa2)) ) ).
% equivp_equiv
tff(fact_3724_asymp__asym__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asymp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
<=> asym(A,R) ) ).
% asymp_asym_eq
tff(fact_3725_asym_Ocases,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
=> ! [A9: A,B9: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B9),A3)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B9),A9),A3) ) ) ).
% asym.cases
tff(fact_3726_asym_Osimps,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
<=> ? [R7: set(product_prod(A,A))] :
( ( A3 = R7 )
& ! [X4: A,Xa2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R7)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R7) ) ) ) ).
% asym.simps
tff(fact_3727_asym_Ointros,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R) )
=> asym(A,R) ) ).
% asym.intros
tff(fact_3728_asymD,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A] :
( asym(A,R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),R) ) ) ).
% asymD
tff(fact_3729_asym__iff,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( asym(A,R)
<=> ! [X4: A,Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R)
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R) ) ) ).
% asym_iff
tff(fact_3730_Inf__multiset_Orsp,axiom,
! [A: $tType] : aa(fun(set(fun(A,nat)),fun(A,nat)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(fun(A,nat)),fun(A,nat)),$o),bNF_rel_fun(set(fun(A,nat)),set(fun(A,nat)),fun(A,nat),fun(A,nat),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_mh(fun(A,nat),$o))),bNF_eq_onp(fun(A,nat),aTP_Lamp_mh(fun(A,nat),$o))),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat))) ).
% Inf_multiset.rsp
tff(fact_3731_empty__transfer,axiom,
! [A: $tType,B: $tType,Aa2: fun(A,fun(B,$o))] : aa(set(B),$o,aa(set(A),fun(set(B),$o),bNF_rel_set(A,B,Aa2),bot_bot(set(A))),bot_bot(set(B))) ).
% empty_transfer
tff(fact_3732_ordLess__iff,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
<=> ( order_well_order_on(A,field2(A,R2),R2)
& order_well_order_on(B,field2(B,R4),R4)
& ~ ? [X_12: fun(B,A)] : bNF_Wellorder_embed(B,A,R4,R2,X_12) ) ) ).
% ordLess_iff
tff(fact_3733_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R2: set(product_prod(B,A))] : range2(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R2)) = aa(set(A),set(A),insert2(A,B2),range2(B,A,R2)) ).
% Range_insert
tff(fact_3734_Range__empty,axiom,
! [B: $tType,A: $tType] : range2(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(A)) ).
% Range_empty
tff(fact_3735_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
( member(A,A3,range2(B,A,R2))
<=> ? [Y3: B] : member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y3),A3),R2) ) ).
% Range_iff
tff(fact_3736_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A))] :
( member(A,B2,range2(B,A,R2))
=> ~ ! [A4: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),B2),R2) ) ).
% RangeE
tff(fact_3737_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> member(B,B2,range2(A,B,R2)) ) ).
% Range.intros
tff(fact_3738_Range_Osimps,axiom,
! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
( member(A,A3,range2(B,A,R2))
<=> ? [A5: B,B4: A] :
( ( A3 = B4 )
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B4),R2) ) ) ).
% Range.simps
tff(fact_3739_Range_Ocases,axiom,
! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
( member(A,A3,range2(B,A,R2))
=> ~ ! [A4: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A4),A3),R2) ) ).
% Range.cases
tff(fact_3740_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> ~ ? [X_13: fun(B,A)] : bNF_Wellorder_embed(B,A,R4,R2,X_13) ) ).
% ordLess_not_embed
tff(fact_3741_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
( ( range2(B,A,R2) = bot_bot(set(A)) )
<=> ( R2 = bot_bot(set(product_prod(B,A))) ) ) ).
% Range_empty_iff
tff(fact_3742_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_We4044943003108391690rdLess(A,B))
=> ( bNF_Wellorder_embed(A,B,R1,R22,F)
=> ( aa(set(A),set(B),image2(A,B,F),field2(A,R1)) != field2(B,R22) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_3743_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R32: set(product_prod(C,C)),F13: fun(A,C),F23: fun(B,C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_We4044943003108391690rdLess(A,B))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R22),R32),bNF_We4044943003108391690rdLess(B,C))
=> ( bNF_Wellorder_embed(A,C,R1,R32,F13)
=> ( bNF_Wellorder_embed(B,C,R22,R32,F23)
=> member(product_prod(set(C),set(C)),aa(set(C),product_prod(set(C),set(C)),aa(set(C),fun(set(C),product_prod(set(C),set(C))),product_Pair(set(C),set(C)),aa(set(A),set(C),image2(A,C,F13),field2(A,R1))),aa(set(B),set(C),image2(B,C,F23),field2(B,R22))),bNF_We413866401316099525erIncl(C,R32)) ) ) ) ) ).
% embed_ordLess_ofilterIncl
tff(fact_3744_Inf__multiset_Otransfer,axiom,
! [A: $tType] : aa(fun(set(multiset(A)),multiset(A)),$o,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(multiset(A)),multiset(A)),$o),bNF_rel_fun(set(fun(A,nat)),set(multiset(A)),fun(A,nat),multiset(A),bNF_rel_set(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),pcr_multiset(A,A,fequal(A))),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat))),complete_Inf_Inf(multiset(A))) ).
% Inf_multiset.transfer
tff(fact_3745_wf__UN,axiom,
! [B: $tType,A: $tType,I4: set(A),R2: fun(A,set(product_prod(B,B)))] :
( ! [I3: A] :
( member(A,I3,I4)
=> wf(B,aa(A,set(product_prod(B,B)),R2,I3)) )
=> ( ! [I3: A,J2: A] :
( member(A,I3,I4)
=> ( member(A,J2,I4)
=> ( ( 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,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),I4))) ) ) ).
% wf_UN
tff(fact_3746_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_3747_Domain__empty,axiom,
! [B: $tType,A: $tType] : domain(A,B,bot_bot(set(product_prod(A,B)))) = bot_bot(set(A)) ).
% Domain_empty
tff(fact_3748_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] : domain(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2)) = aa(set(A),set(A),insert2(A,A3),domain(A,B,R2)) ).
% Domain_insert
tff(fact_3749_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
( member(A,A3,domain(A,B,R2))
=> ~ ! [B3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2) ) ).
% Domain.cases
tff(fact_3750_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
( member(A,A3,domain(A,B,R2))
<=> ? [A5: A,B4: B] :
( ( A3 = A5 )
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4),R2) ) ) ).
% Domain.simps
tff(fact_3751_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R2)
=> member(A,A3,domain(A,B,R2)) ) ).
% Domain.DomainI
tff(fact_3752_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
( member(A,A3,domain(A,B,R2))
=> ~ ! [B3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R2) ) ).
% DomainE
tff(fact_3753_Domain__iff,axiom,
! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
( member(A,A3,domain(A,B,R2))
<=> ? [Y3: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y3),R2) ) ).
% Domain_iff
tff(fact_3754_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R: set(product_prod(A,A)),Y: A] :
( ~ member(A,X,domain(A,A,R))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R))
<=> ( X = Y ) ) ) ).
% Not_Domain_rtrancl
tff(fact_3755_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
( ( domain(A,B,R2) = bot_bot(set(A)) )
<=> ( R2 = bot_bot(set(product_prod(A,B))) ) ) ).
% Domain_empty_iff
tff(fact_3756_Domain__unfold,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : domain(A,B,R2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mi(set(product_prod(A,B)),fun(A,$o),R2)) ).
% Domain_unfold
tff(fact_3757_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_3758_wf__min,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
=> ( ( R != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M3: A] : ~ member(A,M3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),domain(A,A,R)),range2(A,A,R))) ) ) ).
% wf_min
tff(fact_3759_wf__Un,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( wf(A,R2)
=> ( wf(A,S2)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,R2)),range2(A,A,S2)) = bot_bot(set(A)) )
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S2)) ) ) ) ).
% wf_Un
tff(fact_3760_wf__Union,axiom,
! [A: $tType,R: set(set(product_prod(A,A)))] :
( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
=> wf(A,X3) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
=> ! [Xa4: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa4,R)
=> ( ( X3 != Xa4 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),domain(A,A,X3)),range2(A,A,Xa4)) = bot_bot(set(A)) ) ) ) )
=> wf(A,aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)) ) ) ).
% wf_Union
tff(fact_3761_wf__max,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,converse(A,A,R))
=> ( ( R != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M3: A] : ~ member(A,M3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),range2(A,A,R)),domain(A,A,R))) ) ) ).
% wf_max
tff(fact_3762_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R: set(product_prod(A,B))] :
( member(A,X,domain(A,B,R))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),fun_of_rel(A,B,R,X)),R) ) ).
% for_in_RI
tff(fact_3763_ord__to__filter__compat,axiom,
! [A: $tType,R0: set(product_prod(A,A))] : bNF_Wellorder_compat(set(product_prod(A,A)),set(A),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),inf_inf(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),bNF_We4044943003108391690rdLess(A,A)),product_Sigma(set(product_prod(A,A)),set(product_prod(A,A)),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),insert2(set(product_prod(A,A)),R0),bot_bot(set(set(product_prod(A,A)))))),aTP_Lamp_mj(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),R0))),bNF_We413866401316099525erIncl(A,R0),bNF_We8469521843155493636filter(A,R0)) ).
% ord_to_filter_compat
tff(fact_3764_Range__def,axiom,
! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : range2(A,B,X2) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),X2))) ).
% Range_def
tff(fact_3765_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X2: A] :
( aa(A,$o,rangep(B,A,aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X2)
<=> member(A,X2,range2(B,A,R2)) ) ).
% Rangep_Range_eq
tff(fact_3766_compat__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F: fun(A,B)] :
( bNF_Wellorder_compat(A,B,R2,R4,F)
<=> ! [A5: A,B4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),R2)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,A5)),aa(A,B,F,B4)),R4) ) ) ).
% compat_def
tff(fact_3767_in__chain__finite,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [Aa2: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),Aa2)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> member(A,aa(set(A),A,complete_Sup_Sup(A),Aa2),Aa2) ) ) ) ) ).
% in_chain_finite
tff(fact_3768_cofinite__bot,axiom,
! [A: $tType] :
( ( cofinite(A) = bot_bot(filter(A)) )
<=> aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ).
% cofinite_bot
tff(fact_3769_drop__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : bit_se4197421643247451524op_bit(A,N,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).
% drop_bit_of_1
tff(fact_3770_chain__empty,axiom,
! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).
% chain_empty
tff(fact_3771_chain__singleton,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ) ).
% chain_singleton
tff(fact_3772_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_3773_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( aa(nat,$o,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_3774_drop__bit__exp__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M2: nat,N: nat] :
bit_se4197421643247451524op_bit(A,M2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,
aa(A,fun(A,A),times_times(A),
aa($o,A,zero_neq_one_of_bool(A),
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).
% drop_bit_exp_eq
tff(fact_3775_bit__double__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),N)
<=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
& ( N != zero_zero(nat) )
& bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).
% bit_double_iff
tff(fact_3776_cut__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),R: set(product_prod(A,A)),X: A,X2: A] :
aa(A,B,cut(A,B,F,R,X),X2) = $ite(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X),R),aa(A,B,F,X2),undefined(B)) ).
% cut_def
tff(fact_3777_pairwise__alt,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
( pairwise(A,R,S)
<=> ! [X4: A] :
( member(A,X4,S)
=> ! [Xa2: A] :
( member(A,Xa2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert2(A,X4),bot_bot(set(A)))))
=> aa(A,$o,aa(A,fun(A,$o),R,X4),Xa2) ) ) ) ).
% pairwise_alt
tff(fact_3778_bit__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N)
<=> bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).
% bit_minus_1_iff
tff(fact_3779_pairwise__empty,axiom,
! [A: $tType,P: fun(A,fun(A,$o))] : pairwise(A,P,bot_bot(set(A))) ).
% pairwise_empty
tff(fact_3780_bit__minus__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,N: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3)),N)
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
& ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N) ) ) ) ).
% bit_minus_iff
tff(fact_3781_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A3: A,R: set(product_prod(A,A)),F: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3),R)
=> ( aa(A,B,cut(A,B,F,R,A3),X) = aa(A,B,F,X) ) ) ).
% cut_apply
tff(fact_3782_cuts__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R: set(product_prod(A,A)),X: A,G: fun(A,B)] :
( ( cut(A,B,F,R,X) = cut(A,B,G,R,X) )
<=> ! [Y3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R)
=> ( aa(A,B,F,Y3) = aa(A,B,G,Y3) ) ) ) ).
% cuts_eq
tff(fact_3783_pairwise__singleton,axiom,
! [A: $tType,P: fun(A,fun(A,$o)),Aa2: A] : pairwise(A,P,aa(set(A),set(A),insert2(A,Aa2),bot_bot(set(A)))) ).
% pairwise_singleton
tff(fact_3784_bit__mask__sub__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M2: nat,N: nat] :
( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M2)),one_one(A))),N)
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),M2) ) ) ) ).
% bit_mask_sub_iff
tff(fact_3785_above__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] : order_above(A,R2,A3) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A3)) ).
% above_def
tff(fact_3786_ID_Oin__rel,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: A,B2: B] :
( aa(B,$o,aa(A,fun(B,$o),bNF_id_bnf(fun(A,fun(B,$o)),R),A3),B2)
<=> ? [Z4: product_prod(A,B)] :
( member(product_prod(A,B),Z4,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_mk(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),R)))
& ( aa(product_prod(A,B),A,bNF_id_bnf(fun(product_prod(A,B),A),product_fst(A,B)),Z4) = A3 )
& ( aa(product_prod(A,B),B,bNF_id_bnf(fun(product_prod(A,B),B),product_snd(A,B)),Z4) = B2 ) ) ) ).
% ID.in_rel
tff(fact_3787_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_3788_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_3789_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra2: fun(A,fun(A,$o))] :
( ! [Z3: A] :
( member(A,Z3,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))
=> aa(A,$o,aa(A,fun(A,$o),Ra2,Z3),Z3) )
=> aa(A,$o,aa(A,fun(A,$o),bNF_id_bnf(fun(A,fun(A,$o)),Ra2),X),X) ) ).
% ID.rel_refl_strong
tff(fact_3790_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X: A,Y: B,Ra2: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),bNF_id_bnf(fun(A,fun(B,$o)),R),X),Y)
=> ( ! [Z3: A,Yb: B] :
( member(A,Z3,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))
=> ( member(B,Yb,aa(set(B),set(B),insert2(B,Y),bot_bot(set(B))))
=> ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),bNF_id_bnf(fun(A,fun(B,$o)),Ra2),X),Y) ) ) ).
% ID.rel_mono_strong
tff(fact_3791_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya2: A,Y: B,Xa: B,R: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
( ( X = Ya2 )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( member(A,Z3,aa(set(A),set(A),insert2(A,Ya2),bot_bot(set(A))))
=> ( member(B,Yb,aa(set(B),set(B),insert2(B,Xa),bot_bot(set(B))))
=> ( aa(B,$o,aa(A,fun(B,$o),R,Z3),Yb)
<=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z3),Yb) ) ) )
=> ( aa(B,$o,aa(A,fun(B,$o),bNF_id_bnf(fun(A,fun(B,$o)),R),X),Y)
<=> aa(B,$o,aa(A,fun(B,$o),bNF_id_bnf(fun(A,fun(B,$o)),Ra2),Ya2),Xa) ) ) ) ) ).
% ID.rel_cong
tff(fact_3792_is__num_Ocases,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
=> ( ( A3 != one_one(A) )
=> ( ! [X3: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X3) )
=> ~ neg_numeral_is_num(A,X3) )
=> ~ ! [X3: A,Y2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y2) )
=> ( neg_numeral_is_num(A,X3)
=> ~ neg_numeral_is_num(A,Y2) ) ) ) ) ) ) ).
% is_num.cases
tff(fact_3793_subset__mset_OatLeastatMost__empty,axiom,
! [A: $tType,B2: multiset(A),A3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),B2),A3)
=> ( set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.atLeastatMost_empty
tff(fact_3794_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_3795_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( 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_3796_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( 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_3797_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_3798_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_3799_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_3800_subset__mset_OacyclicI__order,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F: fun(A,multiset(B))] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F,B3)),aa(A,multiset(B),F,A4)) )
=> transitive_acyclic(A,R2) ) ).
% subset_mset.acyclicI_order
tff(fact_3801_is__unit__unit__factor,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).
% is_unit_unit_factor
tff(fact_3802_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_3803_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_3804_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_3805_subset__implies__mult,axiom,
! [A: $tType,Aa2: multiset(A),Ba: multiset(A),R2: set(product_prod(A,A))] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),Aa2),Ba)
=> member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),Aa2),Ba),mult(A,R2)) ) ).
% subset_implies_mult
tff(fact_3806_unit__factor__is__unit,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> dvd_dvd(A,unit_f5069060285200089521factor(A,A3),one_one(A)) ) ) ).
% unit_factor_is_unit
tff(fact_3807_unit__factor__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = $ite(
( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) ),
zero_zero(A),
one_one(A) ) ) ).
% unit_factor_gcd
tff(fact_3808_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_3809_unit__factor__Gcd,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
unit_f5069060285200089521factor(A,gcd_Gcd(A,Aa2)) = $ite(gcd_Gcd(A,Aa2) = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% unit_factor_Gcd
tff(fact_3810_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),A3),bot_bot(set(multiset(A))))) ).
% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_3811_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] : set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),minus_minus(set(multiset(A))),set_atLeastAtMost(multiset(A),subseteq_mset(A),A3,B2)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),B2),bot_bot(set(multiset(A))))) ).
% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3812_subset__mset_OgreaterThanLessThan__empty,axiom,
! [A: $tType,L: multiset(A),K: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
=> ( set_gr287244882034783167ssThan(multiset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.greaterThanLessThan_empty
tff(fact_3813_subset__mset_OIio__Int__singleton,axiom,
! [A: $tType,K: multiset(A),X: multiset(A)] :
aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),inf_inf(set(multiset(A))),set_lessThan(multiset(A),subset_mset(A),K)),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A))))) = $ite(aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),X),K),aa(set(multiset(A)),set(multiset(A)),insert2(multiset(A),X),bot_bot(set(multiset(A)))),bot_bot(set(multiset(A)))) ).
% subset_mset.Iio_Int_singleton
tff(fact_3814_subset__mset_OatLeastLessThan__empty,axiom,
! [A: $tType,B2: multiset(A),A3: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B2),A3)
=> ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.atLeastLessThan_empty
tff(fact_3815_subset__mset_OatLeastLessThan__empty__iff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( ( set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A3),B2) ) ).
% subset_mset.atLeastLessThan_empty_iff
tff(fact_3816_subset__mset_OatLeastLessThan__empty__iff2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_atLeastLessThan(multiset(A),subseteq_mset(A),subset_mset(A),A3,B2) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),A3),B2) ) ).
% subset_mset.atLeastLessThan_empty_iff2
tff(fact_3817_subset__mset_OgreaterThanAtMost__empty,axiom,
! [A: $tType,L: multiset(A),K: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),L),K)
=> ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) ) ) ).
% subset_mset.greaterThanAtMost_empty
tff(fact_3818_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
! [A: $tType,K: multiset(A),L: multiset(A)] :
( ( set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) = bot_bot(set(multiset(A))) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).
% subset_mset.greaterThanAtMost_empty_iff
tff(fact_3819_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
! [A: $tType,K: multiset(A),L: multiset(A)] :
( ( bot_bot(set(multiset(A))) = set_gr3752724095348155675AtMost(multiset(A),subseteq_mset(A),subset_mset(A),K,L) )
<=> ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subset_mset(A),K),L) ) ).
% subset_mset.greaterThanAtMost_empty_iff2
tff(fact_3820_subset__mset_Osum__pos,axiom,
! [B: $tType,A: $tType,I4: set(A),F: fun(A,multiset(B))] :
( aa(set(A),$o,finite_finite(A),I4)
=> ( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F,I3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),zero_zero(multiset(B))),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,I4)) ) ) ) ).
% subset_mset.sum_pos
tff(fact_3821_subset__mset_Osum__strict__mono,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,multiset(B)),G: fun(A,multiset(B))] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),aa(A,multiset(B),F,X3)),aa(A,multiset(B),G,X3)) )
=> aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subset_mset(B),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F,Aa2)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),G,Aa2)) ) ) ) ).
% subset_mset.sum_strict_mono
tff(fact_3822_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_3823_Lcm__no__units,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] : gcd_Lcm(A,Aa2) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ml(A,$o)))) ) ).
% Lcm_no_units
tff(fact_3824_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_3825_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_3826_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_3827_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_3828_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_3829_Lcm__1__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
( ( gcd_Lcm(A,Aa2) = one_one(A) )
<=> ! [X4: A] :
( member(A,X4,Aa2)
=> dvd_dvd(A,X4,one_one(A)) ) ) ) ).
% Lcm_1_iff
tff(fact_3830_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_3831_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_3832_normalize__mult__unit__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [B2: A,A3: A] :
( 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_3833_normalize__mult__unit__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( 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_3834_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_3835_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_3836_Lcm__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A] : gcd_Lcm(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% Lcm_singleton
tff(fact_3837_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_3838_Gcd__singleton,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A] : gcd_Gcd(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% Gcd_singleton
tff(fact_3839_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_3840_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A),C2: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),Aa2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,Aa2))) ) ) ) ).
% Lcm_mult
tff(fact_3841_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( dvd_dvd(A,A3,one_one(A))
<=> ( A3 = one_one(A) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
tff(fact_3842_is__unit__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( dvd_dvd(A,A3,one_one(A))
=> ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).
% is_unit_normalize
tff(fact_3843_normalize__1__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
<=> dvd_dvd(A,A3,one_one(A)) ) ) ).
% normalize_1_iff
tff(fact_3844_associated__unit,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> ( dvd_dvd(A,A3,one_one(A))
=> dvd_dvd(A,B2,one_one(A)) ) ) ) ).
% associated_unit
tff(fact_3845_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_3846_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_3847_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_3848_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_3849_Lcm__coprime,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [A4: A,B3: A] :
( member(A,A4,Aa2)
=> ( member(A,B3,Aa2)
=> ( ( A4 != B3 )
=> algebr8660921524188924756oprime(A,A4,B3) ) ) )
=> ( gcd_Lcm(A,Aa2) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_mm(A,A)),Aa2)) ) ) ) ) ) ).
% Lcm_coprime
tff(fact_3850_Lcm__nat__empty,axiom,
gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).
% Lcm_nat_empty
tff(fact_3851_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C2: A,Aa2: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),Aa2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,Aa2))) ) ).
% Gcd_mult
tff(fact_3852_unit__factor__Lcm,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Aa2: set(A)] :
unit_f5069060285200089521factor(A,gcd_Lcm(A,Aa2)) = $ite(gcd_Lcm(A,Aa2) = zero_zero(A),zero_zero(A),one_one(A)) ) ).
% unit_factor_Lcm
tff(fact_3853_Gcd__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A),B2: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),Aa2)) = 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),Aa2))) ) ) ) ).
% Gcd_fin_mult
tff(fact_3854_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_3855_Lcm__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A),B2: A] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),Aa2)) = 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),Aa2))) ) ) ) ).
% Lcm_fin_mult
tff(fact_3856_gcd__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).
% gcd_lcm
tff(fact_3857_Lcm__eq__Max__nat,axiom,
! [M4: set(nat)] :
( aa(set(nat),$o,finite_finite(nat),M4)
=> ( ( M4 != bot_bot(set(nat)) )
=> ( ~ member(nat,zero_zero(nat),M4)
=> ( ! [M3: nat,N4: nat] :
( member(nat,M3,M4)
=> ( member(nat,N4,M4)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N4),M4) ) )
=> ( gcd_Lcm(nat,M4) = lattic643756798349783984er_Max(nat,M4) ) ) ) ) ) ).
% Lcm_eq_Max_nat
tff(fact_3858_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) )
<=> ( dvd_dvd(A,A3,one_one(A))
& dvd_dvd(A,B2,one_one(A)) ) ) ) ).
% lcm_eq_1_iff
tff(fact_3859_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_3860_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_3861_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_3862_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
( dvd_dvd(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),Aa2),one_one(A))
<=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),Aa2) = one_one(A) ) ) ) ).
% is_unit_Lcm_fin_iff
tff(fact_3863_unit__factor__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = $ite(
( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ),
zero_zero(A),
one_one(A) ) ) ).
% unit_factor_lcm
tff(fact_3864_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_3865_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_3866_Lcm__2,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,B2: A] : gcd_Lcm(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).
% Lcm_2
tff(fact_3867_lcm_Osemigroup__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> semigroup(A,gcd_lcm(A)) ) ).
% lcm.semigroup_axioms
tff(fact_3868_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,Aa2: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),insert2(A,A3),Aa2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ).
% Lcm_fin.insert_remove
tff(fact_3869_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,Aa2: set(A)] :
( member(A,A3,Aa2)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),Aa2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).
% Lcm_fin.remove
tff(fact_3870_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_3871_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_3872_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_3873_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_3874_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_3875_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
aa(set(A),A,semiring_gcd_Lcm_fin(A),Aa2) = $ite(aa(set(A),$o,finite_finite(A),Aa2),finite_fold(A,A,gcd_lcm(A),one_one(A),Aa2),zero_zero(A)) ) ).
% Lcm_fin.eq_fold
tff(fact_3876_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_3877_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_3878_mult__lcm__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).
% mult_lcm_left
tff(fact_3879_Lcm__in__lcm__closed__set__nat,axiom,
! [M4: set(nat)] :
( aa(set(nat),$o,finite_finite(nat),M4)
=> ( ( M4 != bot_bot(set(nat)) )
=> ( ! [M3: nat,N4: nat] :
( member(nat,M3,M4)
=> ( member(nat,N4,M4)
=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M3),N4),M4) ) )
=> member(nat,gcd_Lcm(nat,M4),M4) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
tff(fact_3880_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_3881_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_3882_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_3883_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( 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_3884_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_3885_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_3886_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Aa2: set(A)] :
( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),Aa2) = one_one(A) )
<=> ( ! [X4: A] :
( member(A,X4,Aa2)
=> dvd_dvd(A,X4,one_one(A)) )
& aa(set(A),$o,finite_finite(A),Aa2) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_3887_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_3888_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_3889_in__range_Osimps,axiom,
! [H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ! [X4: nat] :
( member(nat,X4,As)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H2)) ) ) ).
% in_range.simps
tff(fact_3890_in__range_Oelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ~ ! [X4: nat] :
( member(nat,X4,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H3)) ) ) ) ) ).
% in_range.elims(1)
tff(fact_3891_in__range_Oelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ~ ! [X2: nat] :
( member(nat,X2,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,H3)) ) ) ) ).
% in_range.elims(2)
tff(fact_3892_in__range_Oelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ! [X3: nat] :
( member(nat,X3,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),lim(product_unit,H3)) ) ) ) ).
% in_range.elims(3)
tff(fact_3893_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),X: A,H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ( ( ref_get(A,H2,R2) = X )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,R2)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,R2)),lim(product_unit,H2)) ) ) ) ).
% sngr_assn_raw.simps
tff(fact_3894_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
tff(fact_3895_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) ) ) ) ).
% sngr_assn_raw.elims(2)
tff(fact_3896_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) ) ) ) ).
% sngr_assn_raw.elims(3)
tff(fact_3897_relH__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: ref(A)] :
( relH(As,H2,H5)
=> ( member(nat,addr_of_ref(A,R2),As)
=> ( ref_get(A,H2,R2) = ref_get(A,H5,R2) ) ) ) ) ).
% relH_ref
tff(fact_3898_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( ( (Y)
<=> ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) )
=> ~ aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)))) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
tff(fact_3899_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ~ ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
tff(fact_3900_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,sngr_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ( ( ref_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H3)) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
tff(fact_3901_relH__set__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),As: set(nat),H2: heap_ext(product_unit),X: A] :
( ~ member(nat,addr_of_ref(A,R2),As)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> relH(As,H2,ref_set(A,R2,X,H2)) ) ) ) ).
% relH_set_ref
tff(fact_3902_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),X: list(A),H2: heap_ext(product_unit),As: set(nat)] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,R2,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
<=> ( ( array_get(A,H2,R2) = X )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,R2)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,R2)),lim(product_unit,H2)) ) ) ) ).
% snga_assn_raw.simps
tff(fact_3903_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( (Y)
<=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
tff(fact_3904_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) ) ) ) ).
% snga_assn_raw.elims(2)
tff(fact_3905_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) ) ) ) ).
% snga_assn_raw.elims(3)
tff(fact_3906_relH__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: array(A)] :
( relH(As,H2,H5)
=> ( member(nat,addr_of_array(A,R2),As)
=> ( array_get(A,H2,R2) = array_get(A,H5,R2) ) ) ) ) ).
% relH_array
tff(fact_3907_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
<=> (Y) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( ( (Y)
<=> ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) )
=> ~ aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)))) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
tff(fact_3908_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ~ ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
tff(fact_3909_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,snga_assn_raw(A,X,Xa),Xb)
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb)))
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))))
=> ( ( array_get(A,H3,X) = Xa )
& ( As3 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X)),bot_bot(set(nat))) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H3)) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
tff(fact_3910_relH__set__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: array(A),As: set(nat),H2: heap_ext(product_unit),X: list(A)] :
( ~ member(nat,addr_of_array(A,R2),As)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> relH(As,H2,array_set(A,R2,X,H2)) ) ) ) ).
% relH_set_array
tff(fact_3911_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [R2: set(product_prod(A,A)),As8: fun(A,B)] :
( bNF_Ca3754400796208372196lChain(A,B,R2,As8)
<=> ! [I2: A,J5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J5),R2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As8,I2)),aa(A,B,As8,J5)) ) ) ) ).
% relChain_def
tff(fact_3912_prod_H__def,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ( groups1962203154675924110t_prod(A,B) = groups_comm_monoid_G(B,A,times_times(B),one_one(B)) ) ) ).
% prod'_def
tff(fact_3913_Gr__def,axiom,
! [B: $tType,A: $tType,Aa2: set(A),F: fun(A,B)] : bNF_Gr(A,B,Aa2,F) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Aa2),F)) ).
% Gr_def
tff(fact_3914_prod_Oreindex__bij__betw__not__neutral,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [S5: set(A),T4: set(B),H2: fun(A,B),S: set(A),T5: set(B),G: fun(B,C)] :
( aa(set(A),$o,finite_finite(A),S5)
=> ( aa(set(B),$o,finite_finite(B),T4)
=> ( bij_betw(A,B,H2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),T4))
=> ( ! [A4: A] :
( member(A,A4,S5)
=> ( aa(B,C,G,aa(A,B,H2,A4)) = one_one(C) ) )
=> ( ! [B3: B] :
( member(B,B3,T4)
=> ( aa(B,C,G,B3) = one_one(C) ) )
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mo(fun(A,B),fun(fun(B,C),fun(A,C)),H2),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T5) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
tff(fact_3915_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Aa2: set(B)] :
( bij_betw(A,B,F,bot_bot(set(A)),Aa2)
=> ( Aa2 = bot_bot(set(B)) ) ) ).
% bij_betw_empty1
tff(fact_3916_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Aa2: set(A)] :
( bij_betw(A,B,F,Aa2,bot_bot(set(B)))
=> ( Aa2 = bot_bot(set(A)) ) ) ).
% bij_betw_empty2
tff(fact_3917_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,Aa2: set(A),F: fun(A,B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx),bNF_Gr(A,B,Aa2,F))
=> member(A,X,Aa2) ) ).
% GrD1
tff(fact_3918_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,Aa2: set(A),F: fun(A,B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx),bNF_Gr(A,B,Aa2,F))
=> ( aa(A,B,F,X) = Fx ) ) ).
% GrD2
tff(fact_3919_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,Aa2: set(A),F: fun(A,B),A13: set(B)] :
( ~ member(A,B2,Aa2)
=> ( ~ member(B,aa(A,B,F,B2),A13)
=> ( bij_betw(A,B,F,Aa2,A13)
<=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A13),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw3
tff(fact_3920_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,Aa2: set(A),F: fun(A,B),A13: set(B)] :
( ~ member(A,B2,Aa2)
=> ( ~ member(B,aa(A,B,F,B2),A13)
=> ( bij_betw(A,B,F,Aa2,A13)
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A13),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw
tff(fact_3921_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Aa2: set(A),Ba: set(B),Ca: set(A),D4: set(B)] :
( bij_betw(A,B,F,Aa2,Ba)
=> ( bij_betw(A,B,F,Ca,D4)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Ba),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ca),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Ba),D4)) ) ) ) ).
% bij_betw_combine
tff(fact_3922_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Aa2: set(A),Ca: set(A),Ba: set(B),D4: set(B)] :
( bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ca),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Ba),D4))
=> ( bij_betw(A,B,F,Ca,D4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ca) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Ba),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F,Aa2,Ba) ) ) ) ) ).
% bij_betw_partition
tff(fact_3923_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),F: fun(A,B),A13: set(B)] :
( order_well_order_on(A,Aa2,R2)
=> ( bij_betw(A,B,F,Aa2,A13)
=> ? [R8: set(product_prod(B,B))] :
( order_well_order_on(B,A13,R8)
& member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R8),bNF_Wellorder_ordIso(A,B)) ) ) ) ).
% Well_order_iso_copy
tff(fact_3924_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),Aa2: set(A),Ca: set(B),G: fun(A,B),Ba: set(A),D4: set(B)] :
( bij_betw(A,B,F,Aa2,Ca)
=> ( bij_betw(A,B,G,Ba,D4)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Aa2),Ba) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Ca),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_ko(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),Aa2),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Ca),D4)) ) ) ) ) ).
% bij_betw_disjoint_Un
tff(fact_3925_infinite__imp__bij__betw2,axiom,
! [A: $tType,Aa2: set(A),A3: A] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ? [H3: fun(A,A)] : bij_betw(A,A,H3,Aa2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw2
tff(fact_3926_infinite__imp__bij__betw,axiom,
! [A: $tType,Aa2: set(A),A3: A] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ? [H3: fun(A,A)] : bij_betw(A,A,H3,Aa2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw
tff(fact_3927_cofinal__def,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( bNF_Ca7293521722713021262ofinal(A,Aa2,R2)
<=> ! [X4: A] :
( member(A,X4,field2(A,R2))
=> ? [Xa2: A] :
( member(A,Xa2,Aa2)
& ( X4 != Xa2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2) ) ) ) ).
% cofinal_def
tff(fact_3928_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: set(A),I4: set(B),Aa2: fun(B,set(C))] :
( ~ aa(set(A),$o,finite_finite(A),Ba)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(B,A))
=> ( ! [X3: B] :
( member(B,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Aa2,X3))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(C,A)) )
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Aa2),I4)))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(C,A)) ) ) ) ).
% card_of_UNION_ordLeq_infinite
tff(fact_3929_card__of__ordLess,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ~ ? [F6: fun(A,B)] :
( inj_on(A,B,F6,Aa2)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),Aa2)),Ba) )
<=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_We4044943003108391690rdLess(B,A)) ) ).
% card_of_ordLess
tff(fact_3930_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
<=> ? [B7: set(B)] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B7),field2(B,R2))
& member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,B7)),bNF_Wellorder_ordIso(A,B))
& member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B7)),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).
% internalize_card_of_ordLeq
tff(fact_3931_card__of__Times2,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ).
% card_of_Times2
tff(fact_3932_card__of__Times1,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ba,aTP_Lamp_fh(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordLeq(B,product_prod(B,A))) ) ).
% card_of_Times1
tff(fact_3933_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] : member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,C)),bNF_Wellorder_Func(A,product_prod(B,C),Aa2,product_Sigma(B,C,Ba,aTP_Lamp_mp(set(C),fun(B,set(C)),Ca))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,Aa2,Ba),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_mq(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Aa2),Ca)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C)))) ).
% Func_Times_Range
tff(fact_3934_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] : member(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),fun(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),product_Pair(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),C),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)),Ca))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),Aa2,bNF_Wellorder_Func(B,C,Ba,Ca)))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C)))) ).
% card_of_Func_Times
tff(fact_3935_card__of__Times3,axiom,
! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))),bNF_Wellorder_ordLeq(A,product_prod(A,A))) ).
% card_of_Times3
tff(fact_3936_card__of__Times__same__infinite,axiom,
! [A: $tType,Aa2: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,Aa2,aTP_Lamp_in(set(A),fun(A,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).
% card_of_Times_same_infinite
tff(fact_3937_card__of__Pow,axiom,
! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(set(A),pow(A,Aa2))),bNF_We4044943003108391690rdLess(A,set(A))) ).
% card_of_Pow
tff(fact_3938_card__of__Pow__Func,axiom,
! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,$o),fun(A,$o)))),bNF_Ca6860139660246222851ard_of(set(A),pow(A,Aa2))),bNF_Ca6860139660246222851ard_of(fun(A,$o),bNF_Wellorder_Func(A,$o,Aa2,top_top(set($o))))),bNF_Wellorder_ordIso(set(A),fun(A,$o))) ).
% card_of_Pow_Func
tff(fact_3939_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] : member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ba,aTP_Lamp_fh(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).
% card_of_Times_commute
tff(fact_3940_infinite__iff__card__of__nat,axiom,
! [A: $tType,Aa2: set(A)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
<=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(nat,A)) ) ).
% infinite_iff_card_of_nat
tff(fact_3941_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of($o,top_top(set($o)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A)))))),bNF_Wellorder_ordIso($o,A)) ) ).
% card_of_bool
tff(fact_3942_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,Aa2: fun(B,set(A)),I4: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),Aa2),I4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I4,Aa2))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ).
% card_of_UNION_Sigma
tff(fact_3943_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),Aa2: fun(A,set(B)),Ba: fun(A,set(C))] :
( ! [X3: A] :
( member(A,X3,I4)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Aa2,X3))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),Ba,X3))),bNF_Wellorder_ordLeq(B,C)) )
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I4,Aa2))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I4,Ba))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C))) ) ).
% card_of_Sigma_mono1
tff(fact_3944_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,Aa2,aTP_Lamp_mr(set(C),fun(A,set(C)),Ca)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,Ba,aTP_Lamp_mp(set(C),fun(B,set(C)),Ca)))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).
% card_of_Times_mono1
tff(fact_3945_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,Ca,aTP_Lamp_ms(set(A),fun(C,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,Ca,aTP_Lamp_mt(set(B),fun(C,set(B)),Ba)))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).
% card_of_Times_mono2
tff(fact_3946_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,Ba: set(A),I4: set(B),Aa2: fun(B,set(C))] :
( ~ aa(set(A),$o,finite_finite(A),Ba)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(B,A))
=> ( ! [X3: B] :
( member(B,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Aa2,X3))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(C,A)) )
=> member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I4,Aa2))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
tff(fact_3947_card__of__refl,axiom,
! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(A,A)) ).
% card_of_refl
tff(fact_3948_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ba,aTP_Lamp_fh(set(A),fun(B,set(A)),Aa2)))),bNF_Wellorder_ordIso(A,product_prod(B,A))) ) ) ) ).
% card_of_Times_infinite_simps(4)
tff(fact_3949_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Wellorder_ordIso(A,product_prod(A,B))) ) ) ) ).
% card_of_Times_infinite_simps(2)
tff(fact_3950_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ba,aTP_Lamp_fh(set(A),fun(B,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ).
% card_of_Times_infinite_simps(3)
tff(fact_3951_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).
% card_of_Times_infinite_simps(1)
tff(fact_3952_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Ba != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(A,B),A))
& member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Ba,aTP_Lamp_fh(set(A),fun(B,set(A)),Aa2)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ).
% card_of_Times_infinite
tff(fact_3953_card__of__image,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Aa2: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(B),set(A),image2(B,A,F),Aa2))),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordLeq(A,B)) ).
% card_of_image
tff(fact_3954_card__of__empty3,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B))
=> ( Aa2 = bot_bot(set(A)) ) ) ).
% card_of_empty3
tff(fact_3955_card__of__empty,axiom,
! [B: $tType,A: $tType,Aa2: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordLeq(A,B)) ).
% card_of_empty
tff(fact_3956_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> ( aa(set(B),$o,finite_finite(B),Ba)
=> aa(set(A),$o,finite_finite(A),Aa2) ) ) ).
% card_of_ordLeq_finite
tff(fact_3957_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> ( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ~ aa(set(B),$o,finite_finite(B),Ba) ) ) ).
% card_of_ordLeq_infinite
tff(fact_3958_card__of__mono1,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),Ba)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordLeq(A,A)) ) ).
% card_of_mono1
tff(fact_3959_card__of__empty2,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B))
=> ( Aa2 = bot_bot(set(A)) ) ) ).
% card_of_empty2
tff(fact_3960_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B)) ).
% card_of_empty_ordIso
tff(fact_3961_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B))
=> ( aa(set(A),$o,finite_finite(A),Aa2)
<=> aa(set(B),$o,finite_finite(B),Ba) ) ) ).
% card_of_ordIso_finite
tff(fact_3962_card__of__mono2,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),bNF_Ca6860139660246222851ard_of(B,field2(B,R4))),bNF_Wellorder_ordLeq(A,B)) ) ).
% card_of_mono2
tff(fact_3963_card__of__cong,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),bNF_Ca6860139660246222851ard_of(B,field2(B,R4))),bNF_Wellorder_ordIso(A,B)) ) ).
% card_of_cong
tff(fact_3964_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Aa2: set(A),Ba: set(B)] :
( inj_on(A,B,F,Aa2)
=> ( ! [A4: A] :
( member(A,A4,Aa2)
=> member(B,aa(A,B,F,A4),Ba) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B)) ) ) ).
% card_of_ordLeqI
tff(fact_3965_ex__bij__betw,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ? [F3: fun(B,A),B8: set(B)] : bij_betw(B,A,F3,B8,Aa2) ) ).
% ex_bij_betw
tff(fact_3966_card__of__least,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( order_well_order_on(A,Aa2,R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,A)) ) ).
% card_of_least
tff(fact_3967_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),R4),bNF_We4044943003108391690rdLess(A,B)) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_3968_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Aa2: set(A),Ba: set(B)] :
( bij_betw(A,B,F,Aa2,Ba)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B)) ) ).
% card_of_ordIsoI
tff(fact_3969_card__of__ordIso,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ? [F6: fun(A,B)] : bij_betw(A,B,F6,Aa2,Ba)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B)) ) ).
% card_of_ordIso
tff(fact_3970_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X: D] :
( ! [Y2: A] : member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),S,Y2))),Bd),bNF_Wellorder_ordLeq(B,C))
=> member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),comp(A,set(B),D,S,Rep),X))),Bd),bNF_Wellorder_ordLeq(B,C)) ) ).
% type_copy_set_bd
tff(fact_3971_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),Ba) = Aa2
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B)) ) ) ).
% card_of_ordLeq2
tff(fact_3972_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,Ba: set(A),F: fun(B,A),Aa2: set(B)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ba),aa(set(B),set(A),image2(B,A,F),Aa2))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordLeq(A,B)) ) ).
% surj_imp_ordLeq
tff(fact_3973_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,Aa2: set(A),B2: B] :
( ( Aa2 != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert2(B,B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A)) ) ).
% card_of_singl_ordLeq
tff(fact_3974_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,Ba: set(A),Aa2: set(B)] :
( ( Ba != bot_bot(set(A)) )
=> ( ~ ? [F6: fun(B,A)] : aa(set(B),set(A),image2(B,A,F6),Aa2) = Ba
<=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_We4044943003108391690rdLess(B,A)) ) ) ).
% card_of_ordLess2
tff(fact_3975_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ca: set(B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ca)),bNF_Wellorder_ordLeq(A,B))
<=> ? [B7: set(B)] :
( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B7),Ca)
& member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,B7)),bNF_Wellorder_ordIso(A,B))
& member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B7)),bNF_Ca6860139660246222851ard_of(B,Ca)),bNF_Wellorder_ordLeq(B,B)) ) ) ).
% internalize_card_of_ordLeq2
tff(fact_3976_card__of__Field__ordLess,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),R2),bNF_Wellorder_ordLeq(A,A)) ) ).
% card_of_Field_ordLess
tff(fact_3977_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,Ba: set(B)] : member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),Ba))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_mu(set(B),fun(fun(A,B),$o),Ba)))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B))) ).
% card_of_Func_UNIV
tff(fact_3978_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,field2(A,R2),aTP_Lamp_mr(set(C),fun(A,set(C)),Ca)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R4),aTP_Lamp_mp(set(C),fun(B,set(C)),Ca)))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).
% ordLeq_Times_mono1
tff(fact_3979_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Aa2: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,Aa2,aTP_Lamp_mv(set(product_prod(A,A)),fun(C,set(A)),R2)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,Aa2,aTP_Lamp_mw(set(product_prod(B,B)),fun(C,set(B)),R4)))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).
% ordLeq_Times_mono2
tff(fact_3980_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ? [F6: fun(A,B)] :
( inj_on(A,B,F6,Aa2)
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),Aa2)),Ba) )
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B)) ) ).
% card_of_ordLeq
tff(fact_3981_regularCard__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca7133664381575040944arCard(A,R2)
<=> ! [K7: set(A)] :
( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K7),field2(A,R2))
& bNF_Ca7293521722713021262ofinal(A,K7,R2) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,K7)),R2),bNF_Wellorder_ordIso(A,A)) ) ) ).
% regularCard_def
tff(fact_3982_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X5: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),image2(fun(C,set(set(A))),set(set(A)),aTP_Lamp_mx(C,fun(fun(C,set(set(A))),set(set(A))),X)),X5))))),Hbd),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(C,set(A),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A)),bNF_collect(C,set(A),X5)),X))),Hbd),bNF_Wellorder_ordLeq(A,B)) ) ).
% comp_set_bd_Union_o_collect
tff(fact_3983_card__of__ordIso__subst,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] :
( ( Aa2 = Ba )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Wellorder_ordIso(A,A)) ) ).
% card_of_ordIso_subst
tff(fact_3984_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] :
( member(A,X4,field2(A,R2))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,X4),aa(A,fun(A,set(A)),aTP_Lamp_kf(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_3985_Card__order__trans,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( X != Y )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( ( Y != Z2 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
=> ( ( X != Z2 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2),R2) ) ) ) ) ) ) ).
% Card_order_trans
tff(fact_3986_ordLeq__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),bNF_Wellorder_ordLeq(A,A)) ) ).
% ordLeq_refl
tff(fact_3987_ordIso__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2),bNF_Wellorder_ordIso(A,A)) ) ).
% ordIso_refl
tff(fact_3988_card__order__on__ordIso,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
=> ( bNF_Ca8970107618336181345der_on(A,Aa2,R4)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R4),bNF_Wellorder_ordIso(A,A)) ) ) ).
% card_order_on_ordIso
tff(fact_3989_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( member(A,A3,field2(A,R2))
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& ( A3 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),R2) ) ) ) ) ).
% infinite_Card_order_limit
tff(fact_3990_dir__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R2: set(product_prod(A,A))] :
( ! [X3: A,Y2: A] :
( ( aa(A,B,F,X3) = aa(A,B,F,Y2) )
<=> ( X3 = Y2 ) )
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_We2720479622203943262_image(A,B,R2,F)),bNF_Wellorder_ordIso(A,B)) ) ) ).
% dir_image
tff(fact_3991_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4) ) ) ).
% Card_order_ordIso2
tff(fact_3992_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordIso(B,A))
=> bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4) ) ) ).
% Card_order_ordIso
tff(fact_3993_card__of__unique,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(A,A)) ) ).
% card_of_unique
tff(fact_3994_card__order__on__def,axiom,
! [A: $tType,Aa2: set(A),R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,Aa2,R2)
<=> ( order_well_order_on(A,Aa2,R2)
& ! [R9: set(product_prod(A,A))] :
( order_well_order_on(A,Aa2,R9)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R9),bNF_Wellorder_ordLeq(A,A)) ) ) ) ).
% card_order_on_def
tff(fact_3995_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( Aa2 != bot_bot(set(B)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,Aa2,aTP_Lamp_my(set(product_prod(A,A)),fun(B,set(A)),R2)))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ) ) ).
% Card_order_Times2
tff(fact_3996_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ba: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( Ba != bot_bot(set(B)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R2),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R2),aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Wellorder_ordLeq(A,product_prod(A,B))) ) ) ).
% Card_order_Times1
tff(fact_3997_Card__order__Times__same__infinite,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,field2(A,R2),aTP_Lamp_mz(set(product_prod(A,A)),fun(A,set(A)),R2)))),R2),bNF_Wellorder_ordLeq(product_prod(A,A),A)) ) ) ).
% Card_order_Times_same_infinite
tff(fact_3998_Card__order__Pow,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),bNF_Ca6860139660246222851ard_of(set(A),pow(A,field2(A,R2)))),bNF_We4044943003108391690rdLess(A,set(A))) ) ).
% Card_order_Pow
tff(fact_3999_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),bNF_Wellorder_ordLeq(A,A)) ) ).
% Card_order_iff_ordLeq_card_of
tff(fact_4000_card__of__Field__ordIso,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),R2),bNF_Wellorder_ordIso(A,A)) ) ).
% card_of_Field_ordIso
tff(fact_4001_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Ca6860139660246222851ard_of(A,field2(A,R2))),bNF_Wellorder_ordIso(A,A)) ) ).
% Card_order_iff_ordIso_card_of
tff(fact_4002_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordIso(A,B))
=> bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) ) ).
% ordIso_card_of_imp_Card_order
tff(fact_4003_exists__minim__Card__order,axiom,
! [A: $tType,R: set(set(product_prod(A,A)))] :
( ( R != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
=> bNF_Ca8970107618336181345der_on(A,field2(A,X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),X3,R)
& ! [Xa3: set(product_prod(A,A))] :
( member(set(product_prod(A,A)),Xa3,R)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).
% exists_minim_Card_order
tff(fact_4004_Card__order__empty,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).
% Card_order_empty
tff(fact_4005_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Wellorder_ordIso(A,B))
=> ( aa(set(A),$o,finite_finite(A),field2(A,R2))
<=> aa(set(B),$o,finite_finite(B),Aa2) ) ) ) ).
% card_of_ordIso_finite_Field
tff(fact_4006_card__of__underS,axiom,
! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(A,A3,field2(A,R2))
=> member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,order_underS(A,R2,A3))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ).
% card_of_underS
tff(fact_4007_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As8: fun(A,set(B)),Ba: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca7133664381575040944arCard(A,R2)
=> ( bNF_Ca3754400796208372196lChain(A,set(B),R2,As8)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As8),field2(A,R2))))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),R2),bNF_We4044943003108391690rdLess(B,A))
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(A,set(B),As8,X3)) ) ) ) ) ) ) ).
% regularCard_UNION
tff(fact_4008_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B2: B] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( field2(A,R2) != bot_bot(set(A)) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert2(B,B2),bot_bot(set(B))))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).
% Card_order_singl_ordLeq
tff(fact_4009_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),Ba))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
tff(fact_4010_card__of__empty1,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
( ( order_well_order_on(A,field2(A,R2),R2)
| bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).
% card_of_empty1
tff(fact_4011_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P5: set(product_prod(B,B))] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( field2(B,P5) != bot_bot(set(B)) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P5),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R2),aTP_Lamp_na(set(product_prod(B,B)),fun(A,set(B)),P5)))),R2),bNF_Wellorder_ordIso(product_prod(A,B),A))
& member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P5),aTP_Lamp_my(set(product_prod(A,A)),fun(B,set(A)),R2)))),R2),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ) ).
% Card_order_Times_infinite
tff(fact_4012_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),Aa2: fun(B,set(C))] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( ! [X3: B] :
( member(B,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Aa2,X3))),R2),bNF_Wellorder_ordLeq(C,A)) )
=> member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I4,Aa2))),R2),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
tff(fact_4013_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),Ba: set(C)] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,Ba)),R2),bNF_Wellorder_ordLeq(C,A))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,Aa2,aTP_Lamp_mp(set(C),fun(B,set(C)),Ba)))),R2),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
tff(fact_4014_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),I4: set(B),Aa2: fun(B,set(C))] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( ! [X3: B] :
( member(B,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Aa2,X3))),R2),bNF_Wellorder_ordLeq(C,A)) )
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Aa2),I4)))),R2),bNF_Wellorder_ordLeq(C,A)) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
tff(fact_4015_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2)
=> ? [X_1: fun(A,B)] : bNF_Gr1419584066657907630d_pred(A,B,Aa2,R2,X_1) ) ) ).
% ex_toCard_pred
tff(fact_4016_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As8: fun(set(A),set(B)),Ba: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As8)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As8),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),R2),bNF_Wellorder_ordLeq(B,A))
=> ? [X3: set(A)] :
( member(set(A),X3,field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(set(A),set(B),As8,X3)) ) ) ) ) ) ) ).
% cardSuc_UNION
tff(fact_4017_isCardSuc__def,axiom,
! [A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(set(A),set(A)))] :
( bNF_Ca6246979054910435723ardSuc(A,R2,R4)
<=> ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R4),R4)
& member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),R4),bNF_We4044943003108391690rdLess(A,set(A)))
& ! [R10: set(product_prod(set(A),set(A)))] :
( ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R10),R10)
& member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),R10),bNF_We4044943003108391690rdLess(A,set(A))) )
=> member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),R4),R10),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).
% isCardSuc_def
tff(fact_4018_toCard__inj,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),X: A,Y: A] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2)
=> ( member(A,X,Aa2)
=> ( member(A,Y,Aa2)
=> ( ( aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),X) = aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),Y) )
<=> ( X = Y ) ) ) ) ) ) ).
% toCard_inj
tff(fact_4019_cardSuc__ordLeq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_Wellorder_ordLeq(A,set(A))) ) ).
% cardSuc_ordLeq
tff(fact_4020_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,Aa2: set(A),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2)
=> bNF_Gr1419584066657907630d_pred(A,B,Aa2,R2,bNF_Greatest_toCard(A,B,Aa2,R2)) ) ) ).
% toCard_pred_toCard
tff(fact_4021_cardSuc__greater,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_We4044943003108391690rdLess(A,set(A))) ) ).
% cardSuc_greater
tff(fact_4022_cardSuc__least,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
=> member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R2)),R4),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).
% cardSuc_least
tff(fact_4023_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_We4044943003108391690rdLess(A,B))
<=> member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R2)),R4),bNF_Wellorder_ordLeq(set(A),B)) ) ) ) ).
% cardSuc_ordLess_ordLeq
tff(fact_4024_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_Ca8387033319878233205ardSuc(B,R4)),bNF_Wellorder_ordLeq(set(A),set(B)))
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).
% cardSuc_mono_ordLeq
tff(fact_4025_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_Ca8387033319878233205ardSuc(B,R4)),bNF_Wellorder_ordIso(set(A),set(B)))
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B)) ) ) ) ).
% cardSuc_invar_ordIso
tff(fact_4026_cardSuc__least__aux,axiom,
! [A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(set(A),set(A)))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R4),R4)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R2),R4),bNF_We4044943003108391690rdLess(A,set(A)))
=> member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),bNF_Ca8387033319878233205ardSuc(A,R2)),R4),bNF_Wellorder_ordLeq(set(A),set(A))) ) ) ) ).
% cardSuc_least_aux
tff(fact_4027_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R4),R4)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(B,B)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),R4),bNF_Ca8387033319878233205ardSuc(A,R2)),bNF_We4044943003108391690rdLess(B,set(A)))
<=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ).
% cardSuc_ordLeq_ordLess
tff(fact_4028_fromCard__toCard,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),B2: A] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2)
=> ( member(A,B2,Aa2)
=> ( bNF_Gr5436034075474128252omCard(A,B,Aa2,R2,aa(A,B,bNF_Greatest_toCard(A,B,Aa2,R2),B2)) = B2 ) ) ) ) ).
% fromCard_toCard
tff(fact_4029_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As8: fun(set(A),set(B)),Ba: set(B)] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As8)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As8),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),R2),bNF_Wellorder_ordLeq(B,A))
=> ? [X3: set(A)] :
( member(set(A),X3,field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)))
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Ba),aa(set(A),set(B),As8,X3)) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
tff(fact_4030_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X: D] :
( bNF_Ca8970107618336181345der_on(A,field2(A,Fbd),Fbd)
=> ( ! [X3: B] : member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X3))),Fbd),bNF_Wellorder_ordLeq(C,A))
=> ( ! [X3: D] : member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X3))),Gbd),bNF_Wellorder_ordLeq(B,E))
=> member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Fset),aa(D,set(B),Gset,X))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd)),bNF_Wellorder_ordLeq(C,product_prod(E,A))) ) ) ) ).
% comp_single_set_bd
tff(fact_4031_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,Aa2: set(A),Ba: set(B)] :
( ( ( A1 != A22 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))),Aa2) )
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).
% card_of_Plus_Times_aux
tff(fact_4032_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] : member(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,C)),product_Sigma(A,sum_sum(B,C),Aa2,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_nb(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Ba),Ca)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,C)),sum_Plus(product_prod(A,B),product_prod(A,C),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)),product_Sigma(A,C,Aa2,aTP_Lamp_mr(set(C),fun(A,set(C)),Ca))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,C)),sum_sum(product_prod(A,B),product_prod(A,C)))) ).
% card_of_Times_Plus_distrib
tff(fact_4033_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),C),sum_Plus(sum_sum(A,B),C,sum_Plus(A,B,Aa2,Ba),Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),Aa2,sum_Plus(B,C,Ba,Ca)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C)))) ).
% card_of_Plus_assoc
tff(fact_4034_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] : member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Ba,Aa2))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).
% card_of_Plus_commute
tff(fact_4035_card__of__Plus1,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ).
% card_of_Plus1
tff(fact_4036_card__of__Plus2,axiom,
! [B: $tType,A: $tType,Ba: set(A),Aa2: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,Ba)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Aa2,Ba))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ).
% card_of_Plus2
tff(fact_4037_card__of__Plus__Times__bool,axiom,
! [A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),aa(set(product_prod(product_prod(A,$o),product_prod(A,$o))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,$o),product_prod(A,$o))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,$o),product_prod(A,$o)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,Aa2,Aa2))),bNF_Ca6860139660246222851ard_of(product_prod(A,$o),product_Sigma(A,$o,Aa2,aTP_Lamp_nc(A,set($o))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,$o))) ).
% card_of_Plus_Times_bool
tff(fact_4038_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),Aa2))),bNF_Wellorder_ordIso(A,sum_sum(B,A))) ).
% card_of_Plus_empty2
tff(fact_4039_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B))) ).
% card_of_Plus_empty1
tff(fact_4040_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,Aa2,Ba))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).
% card_of_Un_Plus_ordLeq
tff(fact_4041_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Aa2: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R2),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Aa2,field2(A,R2)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).
% Card_order_Plus2
tff(fact_4042_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ba: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R2),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R2),Ba))),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).
% Card_order_Plus1
tff(fact_4043_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca4139267488887388095finite(A,R1)
=> bNF_Ca4139267488887388095finite(B,R22) ) ) ).
% cinfinite_mono
tff(fact_4044_cprod__infinite,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Cardinal_cprod(A,A,R2,R2)),R2),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).
% cprod_infinite
tff(fact_4045_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Aa2: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,Aa2,field2(A,R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,Aa2,field2(B,R4)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).
% ordLeq_Plus_mono2
tff(fact_4046_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R2),Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R4),Ca))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).
% ordLeq_Plus_mono1
tff(fact_4047_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),P5: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P5),P7),bNF_Wellorder_ordLeq(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R2),field2(C,P5)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R4),field2(D,P7)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% ordLeq_Plus_mono
tff(fact_4048_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Aa2: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,Aa2,field2(A,R2)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,Aa2,field2(B,R4)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).
% ordIso_Plus_cong2
tff(fact_4049_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R2),Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R4),Ca))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).
% ordIso_Plus_cong1
tff(fact_4050_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),P5: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),R4),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P5),P7),bNF_Wellorder_ordIso(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R2),field2(C,P5)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R4),field2(D,P7)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% ordIso_Plus_cong
tff(fact_4051_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,Ca,Aa2))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,Ca,Ba))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).
% card_of_Plus_mono2
tff(fact_4052_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,Aa2,Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,Ba,Ca))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).
% card_of_Plus_mono1
tff(fact_4053_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C),D4: set(D)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,Ca)),bNF_Ca6860139660246222851ard_of(D,D4)),bNF_Wellorder_ordLeq(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,Aa2,Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,Ba,D4))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% card_of_Plus_mono
tff(fact_4054_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,Ca,Aa2))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,Ca,Ba))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).
% card_of_Plus_cong2
tff(fact_4055_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,Aa2,Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,Ba,Ca))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).
% card_of_Plus_cong1
tff(fact_4056_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,Aa2: set(A),Ba: set(B),Ca: set(C),D4: set(D)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,Ca)),bNF_Ca6860139660246222851ard_of(D,D4)),bNF_Wellorder_ordIso(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,Aa2,Ca))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,Ba,D4))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% card_of_Plus_cong
tff(fact_4057_cprod__com,axiom,
! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] : member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_cprod(A,B,P1,P22)),bNF_Cardinal_cprod(B,A,P22,P1)),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A))) ).
% cprod_com
tff(fact_4058_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R2: set(product_prod(A,A)),X22: A] :
( member(A,X1,field2(A,R2))
=> ( member(A,X22,field2(A,R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& ( X1 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3),R2)
& ( X22 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X3),R2) ) ) ) ) ).
% Cinfinite_limit2
tff(fact_4059_Cinfinite__limit,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] :
( member(A,X,field2(A,R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& ( X != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X3),R2) ) ) ) ).
% Cinfinite_limit
tff(fact_4060_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P5),R2),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R2),bNF_Wellorder_ordLeq(C,B))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P5),P5)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,C,P5,Q3)),R2),bNF_Wellorder_ordLeq(product_prod(A,C),B)) ) ) ) ) ) ).
% cprod_cinfinite_bound
tff(fact_4061_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),P5: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
( bNF_Ca4139267488887388095finite(A,R2)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,C,P5,P7)),bNF_Cardinal_cprod(A,A,R2,R2)),bNF_Wellorder_ordIso(product_prod(B,C),product_prod(A,A)))
=> member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Cardinal_cprod(B,C,P5,P7)),R2),bNF_Wellorder_ordIso(product_prod(B,C),A)) ) ) ) ).
% cprod_dup
tff(fact_4062_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(A,B),A))
& member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Ba,Aa2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).
% card_of_Plus_infinite
tff(fact_4063_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).
% card_of_Plus_infinite1
tff(fact_4064_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ~ aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Ba)),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,Ba,Aa2))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ).
% card_of_Plus_infinite2
tff(fact_4065_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,Ca: set(A),Aa2: set(B),Ba: set(C)] :
( ~ aa(set(A),$o,finite_finite(A),Ca)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),bNF_Ca6860139660246222851ard_of(A,Ca)),bNF_We4044943003108391690rdLess(B,A))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,Ba)),bNF_Ca6860139660246222851ard_of(A,Ca)),bNF_We4044943003108391690rdLess(C,A))
=> member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(A,Ca)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ).
% card_of_Plus_ordLess_infinite
tff(fact_4066_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_Wellorder_ordIso(A,B))
=> ( ( bNF_Ca4139267488887388095finite(A,R1)
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) ) ) ) ).
% Cinfinite_cong
tff(fact_4067_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P5: set(product_prod(B,B))] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P5),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R2),field2(B,P5)))),R2),bNF_Wellorder_ordIso(sum_sum(A,B),A))
& member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,field2(B,P5),field2(A,R2)))),R2),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ) ).
% Card_order_Plus_infinite
tff(fact_4068_Cinfinite__limit__finite,axiom,
! [A: $tType,X5: set(A),R2: set(product_prod(A,A))] :
( aa(set(A),$o,finite_finite(A),X5)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),field2(A,R2))
=> ( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ? [X3: A] :
( member(A,X3,field2(A,R2))
& ! [Xa3: A] :
( member(A,Xa3,X5)
=> ( ( Xa3 != X3 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3),R2) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
tff(fact_4069_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordLeq(C,D))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,D,R1,R22)),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,D))) ) ) ).
% cprod_mono
tff(fact_4070_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P1,Q3)),bNF_Cardinal_cprod(B,C,R1,Q3)),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C))) ) ).
% cprod_mono1
tff(fact_4071_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P22)),bNF_Cardinal_cprod(C,B,Q3,R22)),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B))) ) ).
% cprod_mono2
tff(fact_4072_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,Aa2: set(A),B1: B,B22: B,Ba: set(B)] :
( ( ( A1 != A22 )
& aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))),Aa2) )
=> ( ( ( B1 != B22 )
& aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert2(B,B1),aa(set(B),set(B),insert2(B,B22),bot_bot(set(B))))),Ba) )
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,Aa2,aTP_Lamp_ib(set(B),fun(A,set(B)),Ba)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).
% card_of_Plus_Times
tff(fact_4073_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordIso(C,D))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,D,R1,R22)),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,D))) ) ) ).
% cprod_cong
tff(fact_4074_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P1,P22)),bNF_Cardinal_cprod(B,C,R1,P22)),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,C))) ) ).
% cprod_cong1
tff(fact_4075_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P22)),bNF_Cardinal_cprod(C,B,Q3,R22)),bNF_Wellorder_ordIso(product_prod(C,A),product_prod(C,B))) ) ).
% cprod_cong2
tff(fact_4076_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),Ba: set(C)] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),R2),bNF_Wellorder_ordLeq(B,A))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,Ba)),R2),bNF_Wellorder_ordLeq(C,A))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,Aa2,Ba))),R2),bNF_Wellorder_ordLeq(sum_sum(B,C),A)) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
tff(fact_4077_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),Aa2: set(B),Ba: set(C)] :
( ~ aa(set(A),$o,finite_finite(A),field2(A,R2))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,Aa2)),R2),bNF_We4044943003108391690rdLess(B,A))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,Ba)),R2),bNF_We4044943003108391690rdLess(C,A))
=> member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,Aa2,Ba))),R2),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
tff(fact_4078_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,Aa2: set(A),R2: set(product_prod(B,B)),Ba: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Ba)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))),R2),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).
% Un_Cinfinite_bound
tff(fact_4079_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set(A),R2: set(product_prod(B,B)),Aa2: fun(A,set(C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,I4)),R2),bNF_Wellorder_ordLeq(A,B))
=> ( ! [X3: A] :
( member(A,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),Aa2,X3))),R2),bNF_Wellorder_ordLeq(C,B)) )
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),Aa2),I4)))),R2),bNF_Wellorder_ordLeq(C,B)) ) ) ) ).
% UNION_Cinfinite_bound
tff(fact_4080_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),I4: set(B),Aa2: fun(B,set(C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ! [X3: B] :
( member(B,X3,I4)
=> member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Aa2,X3))),R2),bNF_Wellorder_ordLeq(C,A)) )
=> member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,C,bNF_Ca6860139660246222851ard_of(B,I4),aTP_Lamp_nd(fun(B,set(C)),fun(B,set(product_prod(C,C))),Aa2))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I4),R2)),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A))) ) ) ).
% card_of_Csum_Times'
tff(fact_4081_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),Aa2: fun(A,set(B)),Ba: set(C)] :
( ! [X3: A] :
( member(A,X3,I4)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Aa2,X3))),bNF_Ca6860139660246222851ard_of(C,Ba)),bNF_Wellorder_ordLeq(B,C)) )
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I4),aTP_Lamp_ne(fun(A,set(B)),fun(A,set(product_prod(B,B))),Aa2))),bNF_Cardinal_cprod(A,C,bNF_Ca6860139660246222851ard_of(A,I4),bNF_Ca6860139660246222851ard_of(C,Ba))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C))) ) ).
% card_of_Csum_Times
tff(fact_4082_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,Aa2: set(A),Ba: set(B)] :
( ( sum_Plus(A,B,Aa2,Ba) = bot_bot(set(sum_sum(A,B))) )
<=> ( ( Aa2 = bot_bot(set(A)) )
& ( Ba = bot_bot(set(B)) ) ) ) ).
% Plus_eq_empty_conv
tff(fact_4083_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( ( bNF_Ca4139267488887388095finite(B,S2)
& bNF_Ca8970107618336181345der_on(B,field2(B,S2),S2) )
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,B,R2,S2)),S2),bNF_Wellorder_ordLeq(product_prod(A,B),B)) ) ) ).
% Cfinite_cprod_Cinfinite
tff(fact_4084_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( ( bNF_Ca4139267488887388095finite(B,S2)
& bNF_Ca8970107618336181345der_on(B,field2(B,S2),S2) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),S2),bNF_We4044943003108391690rdLess(A,B)) ) ) ).
% Cfinite_ordLess_Cinfinite
tff(fact_4085_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P5: set(product_prod(B,B))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( ( ~ member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P5),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B))
& bNF_Ca8970107618336181345der_on(B,field2(B,P5),P5) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P5),R2),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Cardinal_cprod(A,B,R2,P5)),R2),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).
% cprod_infinite1'
tff(fact_4086_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,P1),P1) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,P22),P22)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),P22),bNF_Cardinal_cprod(A,B,P1,P22)),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ) ).
% ordLeq_cprod2
tff(fact_4087_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> ( bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22))
& bNF_Ca8970107618336181345der_on(product_prod(A,B),field2(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22)),bNF_Cardinal_cprod(A,B,R1,R22)) ) ) ) ).
% Cinfinite_cprod2
tff(fact_4088_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22)) ) ) ).
% cinfinite_cprod2
tff(fact_4089_czero__def,axiom,
! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).
% czero_def
tff(fact_4090_czero__ordIso,axiom,
! [B: $tType,A: $tType] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Cardinal_czero(A)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)) ).
% czero_ordIso
tff(fact_4091_cinfinite__not__czero,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca4139267488887388095finite(A,R2)
=> ~ member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)) ) ).
% cinfinite_not_czero
tff(fact_4092_czeroE,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
=> ( field2(A,R2) = bot_bot(set(A)) ) ) ).
% czeroE
tff(fact_4093_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,Aa2: set(A)] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
<=> ( Aa2 = bot_bot(set(A)) ) ) ).
% card_of_ordIso_czero_iff_empty
tff(fact_4094_czeroI,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( field2(A,R2) = bot_bot(set(A)) )
=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)) ) ) ).
% czeroI
tff(fact_4095_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( field2(A,R2) != bot_bot(set(A)) ) ) ).
% Cnotzero_imp_not_empty
tff(fact_4096_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3)
=> ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R2),Q3),bNF_Wellorder_ordLeq(A,B))
=> ( ~ member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),Q3),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B))
& bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3) ) ) ) ) ).
% Cnotzero_mono
tff(fact_4097_Cinfinite__Cnotzero,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) ) ) ).
% Cinfinite_Cnotzero
tff(fact_4098_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))) ) ).
% Cnotzero_UNIV
tff(fact_4099_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),R2),bNF_Wellorder_ordLeq(product_unit,A)) ) ).
% cone_ordLeq_Cnotzero
tff(fact_4100_cexp__mono,axiom,
! [E: $tType,F2: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordLeq(C,D))
=> ( ( member(product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(C,C)),fun(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),product_Pair(set(product_prod(C,C)),set(product_prod(E,E))),P22),bNF_Cardinal_czero(E)),bNF_Wellorder_ordIso(C,E))
=> member(product_prod(set(product_prod(D,D)),set(product_prod(F2,F2))),aa(set(product_prod(F2,F2)),product_prod(set(product_prod(D,D)),set(product_prod(F2,F2))),aa(set(product_prod(D,D)),fun(set(product_prod(F2,F2)),product_prod(set(product_prod(D,D)),set(product_prod(F2,F2)))),product_Pair(set(product_prod(D,D)),set(product_prod(F2,F2))),R22),bNF_Cardinal_czero(F2)),bNF_Wellorder_ordIso(D,F2)) )
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,P22),P22)
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ) ).
% cexp_mono
tff(fact_4101_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( member(product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(A,A)),fun(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),product_Pair(set(product_prod(A,A)),set(product_prod(D,D))),P22),bNF_Cardinal_czero(D)),bNF_Wellorder_ordIso(A,D))
=> member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),R22),bNF_Cardinal_czero(E)),bNF_Wellorder_ordIso(B,E)) )
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P22),P22)
=> member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ) ).
% cexp_mono2
tff(fact_4102_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P22),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,P22),P22) )
=> member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).
% cexp_mono2_Cnotzero
tff(fact_4103_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(C,C)),R32: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1)
=> member(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),fun(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),product_Pair(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),bNF_Cardinal_cexp(fun(C,A),B,bNF_Cardinal_cexp(A,C,R1,R22),R32)),bNF_Cardinal_cexp(A,product_prod(C,B),R1,bNF_Cardinal_cprod(C,B,R22,R32))),bNF_Wellorder_ordIso(fun(B,fun(C,A)),fun(product_prod(C,B),A))) ) ).
% cexp_cprod
tff(fact_4104_cexp__cone,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(fun(product_unit,A),fun(product_unit,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),bNF_Cardinal_cexp(A,product_unit,R2,bNF_Cardinal_cone)),R2),bNF_Wellorder_ordIso(fun(product_unit,A),A)) ) ).
% cexp_cone
tff(fact_4105_cone__not__czero,axiom,
! [A: $tType] : ~ member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(product_unit,A)) ).
% cone_not_czero
tff(fact_4106_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(B,B)),S2: set(product_prod(C,C)),T2: set(product_prod(A,A))] : member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R2,S2),T2)),bNF_Cardinal_cprod(fun(A,B),fun(A,C),bNF_Cardinal_cexp(B,A,R2,T2),bNF_Cardinal_cexp(C,A,S2,T2))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C)))) ).
% cprod_cexp
tff(fact_4107_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R32: set(product_prod(C,C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1)
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> ( ( ~ member(product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),product_Pair(set(product_prod(C,C)),set(product_prod(C,C))),R32),bNF_Cardinal_czero(C)),bNF_Wellorder_ordIso(C,C))
& bNF_Ca8970107618336181345der_on(C,field2(C,R32),R32) )
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),R32),R22),bNF_Wellorder_ordLeq(C,B))
=> member(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),bNF_Cardinal_cexp(fun(B,A),C,bNF_Cardinal_cexp(A,B,R1,R22),R32)),bNF_Cardinal_cexp(A,B,R1,R22)),bNF_Wellorder_ordIso(fun(C,fun(B,A)),fun(B,A))) ) ) ) ) ).
% cexp_cprod_ordLeq
tff(fact_4108_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordLeq(C,D))
=> ( ( ( field2(C,P22) = bot_bot(set(C)) )
=> ( field2(D,R22) = bot_bot(set(D)) ) )
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ).
% cexp_mono'
tff(fact_4109_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P1,Q3)),bNF_Cardinal_cexp(B,C,R1,Q3)),bNF_Wellorder_ordLeq(fun(C,A),fun(C,B))) ) ) ).
% cexp_mono1
tff(fact_4110_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordLeq(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( ( field2(A,P22) = bot_bot(set(A)) )
=> ( field2(B,R22) = bot_bot(set(B)) ) )
=> member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).
% cexp_mono2'
tff(fact_4111_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),Q3),bNF_Cardinal_cexp(B,A,Q3,R2)),bNF_Wellorder_ordLeq(B,fun(A,B))) ) ) ).
% ordLeq_cexp1
tff(fact_4112_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordIso(C,D))
=> ( bNF_Ca8970107618336181345der_on(D,field2(D,R22),R22)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,P22),P22)
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordIso(fun(C,A),fun(D,B))) ) ) ) ) ).
% cexp_cong
tff(fact_4113_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P1,Q3)),bNF_Cardinal_cexp(B,C,R1,Q3)),bNF_Wellorder_ordIso(fun(C,A),fun(C,B))) ) ) ).
% cexp_cong1
tff(fact_4114_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordIso(A,B))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P22),P22)
=> member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P22)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordIso(fun(A,C),fun(B,C))) ) ) ) ).
% cexp_cong2
tff(fact_4115_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq($o,A))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2)
=> member(product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),R2),bNF_Cardinal_cexp(A,B,Q3,R2)),bNF_Wellorder_ordLeq(B,fun(B,A))) ) ) ).
% ordLeq_cexp2
tff(fact_4116_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S2: set(product_prod(A,A)),T2: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,S2)
& bNF_Ca8970107618336181345der_on(A,field2(A,S2),S2) )
=> ( ( bNF_Ca4139267488887388095finite(B,T2)
& bNF_Ca8970107618336181345der_on(B,field2(B,T2),T2) )
=> member(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),aa(set(product_prod(fun(B,$o),fun(B,$o))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),aa(set(product_prod(fun(B,A),fun(B,A))),fun(set(product_prod(fun(B,$o),fun(B,$o))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o))))),product_Pair(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,$o),fun(B,$o)))),bNF_Cardinal_cexp(A,B,S2,T2)),bNF_Cardinal_cexp($o,B,bNF_Cardinal_ctwo,T2)),bNF_Wellorder_ordLeq(fun(B,A),fun(B,$o))) ) ) ).
% Cfinite_cexp_Cinfinite
tff(fact_4117_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq($o,A))
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2))
& bNF_Ca8970107618336181345der_on(fun(B,A),field2(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2)),bNF_Cardinal_cexp(A,B,Q3,R2)) ) ) ) ).
% Cinfinite_cexp
tff(fact_4118_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
( member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq($o,A))
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R2)) ) ) ).
% cinfinite_cexp
tff(fact_4119_ctwo__Cnotzero,axiom,
( ~ member(product_prod(set(product_prod($o,$o)),set(product_prod($o,$o))),aa(set(product_prod($o,$o)),product_prod(set(product_prod($o,$o)),set(product_prod($o,$o))),aa(set(product_prod($o,$o)),fun(set(product_prod($o,$o)),product_prod(set(product_prod($o,$o)),set(product_prod($o,$o)))),product_Pair(set(product_prod($o,$o)),set(product_prod($o,$o))),bNF_Cardinal_ctwo),bNF_Cardinal_czero($o)),bNF_Wellorder_ordIso($o,$o))
& bNF_Ca8970107618336181345der_on($o,field2($o,bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).
% ctwo_Cnotzero
tff(fact_4120_ordLess__ctwo__cexp,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),aa(set(product_prod(A,A)),fun(set(product_prod(fun(A,$o),fun(A,$o))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o))))),product_Pair(set(product_prod(A,A)),set(product_prod(fun(A,$o),fun(A,$o)))),R2),bNF_Cardinal_cexp($o,A,bNF_Cardinal_ctwo,R2)),bNF_We4044943003108391690rdLess(A,fun(A,$o))) ) ).
% ordLess_ctwo_cexp
tff(fact_4121_ctwo__not__czero,axiom,
! [A: $tType] : ~ member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso($o,A)) ).
% ctwo_not_czero
tff(fact_4122_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R2),bNF_We4044943003108391690rdLess($o,A)) ) ).
% ctwo_ordLess_Cinfinite
tff(fact_4123_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod($o,$o)),fun(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A)))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R2),bNF_Wellorder_ordLeq($o,A)) ) ).
% ctwo_ordLeq_Cinfinite
tff(fact_4124_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),P5: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
( bNF_Ca4139267488887388095finite(A,R2)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,C,P5,P7)),bNF_Cardinal_csum(A,A,R2,R2)),bNF_Wellorder_ordIso(sum_sum(B,C),sum_sum(A,A)))
=> member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Cardinal_csum(B,C,P5,P7)),R2),bNF_Wellorder_ordIso(sum_sum(B,C),A)) ) ) ) ).
% csum_dup
tff(fact_4125_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ~ member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Cardinal_csum(A,B,R1,R22)),bNF_Cardinal_czero(sum_sum(A,B))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B)))
& bNF_Ca8970107618336181345der_on(sum_sum(A,B),field2(sum_sum(A,B),bNF_Cardinal_csum(A,B,R1,R22)),bNF_Cardinal_csum(A,B,R1,R22)) ) ) ).
% csum_Cnotzero1
tff(fact_4126_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: set(product_prod(A,A)),R2: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P5),R2),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R2),bNF_Wellorder_ordLeq(C,B))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P5),P5)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( bNF_Ca4139267488887388095finite(B,R2)
& bNF_Ca8970107618336181345der_on(B,field2(B,R2),R2) )
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),bNF_Cardinal_csum(A,C,P5,Q3)),R2),bNF_Wellorder_ordLeq(sum_sum(A,C),B)) ) ) ) ) ) ).
% csum_cinfinite_bound
tff(fact_4127_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R2)
& bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2) )
=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),R2),bNF_Wellorder_ordLeq(nat,A)) ) ).
% natLeq_ordLeq_cinfinite
tff(fact_4128_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R32: set(product_prod(C,C))] : member(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),fun(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),bNF_Cardinal_csum(product_prod(A,B),product_prod(A,C),bNF_Cardinal_cprod(A,B,R1,R22),bNF_Cardinal_cprod(A,C,R1,R32))),bNF_Cardinal_cprod(A,sum_sum(B,C),R1,bNF_Cardinal_csum(B,C,R22,R32))),bNF_Wellorder_ordIso(sum_sum(product_prod(A,B),product_prod(A,C)),product_prod(A,sum_sum(B,C)))) ).
% cprod_csum_distrib1
tff(fact_4129_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B)),P32: set(product_prod(C,C))] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Cardinal_csum(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,P1,P22),P32)),bNF_Cardinal_csum(A,sum_sum(B,C),P1,bNF_Cardinal_csum(B,C,P22,P32))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C)))) ).
% csum_assoc
tff(fact_4130_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R32: set(product_prod(C,C)),R42: set(product_prod(D,D))] : member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),fun(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),bNF_Cardinal_csum(sum_sum(A,B),sum_sum(C,D),bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_csum(C,D,R32,R42))),bNF_Cardinal_csum(sum_sum(A,C),sum_sum(B,D),bNF_Cardinal_csum(A,C,R1,R32),bNF_Cardinal_csum(B,D,R22,R42))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))) ).
% csum_csum
tff(fact_4131_ctwo__ordLess__natLeq,axiom,
member(product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat))),aa(set(product_prod($o,$o)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod($o,$o)),set(product_prod(nat,nat)))),product_Pair(set(product_prod($o,$o)),set(product_prod(nat,nat))),bNF_Cardinal_ctwo),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess($o,nat)) ).
% ctwo_ordLess_natLeq
tff(fact_4132_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] : member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),aa(set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B)))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun($o,sum_sum(A,B)),fun($o,sum_sum(A,B))))),bNF_Cardinal_cprod(A,B,R1,R22)),bNF_Cardinal_cexp(sum_sum(A,B),$o,bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_ctwo)),bNF_Wellorder_ordLeq(product_prod(A,B),fun($o,sum_sum(A,B)))) ).
% cprod_csum_cexp
tff(fact_4133_csum__com,axiom,
! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] : member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Cardinal_csum(A,B,P1,P22)),bNF_Cardinal_csum(B,A,P22,P1)),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A))) ).
% csum_com
tff(fact_4134_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(B,B)),T2: set(product_prod(C,C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R2),R2)
=> ( ( bNF_Cardinal_cfinite(B,S2)
& bNF_Ca8970107618336181345der_on(B,field2(B,S2),S2) )
=> ( ( bNF_Ca4139267488887388095finite(C,T2)
& bNF_Ca8970107618336181345der_on(C,field2(C,T2),T2) )
=> member(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),aa(set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),aa(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),fun(set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o)))))),product_Pair(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,$o)),fun(C,sum_sum(A,$o))))),bNF_Cardinal_cexp(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,R2,S2),T2)),bNF_Cardinal_cexp(sum_sum(A,$o),C,bNF_Cardinal_csum(A,$o,R2,bNF_Cardinal_ctwo),T2)),bNF_Wellorder_ordLeq(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,$o)))) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
tff(fact_4135_card__of__Field__natLeq,axiom,
member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,field2(nat,bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq),bNF_Wellorder_ordIso(nat,nat)) ).
% card_of_Field_natLeq
tff(fact_4136_finite__iff__ordLess__natLeq,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
<=> member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(A,A)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess(A,nat)) ) ).
% finite_iff_ordLess_natLeq
tff(fact_4137_Un__csum,axiom,
! [A: $tType,Aa2: set(A),Ba: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Aa2),Ba))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,Aa2),bNF_Ca6860139660246222851ard_of(A,Ba))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).
% Un_csum
tff(fact_4138_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,P1),P1)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),P1),bNF_Cardinal_csum(A,B,P1,P22)),bNF_Wellorder_ordLeq(A,sum_sum(A,B))) ) ).
% ordLeq_csum1
tff(fact_4139_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P22: set(product_prod(A,A)),P1: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,P22),P22)
=> member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),P22),bNF_Cardinal_csum(B,A,P1,P22)),bNF_Wellorder_ordLeq(A,sum_sum(B,A))) ) ).
% ordLeq_csum2
tff(fact_4140_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Cardinal_csum(C,A,Q3,P22)),bNF_Cardinal_csum(C,B,Q3,R22)),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B))) ) ).
% csum_cong2
tff(fact_4141_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Cardinal_csum(A,C,P1,Q3)),bNF_Cardinal_csum(B,C,R1,Q3)),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C))) ) ).
% csum_cong1
tff(fact_4142_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordIso(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordIso(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R1,R22)),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% csum_cong
tff(fact_4143_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P22),R22),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Cardinal_csum(C,A,Q3,P22)),bNF_Cardinal_csum(C,B,Q3,R22)),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B))) ) ).
% csum_mono2
tff(fact_4144_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Cardinal_csum(A,C,P1,Q3)),bNF_Cardinal_csum(B,C,R1,Q3)),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C))) ) ).
% csum_mono1
tff(fact_4145_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P1),R1),bNF_Wellorder_ordLeq(A,B))
=> ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P22),R22),bNF_Wellorder_ordLeq(C,D))
=> member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R1,R22)),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D))) ) ) ).
% csum_mono
tff(fact_4146_card__of__nat,axiom,
member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca8665028551170535155natLeq),bNF_Wellorder_ordIso(nat,nat)) ).
% card_of_nat
tff(fact_4147_Plus__csum,axiom,
! [B: $tType,A: $tType,Aa2: set(A),Ba: set(B)] : member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,Aa2,Ba))),bNF_Cardinal_csum(A,B,bNF_Ca6860139660246222851ard_of(A,Aa2),bNF_Ca6860139660246222851ard_of(B,Ba))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B))) ).
% Plus_csum
tff(fact_4148_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,Aa2: set(A)] :
~ ( aa(set(A),$o,finite_finite(A),Aa2)
<=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,Aa2)),bNF_Wellorder_ordLeq(nat,A)) ) ).
% infinite_iff_natLeq_ordLeq
tff(fact_4149_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set(product_prod(A,A)),R2: set(product_prod(B,B)),S2: set(product_prod(C,C))] :
( ( bNF_Ca4139267488887388095finite(A,T2)
& bNF_Ca8970107618336181345der_on(A,field2(A,T2),T2) )
=> member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R2,S2),T2)),bNF_Cardinal_cexp(sum_sum(B,C),A,bNF_Cardinal_csum(B,C,R2,S2),T2)),bNF_Wellorder_ordLeq(fun(A,product_prod(B,C)),fun(A,sum_sum(B,C)))) ) ).
% cprod_cexp_csum_cexp_Cinfinite
tff(fact_4150_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A))
=> ( ( bNF_Ca4139267488887388095finite(B,R1)
| bNF_Ca4139267488887388095finite(A,R22) )
=> member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Cardinal_csum(B,A,R1,R22)),R22),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).
% csum_absorb2'
tff(fact_4151_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22)
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A))
=> ( ( bNF_Ca4139267488887388095finite(B,R1)
| bNF_Ca4139267488887388095finite(A,R22) )
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Cardinal_csum(A,B,R22,R1)),R22),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ) ).
% csum_absorb1'
tff(fact_4152_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( ( bNF_Ca4139267488887388095finite(A,R22)
& bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22) )
=> ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A))
=> member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Cardinal_csum(A,B,R22,R1)),R22),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).
% csum_absorb1
tff(fact_4153_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_max(A),aTP_Lamp_nf(A,fun(A,$o)),aTP_Lamp_ng(A,fun(A,$o))) ) ).
% max.semilattice_order_axioms
tff(fact_4154_iso__iff2,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F: fun(A,B)] :
( bNF_Wellorder_iso(A,B,R2,R4,F)
<=> ( bij_betw(A,B,F,field2(A,R2),field2(B,R4))
& ! [X4: A] :
( member(A,X4,field2(A,R2))
=> ! [Xa2: A] :
( member(A,Xa2,field2(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2)
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X4)),aa(A,B,F,Xa2)),R4) ) ) ) ) ) ).
% iso_iff2
tff(fact_4155_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_4156_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice_order(A,sup_sup(A),aTP_Lamp_jg(A,fun(A,$o)),aTP_Lamp_jh(A,fun(A,$o))) ) ).
% sup.semilattice_order_axioms
tff(fact_4157_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F: fun(A,B)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( bNF_Wellorder_iso(A,B,R2,R4,F)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X)),aa(A,B,F,Y)),R4) ) ) ).
% iso_forward
tff(fact_4158_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).
% semilattice_order.strict_coboundedI2
tff(fact_4159_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).
% semilattice_order.strict_coboundedI1
tff(fact_4160_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
tff(fact_4161_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).
% semilattice_order.strict_boundedE
tff(fact_4162_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).
% semilattice_order.coboundedI2
tff(fact_4163_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) ) ) ).
% semilattice_order.coboundedI1
tff(fact_4164_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
& aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).
% semilattice_order.bounded_iff
tff(fact_4165_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb_iff2
tff(fact_4166_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb_iff1
tff(fact_4167_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) ) ).
% semilattice_order.cobounded2
tff(fact_4168_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),A3) ) ).
% semilattice_order.cobounded1
tff(fact_4169_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
<=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.order_iff
tff(fact_4170_semilattice__order_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ) ).
% semilattice_order.boundedI
tff(fact_4171_semilattice__order_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).
% semilattice_order.boundedE
tff(fact_4172_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb4
tff(fact_4173_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb3
tff(fact_4174_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb2
tff(fact_4175_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb1
tff(fact_4176_semilattice__order_OorderI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2) ) ) ).
% semilattice_order.orderI
tff(fact_4177_semilattice__order_OorderE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.orderE
tff(fact_4178_semilattice__order_Omono,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,C2: A,B2: A,D3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),D3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),aa(A,A,aa(A,fun(A,A),F,C2),D3)) ) ) ) ).
% semilattice_order.mono
tff(fact_4179_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semilattice_order(A,F,Less_eq,Less)
=> semilattice(A,F) ) ).
% semilattice_order.axioms(1)
tff(fact_4180_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ).
% semilattice_neutr_order.axioms(2)
tff(fact_4181_semilattice__neutr__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
<=> ( semilattice_neutr(A,F,Z2)
& semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_neutr_order_def
tff(fact_4182_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semilattice_neutr(A,F,Z2)
=> ( semilattice_order(A,F,Less_eq,Less)
=> semila1105856199041335345_order(A,F,Z2,Less_eq,Less) ) ) ).
% semilattice_neutr_order.intro
tff(fact_4183_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_4184_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_4185_semilattice__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semilattice(A,F)
=> ( semila6385135966242565138axioms(A,F,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_order.intro
tff(fact_4186_semilattice__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semilattice_order(A,F,Less_eq,Less)
<=> ( semilattice(A,F)
& semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_def
tff(fact_4187_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_4188_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),X: A,Aa2: set(A),Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),Aa2)),S)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( finite_folding_F(A,B,F,Z2,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(B,B,aa(A,fun(B,B),F,X),finite_folding_F(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% folding_on.insert_remove
tff(fact_4189_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),F: fun(A,fun(A,A)),Less: fun(A,fun(A,$o))] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
<=> ( A4 = aa(A,A,aa(A,fun(A,A),F,A4),B3) ) )
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
<=> ( ( A4 = aa(A,A,aa(A,fun(A,A),F,A4),B3) )
& ( A4 != B3 ) ) )
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_axioms.intro
tff(fact_4190_semilattice__order__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semila6385135966242565138axioms(A,F,Less_eq,Less)
<=> ( ! [A5: A,B4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B4)
<=> ( A5 = aa(A,A,aa(A,fun(A,A),F,A5),B4) ) )
& ! [A5: A,B4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B4)
<=> ( ( A5 = aa(A,A,aa(A,fun(A,A),F,A5),B4) )
& ( A5 != B4 ) ) ) ) ) ).
% semilattice_order_axioms_def
tff(fact_4191_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set(A),F: fun(A,fun(B,B)),Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( finite_folding_F(A,B,F,Z2,bot_bot(set(A))) = Z2 ) ) ).
% folding_on.empty
tff(fact_4192_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( semilattice_order(A,F,Less_eq,Less)
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ).
% semilattice_order.axioms(2)
tff(fact_4193_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(A,fun(B,B)),Aa2: set(A),X: A,Z2: B] :
( finite_folding_on(A,B,S,F)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Aa2),S)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( finite_folding_F(A,B,F,Z2,Aa2) = aa(B,B,aa(A,fun(B,B),F,X),finite_folding_F(A,B,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ) ).
% folding_on.remove
tff(fact_4194_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F4: filter(A),G6: filter(B),H8: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G6),H8) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_ni(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G6,H8))) ).
% prod_filter_assoc
tff(fact_4195_sndOp__def,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(C,fun(A,$o)),Q: fun(A,fun(B,$o)),Ac: product_prod(C,B)] : bNF_sndOp(C,A,B,P,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P,Q,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).
% sndOp_def
tff(fact_4196_fstOp__def,axiom,
! [B: $tType,C: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(B,fun(C,$o)),Ac: product_prod(A,C)] : bNF_fstOp(A,B,C,P,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P,Q,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).
% fstOp_def
tff(fact_4197_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F4: filter(A),F: fun(B,A)] :
( ( F4 != bot_bot(filter(A)) )
=> ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( filtercomap(B,A,F,F4) != bot_bot(filter(B)) ) ) ) ).
% filtercomap_neq_bot_surj
tff(fact_4198_filtermap__bot,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : filtermap(B,A,F,bot_bot(filter(B))) = bot_bot(filter(A)) ).
% filtermap_bot
tff(fact_4199_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : filtercomap(A,B,F,bot_bot(filter(B))) = bot_bot(filter(A)) ).
% filtercomap_bot
tff(fact_4200_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F: fun(B,A),F4: filter(B)] :
( ( filtermap(B,A,F,F4) = bot_bot(filter(A)) )
<=> ( F4 = bot_bot(filter(B)) ) ) ).
% filtermap_bot_iff
tff(fact_4201_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),G: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_iv(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G),F4)),prod_filter(A,B,filtermap(C,A,F,F4),filtermap(C,B,G,F4))) ).
% filtermap_Pair
tff(fact_4202_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F4: filter(A),F: fun(B,A)] :
( ! [P3: fun(A,$o)] :
( eventually(A,P3,F4)
=> ? [X2: B] : aa(A,$o,P3,aa(B,A,F,X2)) )
=> ( filtercomap(B,A,F,F4) != bot_bot(filter(B)) ) ) ).
% filtercomap_neq_bot
tff(fact_4203_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F4: filter(A),X: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),insert2(B,X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))),X),F4) ).
% prod_filter_principal_singleton2
tff(fact_4204_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F4) ).
% prod_filter_principal_singleton
tff(fact_4205_trancl__def,axiom,
! [A: $tType,X2: set(product_prod(A,A))] : transitive_trancl(A,X2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_tranclp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),X2)))) ).
% trancl_def
tff(fact_4206_reflp__refl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( reflp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> refl_on(A,top_top(set(A)),R2) ) ).
% reflp_refl_eq
tff(fact_4207_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( ! [A4: A,B3: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) )
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_tranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% tranclp_induct2
tff(fact_4208_bot__eq__principal__empty,axiom,
! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).
% bot_eq_principal_empty
tff(fact_4209_principal__eq__bot__iff,axiom,
! [A: $tType,X5: set(A)] :
( ( principal(A,X5) = bot_bot(filter(A)) )
<=> ( X5 = bot_bot(set(A)) ) ) ).
% principal_eq_bot_iff
tff(fact_4210_tranclp__trancl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X2: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X2),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),transitive_trancl(A,R2)) ) ).
% tranclp_trancl_eq
tff(fact_4211_Nitpick_Otranclp__unfold,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),A3: A,B2: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_tranclp(A,R2),A3),B2)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% Nitpick.tranclp_unfold
tff(fact_4212_finite__subsets__at__top__finite,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( finite5375528669736107172at_top(A,Aa2) = principal(set(A),aa(set(set(A)),set(set(A)),insert2(set(A),Aa2),bot_bot(set(set(A))))) ) ) ).
% finite_subsets_at_top_finite
tff(fact_4213_filterlim__base__iff,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,I4: set(A),F4: fun(A,set(B)),F: fun(B,C),G6: fun(D,set(C)),J3: set(D)] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I3: A] :
( member(A,I3,I4)
=> ! [J2: A] :
( member(A,J2,I4)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J2))
| aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I3)) ) ) )
=> ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_nj(fun(D,set(C)),fun(D,filter(C)),G6)),J3)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_nk(fun(A,set(B)),fun(A,filter(B)),F4)),I4)))
<=> ! [X4: D] :
( member(D,X4,J3)
=> ? [Xa2: A] :
( member(A,Xa2,I4)
& ! [Xb2: B] :
( member(B,Xb2,aa(A,set(B),F4,Xa2))
=> member(C,aa(B,C,F,Xb2),aa(D,set(C),G6,X4)) ) ) ) ) ) ) ).
% filterlim_base_iff
tff(fact_4214_fun__of__rel__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R,X) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),R),X)) ).
% fun_of_rel_def
tff(fact_4215_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ks(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
% Eps_case_prod_eq
tff(fact_4216_some__insert__self,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( aa(set(A),set(A),insert2(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),S))),S) = S ) ) ).
% some_insert_self
tff(fact_4217_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,Aa2: set(A)] : finite5375528669736107172at_top(A,Aa2) != bot_bot(filter(set(A))) ).
% finite_subsets_at_top_neq_bot
tff(fact_4218_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G6: filter(B),F4: filter(A),G: fun(A,C),H8: filter(C)] :
( filterlim(A,B,F,G6,F4)
=> ( filterlim(A,C,G,H8,F4)
=> filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_nl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F),G),prod_filter(B,C,G6,H8),F4) ) ) ).
% filterlim_Pair
tff(fact_4219_some__elem,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),S)),S) ) ).
% some_elem
tff(fact_4220_some__in__eq,axiom,
! [A: $tType,Aa2: set(A)] :
( member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),Aa2)),Aa2)
<=> ( Aa2 != bot_bot(set(A)) ) ) ).
% some_in_eq
tff(fact_4221_split__paired__Eps,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_nm(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).
% split_paired_Eps
tff(fact_4222_small__lazy_H_Ocases,axiom,
! [X: product_prod(int,int)] :
~ ! [D2: int,I3: int] : X != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I3) ).
% small_lazy'.cases
tff(fact_4223_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2))) ).
% le_prod_encode_2
tff(fact_4224_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2))) ).
% le_prod_encode_1
tff(fact_4225_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Aa2: fun(A,fun(B,$o)),Ba: fun(C,fun(D,$o))] : aa(fun(B,fun(D,product_prod(B,D))),$o,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),$o),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),Aa2,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),Ba,basic_rel_prod(A,B,C,D,Aa2,Ba))),product_Pair(A,C)),product_Pair(B,D)) ).
% Pair_transfer
tff(fact_4226_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A3: A,B2: C,C2: B,D3: D] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C2),D3))
<=> ( aa(B,$o,aa(A,fun(B,$o),R12,A3),C2)
& aa(D,$o,aa(C,fun(D,$o),R23,B2),D3) ) ) ).
% rel_prod_inject
tff(fact_4227_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: fun(A,fun(B,$o)),A3: A,B2: B,R23: fun(C,fun(D,$o)),C2: C,D3: D] :
( aa(B,$o,aa(A,fun(B,$o),R12,A3),B2)
=> ( aa(D,$o,aa(C,fun(D,$o),R23,C2),D3)
=> aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D3)) ) ) ).
% rel_prod.intros
tff(fact_4228_rel__prod_Osimps,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),A1),A22)
<=> ? [A5: A,B4: B,C4: C,D5: D] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C4) )
& ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B4),D5) )
& aa(B,$o,aa(A,fun(B,$o),R12,A5),B4)
& aa(D,$o,aa(C,fun(D,$o),R23,C4),D5) ) ) ).
% rel_prod.simps
tff(fact_4229_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( aa(product_prod(B,D),$o,aa(product_prod(A,C),fun(product_prod(B,D),$o),basic_rel_prod(A,B,C,D,R12,R23),A1),A22)
=> ~ ! [A4: A,B3: B,C3: C] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),C3) )
=> ! [D2: D] :
( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B3),D2) )
=> ( aa(B,$o,aa(A,fun(B,$o),R12,A4),B3)
=> ~ aa(D,$o,aa(C,fun(D,$o),R23,C3),D2) ) ) ) ) ).
% rel_prod.cases
tff(fact_4230_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top.axioms(2)
tff(fact_4231_ID_Opred__set,axiom,
! [A: $tType,P: fun(A,$o),X2: A] :
( aa(A,$o,bNF_id_bnf(fun(A,$o),P),X2)
<=> ! [Xa2: A] :
( member(A,Xa2,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))
=> aa(A,$o,P,Xa2) ) ) ).
% ID.pred_set
tff(fact_4232_wfP__SUP,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,fun(B,$o)))] :
( ! [I3: A] : wfP(B,aa(A,fun(B,fun(B,$o)),R2,I3))
=> ( ! [I3: A,J2: A] :
( ( aa(A,fun(B,fun(B,$o)),R2,I3) != aa(A,fun(B,fun(B,$o)),R2,J2) )
=> ( aa(fun(B,$o),fun(B,$o),aa(fun(B,$o),fun(fun(B,$o),fun(B,$o)),inf_inf(fun(B,$o)),domainp(B,B,aa(A,fun(B,fun(B,$o)),R2,I3))),rangep(B,B,aa(A,fun(B,fun(B,$o)),R2,J2))) = bot_bot(fun(B,$o)) ) )
=> wfP(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Sup_Sup(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),top_top(set(A))))) ) ) ).
% wfP_SUP
tff(fact_4233_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
& ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs2)),divide_divide(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).
% mergesort_by_rel_split_length
tff(fact_4234_inf1I,axiom,
! [A: $tType,Aa2: fun(A,$o),X: A,Ba: fun(A,$o)] :
( aa(A,$o,Aa2,X)
=> ( aa(A,$o,Ba,X)
=> aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),Aa2),Ba),X) ) ) ).
% inf1I
tff(fact_4235_inf1E,axiom,
! [A: $tType,Aa2: fun(A,$o),Ba: fun(A,$o),X: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),Aa2),Ba),X)
=> ~ ( aa(A,$o,Aa2,X)
=> ~ aa(A,$o,Ba,X) ) ) ).
% inf1E
tff(fact_4236_inf1D1,axiom,
! [A: $tType,Aa2: fun(A,$o),Ba: fun(A,$o),X: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),Aa2),Ba),X)
=> aa(A,$o,Aa2,X) ) ).
% inf1D1
tff(fact_4237_inf1D2,axiom,
! [A: $tType,Aa2: fun(A,$o),Ba: fun(A,$o),X: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),Aa2),Ba),X)
=> aa(A,$o,Ba,X) ) ).
% inf1D2
tff(fact_4238_ordering__top__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
( ordering_top_axioms(A,Less_eq,Top)
<=> ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),Top) ) ).
% ordering_top_axioms_def
tff(fact_4239_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Top: A] :
( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top_axioms.intro
tff(fact_4240_ID_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,$o),X: A,Pa: fun(A,$o)] :
( aa(A,$o,bNF_id_bnf(fun(A,$o),P),X)
=> ( ! [Z3: A] :
( member(A,Z3,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))
=> ( aa(A,$o,P,Z3)
=> aa(A,$o,Pa,Z3) ) )
=> aa(A,$o,bNF_id_bnf(fun(A,$o),Pa),X) ) ) ).
% ID.pred_mono_strong
tff(fact_4241_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya2: A,P: fun(A,$o),Pa: fun(A,$o)] :
( ( X = Ya2 )
=> ( ! [Z3: A] :
( member(A,Z3,aa(set(A),set(A),insert2(A,Ya2),bot_bot(set(A))))
=> ( aa(A,$o,P,Z3)
<=> aa(A,$o,Pa,Z3) ) )
=> ( aa(A,$o,bNF_id_bnf(fun(A,$o),P),X)
<=> aa(A,$o,bNF_id_bnf(fun(A,$o),Pa),Ya2) ) ) ) ).
% ID.pred_cong
tff(fact_4242_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X2: A] :
( aa(A,$o,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X2)
<=> member(A,X2,domain(A,B,R2)) ) ).
% Domainp_Domain_eq
tff(fact_4243_Domain__def,axiom,
! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : domain(A,B,X2) = aa(fun(A,$o),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),X2))) ).
% Domain_def
tff(fact_4244_wfP__wf__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( wfP(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> wf(A,R2) ) ).
% wfP_wf_eq
tff(fact_4245_nth__step__trancl,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),N: nat,M2: nat] :
( ! [N4: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N4))),aa(nat,A,nth(A,Xs),N4)),R) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),aa(nat,A,nth(A,Xs),M2)),transitive_trancl(A,R)) ) ) ) ).
% nth_step_trancl
tff(fact_4246_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))),mset(A,aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),Xs)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Xs1))),mset(A,Xs2)) ).
% mset_mergesort_by_rel_split
tff(fact_4247_lenlex__length,axiom,
! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
=> aa(nat,$o,aa(nat,fun(nat,$o),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_4248_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns),lenlex(A,R2))
<=> ( Ns != nil(A) ) ) ).
% Nil_lenlex_iff1
tff(fact_4249_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A)),lenlex(A,R2)) ).
% Nil_lenlex_iff2
tff(fact_4250_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2) ).
% mergesort_by_rel_split.simps(1)
tff(fact_4251_lenlex__irreflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R2)) ) ).
% lenlex_irreflexive
tff(fact_4252_lenlex__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lenlex(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lenlex(A,R2))
=> ( trans(A,R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Z2),lenlex(A,R2)) ) ) ) ).
% lenlex_trans
tff(fact_4253_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),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_4254_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_nn(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum_funpow
tff(fact_4255_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_no(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum
tff(fact_4256_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list(A),N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),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_4257_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),nil(B)) = zero_zero(A) ) ).
% horner_sum_simps(1)
tff(fact_4258_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),$o))] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),A0),A1))
=> ( ! [R11: fun(A,fun(A,$o)),Xs3: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R11),Xs3))
=> ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R11),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
=> ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs3)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R11),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3))) )
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R11),Xs3) ) ) )
=> aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,A0),A1) ) ) ).
% mergesort_by_rel.pinduct
tff(fact_4259_mergesort__by__rel_Osimps,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
mergesort_by_rel(A,R,Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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_4260_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
( ( mergesort_by_rel(A,X,Xa) = Y )
=> ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa,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_4261_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).
% nth_zip
tff(fact_4262_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
~ ! [R11: fun(A,fun(A,$o)),Xs3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R11),Xs3) ).
% mergesort_by_rel.cases
tff(fact_4263_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: list(A)] :
( ( mergesort_by_rel(A,X,Xa) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xa,merges9089515139780605204_merge(A,X,mergesort_by_rel(A,X,aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),mergesort_by_rel(A,X,aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))))) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa)) ) ) ) ).
% mergesort_by_rel.pelims
tff(fact_4264_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R),Xs))
=> ( mergesort_by_rel(A,R,Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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_4265_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),X1: A,X22: A,Xs: list(A)] : mergesort_by_rel(A,R,aa(list(A),list(A),cons(A,X1),aa(list(A),list(A),cons(A,X22),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_np(fun(A,fun(A,$o)),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),cons(A,X1),nil(A))),aa(list(A),list(A),cons(A,X22),nil(A))),Xs)) ).
% mergesort_by_rel_simps(3)
tff(fact_4266_set__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : set2(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_nq(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).
% set_zip
tff(fact_4267_lenlex__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_nr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% lenlex_conv
tff(fact_4268_zipf__zip,axiom,
! [A: $tType,B: $tType,L1: list(A),L22: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
=> ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L1,L22) = zip(A,B,L1,L22) ) ) ).
% zipf_zip
tff(fact_4269_set__empty,axiom,
! [A: $tType,Xs: list(A)] :
( ( set2(A,Xs) = bot_bot(set(A)) )
<=> ( Xs = nil(A) ) ) ).
% set_empty
tff(fact_4270_set__empty2,axiom,
! [A: $tType,Xs: list(A)] :
( ( bot_bot(set(A)) = set2(A,Xs) )
<=> ( Xs = nil(A) ) ) ).
% set_empty2
tff(fact_4271_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),cons(A,X),Xs),aa(list(B),list(B),cons(B,Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),zip(A,B,Xs,Ys)) ).
% zip_Cons_Cons
tff(fact_4272_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),cons(B,X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs))) ) ).
% horner_sum_simps(2)
tff(fact_4273_enumerate__simps_I2_J,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : enumerate(A,N,aa(list(A),list(A),cons(A,X),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),cons(product_prod(nat,A),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),N),X)),enumerate(A,aa(nat,nat,suc,N),Xs)) ).
% enumerate_simps(2)
tff(fact_4274_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),lex(A,R2))
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
& ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
| ( ( X = Y )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2)) ) ) ) ).
% Cons_in_lex
tff(fact_4275_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( member(B,Y,set2(B,Ys))
=> ~ ! [X3: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),set2(product_prod(A,B),zip(A,B,Xs,Ys))) ) ) ).
% in_set_impl_in_set_zip2
tff(fact_4276_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( member(A,X,set2(A,Xs))
=> ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),set2(product_prod(A,B),zip(A,B,Xs,Ys))) ) ) ).
% in_set_impl_in_set_zip1
tff(fact_4277_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),zip(A,B,Xs,Ys)))
=> member(B,Y,set2(B,Ys)) ) ).
% set_zip_rightD
tff(fact_4278_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),zip(A,B,Xs,Ys)))
=> member(A,X,set2(A,Xs)) ) ).
% set_zip_leftD
tff(fact_4279_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),Xy),Xys) )
=> ~ ! [X3: A,Xs4: list(A)] :
( ( Xs = aa(list(A),list(A),cons(A,X3),Xs4) )
=> ! [Y2: B,Ys2: list(B)] :
( ( Ys = aa(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),X3),Y2) )
=> ( Xys != zip(A,B,Xs4,Ys2) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_4280_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),zip(A,B,Xs,Ys)))
=> ~ ( member(A,X,set2(A,Xs))
=> ~ member(B,Y,set2(B,Ys)) ) ) ).
% in_set_zipE
tff(fact_4281_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),set2(product_prod(A,A),zip(A,A,Xs,Xs)))
<=> ( member(A,A3,set2(A,Xs))
& ( A3 = B2 ) ) ) ).
% zip_same
tff(fact_4282_empty__set,axiom,
! [A: $tType] : bot_bot(set(A)) = set2(A,nil(A)) ).
% empty_set
tff(fact_4283_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)
=> ~ ! [X3: A,Xs3: 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),cons(A,X3),Xs3)),Ys3) ) ).
% subset_eq_mset_impl.cases
tff(fact_4284_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)
=> ( ! [Xs3: 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)),Xs3),nil(A))
=> ~ ! [X3: A,Xs3: 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),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3)) ) ) ).
% shuffles.cases
tff(fact_4285_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
( ! [F3: 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))),F3),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))
=> ~ ! [F3: fun(A,B),A4: A,As3: 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))),F3),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),cons(A,A4),As3)),Bs2)) ) ).
% map_tailrec_rev.cases
tff(fact_4286_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
( ! [Xs12: list(A),Xs22: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),nil(A))
=> ( ! [Xs12: list(A),Xs22: list(A),X3: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),cons(A,X3),nil(A)))
=> ~ ! [Xs12: list(A),Xs22: list(A),X12: A,X23: A,Xs3: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),cons(A,X12),aa(list(A),list(A),cons(A,X23),Xs3))) ) ) ).
% mergesort_by_rel_split.cases
tff(fact_4287_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
( ! [R11: fun(A,fun(A,$o)),X3: A,Xs3: list(A),Y2: A,Ys3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3)))
=> ( ! [R11: fun(A,fun(A,$o)),Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A)))
=> ~ ! [R11: fun(A,fun(A,$o)),V2: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),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),cons(A,V2),Va))) ) ) ).
% mergesort_by_rel_merge.cases
tff(fact_4288_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
( ! [R11: fun(A,fun(A,$o)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
=> ~ ! [R11: fun(A,fun(A,$o)),Sl: list(A),X3: A,Xs3: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),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),cons(A,X3),Xs3))) ) ).
% quicksort_by_rel.cases
tff(fact_4289_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
( ! [P3: fun(A,$o),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P3),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
=> ~ ! [P3: fun(A,$o),Yes: list(A),No: list(A),X3: A,Xs3: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P3),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),cons(A,X3),Xs3))) ) ).
% partition_rev.cases
tff(fact_4290_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
( ! [P3: fun(A,fun(B,$o))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
=> ( ! [P3: fun(A,fun(B,$o)),A4: A,As3: list(A),B3: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As3)),aa(list(B),list(B),cons(B,B3),Bs2)))
=> ( ! [P3: fun(A,fun(B,$o)),V2: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,V2),Va)),nil(B)))
=> ~ ! [P3: fun(A,fun(B,$o)),V2: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),cons(B,V2),Va))) ) ) ) ).
% list_all_zip.cases
tff(fact_4291_merge_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: product_prod(list(A),list(A))] :
( ! [L23: 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)),L23)
=> ( ! [V2: 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),cons(A,V2),Va)),nil(A))
=> ~ ! [X12: A,L12: list(A),X23: A,L23: 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),cons(A,X12),L12)),aa(list(A),list(A),cons(A,X23),L23)) ) ) ) ).
% merge.cases
tff(fact_4292_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
( ! [F3: 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))),F3),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)))
=> ( ! [F3: fun(A,fun(B,C)),A4: A,As3: list(A),B3: 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))),F3),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),cons(A,A4),As3)),aa(list(B),list(B),cons(B,B3),Bs2)))
=> ( ! [A4: fun(A,fun(B,C)),V2: 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))),A4),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),cons(A,V2),Va)),nil(B)))
=> ~ ! [A4: fun(A,fun(B,C)),V2: 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))),A4),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),cons(B,V2),Va))) ) ) ) ).
% zipf.cases
tff(fact_4293_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: product_prod(fun(A,B),list(A))] :
( ! [F3: fun(A,B),X3: 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)),F3),aa(list(A),list(A),cons(A,X3),nil(A)))
=> ( ! [F3: fun(A,B),X3: A,Y2: A,Zs: 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)),F3),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Zs)))
=> ~ ! [A4: 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)),A4),nil(A)) ) ) ) ).
% arg_min_list.cases
tff(fact_4294_successively_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
( ! [P3: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P3),nil(A))
=> ( ! [P3: fun(A,fun(A,$o)),X3: A] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P3),aa(list(A),list(A),cons(A,X3),nil(A)))
=> ~ ! [P3: fun(A,fun(A,$o)),X3: A,Y2: A,Xs3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P3),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3))) ) ) ).
% successively.cases
tff(fact_4295_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
( ! [P3: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P3),nil(A))
=> ~ ! [P3: fun(A,fun(A,$o)),X3: A,Ys3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P3),aa(list(A),list(A),cons(A,X3),Ys3)) ) ).
% sorted_wrt.cases
tff(fact_4296_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),X1: A,X22: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),cons(A,X1),aa(list(A),list(A),cons(A,X22),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X1),Xs1)),aa(list(A),list(A),cons(A,X22),Xs2)),Xs) ).
% mergesort_by_rel_split.simps(3)
tff(fact_4297_Nil__notin__lex,axiom,
! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys),lex(A,R2)) ).
% Nil_notin_lex
tff(fact_4298_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),lex(A,R2)) ).
% Nil2_notin_lex
tff(fact_4299_lexl__not__refl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: list(A)] :
( irrefl(A,R2)
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),X),lex(A,R2)) ) ).
% lexl_not_refl
tff(fact_4300_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X,Xa) = Y )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) ) ) )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),cons(A,X3),nil(A)) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs12)),Xs22) ) ) )
=> ~ ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X12: A,X23: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X12),aa(list(A),list(A),cons(A,X23),Xs3)) )
=> ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X12),Xs12)),aa(list(A),list(A),cons(A,X23),Xs22)),Xs3) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
tff(fact_4301_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),Xs2: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs2),aa(list(A),list(A),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),cons(A,X),Xs1)),Xs2) ).
% mergesort_by_rel_split.simps(2)
tff(fact_4302_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,M2),Ms)),aa(list(A),list(A),cons(A,N),Ns)),lenlex(A,R2))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
| ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),N),R2) )
| ( ( M2 = N )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2)) ) ) ) ).
% Cons_lenlex_iff
tff(fact_4303_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_ns(list(A),product_prod(nat,list(A)))) ).
% lenlex_def
tff(fact_4304_Pow__set_I1_J,axiom,
! [A: $tType] : pow(A,set2(A,nil(A))) = aa(set(set(A)),set(set(A)),insert2(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).
% Pow_set(1)
tff(fact_4305_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ! [X3: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ! [Y2: A,Ys3: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,Y2),Ys3) )
=> ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),X,X3),Y2),aa(list(A),list(A),cons(A,X3),merges9089515139780605204_merge(A,X,Xs3,aa(list(A),list(A),cons(A,Y2),Ys3))),aa(list(A),list(A),cons(A,Y2),merges9089515139780605204_merge(A,X,aa(list(A),list(A),cons(A,X3),Xs3),Ys3))) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3)))) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,V2),Va) )
=> ( ( Y = aa(list(A),list(A),cons(A,V2),Va) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),cons(A,V2),Va)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
tff(fact_4306_zipf_Opelims,axiom,
! [A: $tType,B: $tType,C: $tType,X: fun(B,fun(C,A)),Xa: list(B),Xb: list(C),Y: list(A)] :
( ( zipf(B,C,A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),Xa),Xb)))
=> ( ( ( Xa = nil(B) )
=> ( ( Xb = nil(C) )
=> ( ( Y = nil(A) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),nil(C)))) ) ) )
=> ( ! [A4: B,As3: list(B)] :
( ( Xa = aa(list(B),list(B),cons(B,A4),As3) )
=> ! [B3: C,Bs2: list(C)] :
( ( Xb = aa(list(C),list(C),cons(C,B3),Bs2) )
=> ( ( Y = aa(list(A),list(A),cons(A,aa(C,A,aa(B,fun(C,A),X,A4),B3)),zipf(B,C,A,X,As3,Bs2)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),cons(B,A4),As3)),aa(list(C),list(C),cons(C,B3),Bs2)))) ) ) )
=> ( ! [V2: B,Va: list(B)] :
( ( Xa = aa(list(B),list(B),cons(B,V2),Va) )
=> ( ( Xb = nil(C) )
=> ( ( Y = undefined(list(A)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),aa(list(B),list(B),cons(B,V2),Va)),nil(C)))) ) ) )
=> ~ ( ( Xa = nil(B) )
=> ! [V2: C,Va: list(C)] :
( ( Xb = aa(list(C),list(C),cons(C,V2),Va) )
=> ( ( Y = undefined(list(A)) )
=> ~ aa(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),$o,accp(product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),zipf_rel(B,C,A)),aa(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C))),aa(fun(B,fun(C,A)),fun(product_prod(list(B),list(C)),product_prod(fun(B,fun(C,A)),product_prod(list(B),list(C)))),product_Pair(fun(B,fun(C,A)),product_prod(list(B),list(C))),X),aa(list(C),product_prod(list(B),list(C)),aa(list(B),fun(list(C),product_prod(list(B),list(C))),product_Pair(list(B),list(C)),nil(B)),aa(list(C),list(C),cons(C,V2),Va)))) ) ) ) ) ) ) ) ) ).
% zipf.pelims
tff(fact_4307_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X4: product_prod(A,B)] :
( member(product_prod(A,B),X4,set2(product_prod(A,B),zip(A,B,Xs,Ys)))
=> aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4) ) ) ) ).
% listrel_iff_zip
tff(fact_4308_set__Cons__sing__Nil,axiom,
! [A: $tType,Aa2: set(A)] : set_Cons(A,Aa2,aa(set(list(A)),set(list(A)),insert2(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_nt(A,list(A))),Aa2) ).
% set_Cons_sing_Nil
tff(fact_4309_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),listrel(A,A,transitive_rtrancl(A,R2))) ).
% listrel_rtrancl_refl
tff(fact_4310_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)),insert2(list(B),nil(B)),bot_bot(set(list(B))))) = aa(set(list(A)),set(list(A)),insert2(list(A),nil(A)),bot_bot(set(list(A)))) ).
% listrel_Nil
tff(fact_4311_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list(A),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B)),listrel(A,B,R2))
=> ( Xs = nil(A) ) ) ).
% listrel_Nil2
tff(fact_4312_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs),listrel(A,B,R2))
=> ( Xs = nil(B) ) ) ).
% listrel_Nil1
tff(fact_4313_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)),listrel(A,B,R2)) ).
% listrel.Nil
tff(fact_4314_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).
% listrel_eq_len
tff(fact_4315_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs2: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R2)))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),listrel(A,A,transitive_rtrancl(A,R2)))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),listrel(A,A,transitive_rtrancl(A,R2))) ) ) ).
% listrel_rtrancl_trans
tff(fact_4316_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)),insert2(list(B),aa(list(B),list(B),cons(B,X),Xs)),bot_bot(set(list(B))))) = set_Cons(A,image(B,A,R2,aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))),image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),insert2(list(B),Xs),bot_bot(set(list(B)))))) ).
% listrel_Cons
tff(fact_4317_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R2)
=> ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(B),list(B),cons(B,Y),Ys)),listrel(A,B,R2)) ) ) ).
% listrel.Cons
tff(fact_4318_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,Y),Ys)),Xs),listrel(A,B,R2))
=> ~ ! [Y2: B,Ys3: list(B)] :
( ( Xs = aa(list(B),list(B),cons(B,Y2),Ys3) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y2),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys3),listrel(A,B,R2)) ) ) ) ).
% listrel_Cons1
tff(fact_4319_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),cons(B,Y),Ys)),listrel(A,B,R2))
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys),listrel(A,B,R2)) ) ) ) ).
% listrel_Cons2
tff(fact_4320_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs5: list(A),Ys4: list(B)] :
( ( A1 = aa(list(A),list(A),cons(A,X4),Xs5) )
& ( A22 = aa(list(B),list(B),cons(B,Y3),Ys4) )
& member(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)
& member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs5),Ys4),listrel(A,B,R2)) ) ) ) ).
% listrel.simps
tff(fact_4321_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X3: A,Y2: B,Xs3: list(A)] :
( ( A1 = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ! [Ys3: list(B)] :
( ( A22 = aa(list(B),list(B),cons(B,Y2),Ys3) )
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2),R2)
=> ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys3),listrel(A,B,R2)) ) ) ) ) ) ).
% listrel.cases
tff(fact_4322_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N2)),aa(nat,B,nth(B,Ys),N2)),R2) ) ) ) ).
% listrel_iff_nth
tff(fact_4323_listset_Osimps_I1_J,axiom,
! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),insert2(list(A),nil(A)),bot_bot(set(list(A)))) ).
% listset.simps(1)
tff(fact_4324_part__code_I2_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),Pivot: B,X: A,Xs: list(A)] : linorder_part(A,B,F,Pivot,aa(list(A),list(A),cons(A,X),Xs)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),aa(fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_nv(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),F),Pivot),X)),linorder_part(A,B,F,Pivot,Xs)) ) ).
% part_code(2)
tff(fact_4325_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( linord4507533701916653071of_set(A,Aa2) = aa(list(A),list(A),cons(A,lattic643756798350308766er_Min(A,Aa2)),linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,lattic643756798350308766er_Min(A,Aa2)),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set_nonempty
tff(fact_4326_upto_Opsimps,axiom,
! [I: int,J: int] :
( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
=> ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).
% upto.psimps
tff(fact_4327_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_4328_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( linord4507533701916653071of_set(A,Aa2) = nil(A) )
<=> ( Aa2 = bot_bot(set(A)) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4329_part__code_I1_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),Pivot: B] : linorder_part(A,B,F,Pivot,nil(A)) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),nil(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A))) ) ).
% part_code(1)
tff(fact_4330_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list(int)] :
( ( upto(X,Xa) = Y )
=> ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa))
=> ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa),aa(list(int),list(int),cons(int,X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)),nil(int)) )
=> ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa)) ) ) ) ).
% upto.pelims
tff(fact_4331_list__encode_Oelims,axiom,
! [X: list(nat),Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( ( ( X = nil(nat) )
=> ( Y != zero_zero(nat) ) )
=> ~ ! [X3: nat,Xs3: list(nat)] :
( ( X = aa(list(nat),list(nat),cons(nat,X3),Xs3) )
=> ( Y != aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),nat_list_encode(Xs3)))) ) ) ) ) ).
% list_encode.elims
tff(fact_4332_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: fun(A,B),Xa: list(A),Y: A] :
( ( arg_min_list(A,B,X,Xa) = Y )
=> ( aa(product_prod(fun(A,B),list(A)),$o,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))
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),cons(A,X3),nil(A)) )
=> ( ( Y = X3 )
=> ~ aa(product_prod(fun(A,B),list(A)),$o,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),cons(A,X3),nil(A)))) ) )
=> ( ! [X3: A,Y2: A,Zs: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Zs)) )
=> ( ( Y = $let(
m: A,
m:= arg_min_list(A,B,X,aa(list(A),list(A),cons(A,Y2),Zs)),
$ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,m)),X3,m) ) )
=> ~ aa(product_prod(fun(A,B),list(A)),$o,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),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Zs)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ aa(product_prod(fun(A,B),list(A)),$o,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_4333_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list(A)] : set2(A,removeAll(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% set_removeAll
tff(fact_4334_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( linord4507533701916653071of_set(A,aa(set(A),set(A),insert2(A,X),Aa2)) = linorder_insort_key(A,A,aTP_Lamp_nw(A,A),X,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4335_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list(nat)] : nat_list_encode(aa(list(nat),list(nat),cons(nat,X),Xs)) = aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),nat_list_encode(Xs)))) ).
% list_encode.simps(2)
tff(fact_4336_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( linord4507533701916653071of_set(A,Aa2) = linorder_insort_key(A,A,aTP_Lamp_nw(A,A),X,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
tff(fact_4337_lex__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_nx(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% lex_conv
tff(fact_4338_listrel__def,axiom,
! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : listrel(A,B,X2) = aa(fun(product_prod(list(A),list(B)),$o),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),$o)),fun(product_prod(list(A),list(B)),$o),product_case_prod(list(A),list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),X2)))) ).
% listrel_def
tff(fact_4339_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( finite_card(list(A),shuffles(A,Xs,Ys)) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).
% card_disjoint_shuffles
tff(fact_4340_list__encode_Opelims,axiom,
! [X: list(nat),Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),X)
=> ( ( ( X = nil(nat) )
=> ( ( Y = zero_zero(nat) )
=> ~ aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),nil(nat)) ) )
=> ~ ! [X3: nat,Xs3: list(nat)] :
( ( X = aa(list(nat),list(nat),cons(nat,X3),Xs3) )
=> ( ( Y = aa(nat,nat,suc,nat_prod_encode(aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),nat_list_encode(Xs3)))) )
=> ~ aa(list(nat),$o,accp(list(nat),nat_list_encode_rel),aa(list(nat),list(nat),cons(nat,X3),Xs3)) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_4341_lenlex__append2,axiom,
! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
( irrefl(A,R)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys)),lenlex(A,R))
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lenlex(A,R)) ) ) ).
% lenlex_append2
tff(fact_4342_append_Osemigroup__axioms,axiom,
! [A: $tType] : semigroup(list(A),append(A)) ).
% append.semigroup_axioms
tff(fact_4343_lex__append__leftI,axiom,
! [A: $tType,Ys: list(A),Zs2: list(A),R2: set(product_prod(A,A)),Xs: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lex(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lex(A,R2)) ) ).
% lex_append_leftI
tff(fact_4344_lex__append__left__iff,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lex(A,R2))
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lex(A,R2)) ) ) ).
% lex_append_left_iff
tff(fact_4345_lex__append__leftD,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lex(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lex(A,R2)) ) ) ).
% lex_append_leftD
tff(fact_4346_lex__append__rightI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),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_4347_lenlex__append1,axiom,
! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs),lenlex(A,R))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),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_4348_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),insert2(list(A),Xs),bot_bot(set(list(A)))) ).
% shuffles.simps(2)
tff(fact_4349_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),insert2(list(A),Ys),bot_bot(set(list(A)))) ).
% shuffles.simps(1)
tff(fact_4350_shuffles_Oelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( shuffles(A,X,Xa) = Y )
=> ( ( ( X = nil(A) )
=> ( Y != aa(set(list(A)),set(list(A)),insert2(list(A),Xa),bot_bot(set(list(A)))) ) )
=> ( ( ( Xa = nil(A) )
=> ( Y != aa(set(list(A)),set(list(A)),insert2(list(A),X),bot_bot(set(list(A)))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ! [Y2: A,Ys3: list(A)] :
( ( Xa = aa(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))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,X3)),shuffles(A,Xs3,aa(list(A),list(A),cons(A,Y2),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,Y2)),shuffles(A,aa(list(A),list(A),cons(A,X3),Xs3),Ys3))) ) ) ) ) ) ) ).
% shuffles.elims
tff(fact_4351_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Ys))) ) ).
% horner_sum_append
tff(fact_4352_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X2: list(A),Xa3: list(B)] :
( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X2),Xa3)
<=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X2),Xa3),listrel(A,B,R2)) ) ).
% listrelp_listrel_eq
tff(fact_4353_lexn__conv,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] : lexn(A,R2,N) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ny(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),N))) ).
% lexn_conv
tff(fact_4354_lexord__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_nz(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% lexord_def
tff(fact_4355_listrel1__def,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_oa(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).
% listrel1_def
tff(fact_4356_lists__empty,axiom,
! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),insert2(list(A),nil(A)),bot_bot(set(list(A)))) ).
% lists_empty
tff(fact_4357_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),listrel1(A,R2))
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
& ( Xs = Ys ) )
| ( ( X = Y )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_4358_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,A3),X)),aa(list(A),list(A),cons(A,B2),Y)),lexord(A,R2))
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
| ( ( A3 = B2 )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2)) ) ) ) ).
% lexord_cons_cons
tff(fact_4359_lexord__Nil__left,axiom,
! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y),lexord(A,R2))
<=> ? [A5: A,X4: list(A)] : Y = aa(list(A),list(A),cons(A,A5),X4) ) ).
% lexord_Nil_left
tff(fact_4360_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
( irrefl(A,R2)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lexord(A,R2))
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lexord(A,R2)) ) ) ).
% lexord_same_pref_if_irrefl
tff(fact_4361_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).
% lexn.simps(1)
tff(fact_4362_listrel1I2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,X),Ys)),listrel1(A,R2)) ) ).
% listrel1I2
tff(fact_4363_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2)))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,X),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).
% rtrancl_listrel1_ConsI1
tff(fact_4364_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),listrel1(A,R2)) ).
% not_listrel1_Nil
tff(fact_4365_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs),listrel1(A,R2)) ).
% not_Nil_listrel1
tff(fact_4366_listrel1__eq__len,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).
% listrel1_eq_len
tff(fact_4367_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),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_4368_append__listrel1I,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
( ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
& ( Us = Vs ) )
| ( ( Xs = Ys )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs),listrel1(A,R2)) ) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(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_4369_lexord__linear,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X: list(A),Y: list(A)] :
( ! [A4: A,B3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3),R2)
| ( A4 = B3 )
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4),R2) )
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2))
| ( X = Y )
| member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X),lexord(A,R2)) ) ) ).
% lexord_linear
tff(fact_4370_lexord__irreflexive,axiom,
! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2)
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lexord(A,R2)) ) ).
% lexord_irreflexive
tff(fact_4371_lexord__Nil__right,axiom,
! [A: $tType,X: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A)),lexord(A,R2)) ).
% lexord_Nil_right
tff(fact_4372_in__listrel1__converse,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),listrel1(A,converse(A,A,R2)))
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),converse(list(A),list(A),listrel1(A,R2))) ) ).
% in_listrel1_converse
tff(fact_4373_lexord__append__leftI,axiom,
! [A: $tType,U: list(A),V: list(A),R2: set(product_prod(A,A)),X: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V)),lexord(A,R2)) ) ).
% lexord_append_leftI
tff(fact_4374_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).
% rtrancl_listrel1_if_listrel
tff(fact_4375_lexord__trans,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A)),Z2: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lexord(A,R2))
=> ( trans(A,R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Z2),lexord(A,R2)) ) ) ) ).
% lexord_trans
tff(fact_4376_lexord__asymmetric,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: list(A),B2: list(A)] :
( asym(A,R)
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A3),B2),lexord(A,R))
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),B2),A3),lexord(A,R)) ) ) ).
% lexord_asymmetric
tff(fact_4377_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,Y),Xs)),listrel1(A,R2)) ) ).
% listrel1I1
tff(fact_4378_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),Ys),listrel1(A,R2))
=> ( ! [Y2: A] :
( ( Ys = aa(list(A),list(A),cons(A,Y2),Xs) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2),R2) )
=> ~ ! [Zs: list(A)] :
( ( Ys = aa(list(A),list(A),cons(A,X),Zs) )
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1E1
tff(fact_4379_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),cons(A,Y),Ys)),listrel1(A,R2))
=> ( ! [X3: A] :
( ( Xs = aa(list(A),list(A),cons(A,X3),Ys) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2) )
=> ~ ! [Zs: list(A)] :
( ( Xs = aa(list(A),list(A),cons(A,Y),Zs) )
=> ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs),Ys),listrel1(A,R2)) ) ) ) ).
% Cons_listrel1E2
tff(fact_4380_lexord__partial__trans,axiom,
! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs2: list(A)] :
( ! [X3: A,Y2: A,Z3: A] :
( member(A,X3,set2(A,Xs))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3),R2)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3),R2) ) ) )
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lexord(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),lexord(A,R2)) ) ) ) ).
% lexord_partial_trans
tff(fact_4381_lexord__append__leftD,axiom,
! [A: $tType,X: list(A),U: list(A),V: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V)),lexord(A,R2))
=> ( ! [A4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R2)) ) ) ).
% lexord_append_leftD
tff(fact_4382_lexord__append__rightI,axiom,
! [A: $tType,Y: list(A),X: list(A),R2: set(product_prod(A,A))] :
( ? [B9: A,Z5: list(A)] : Y = aa(list(A),list(A),cons(A,B9),Z5)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),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_4383_lexord__sufE,axiom,
! [A: $tType,Xs: list(A),Zs2: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),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)),Zs2) = aa(list(A),nat,size_size(list(A)),Qs) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R2)) ) ) ) ) ).
% lexord_sufE
tff(fact_4384_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R2))) ) ).
% listrel_reflcl_if_listrel1
tff(fact_4385_lexord__lex,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lex(A,R2))
<=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),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_4386_lexn__length,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),N: nat] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),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_4387_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
=> ~ ! [X3: A,Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y2),R2)
=> ! [Us2: list(A),Vs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,X3),Vs2)) )
=> ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Y2),Vs2)) ) ) ) ) ).
% listrel1E
tff(fact_4388_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)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2)
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(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),cons(A,Y),Vs)) )
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).
% listrel1I
tff(fact_4389_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,R2))
=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2)))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_4390_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)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),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),cons(A,B2),Y))),lexord(A,R2)) ) ).
% lexord_append_left_rightI
tff(fact_4391_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs2)),lexord(A,R2))
<=> ( ? [X4: A] :
( member(A,X4,set2(A,Xs))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2) )
| member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2),lexord(A,R2)) ) ) ).
% lexord_same_pref_iff
tff(fact_4392_lexord__sufI,axiom,
! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V: list(A),Z2: list(A)] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2),lexord(A,R2))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z2)),lexord(A,R2)) ) ) ).
% lexord_sufI
tff(fact_4393_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),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),cons(A,Y),nil(A)))),listrel1(A,R2))
<=> ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
& ( X = Y ) )
| ( ( Xs = Ys )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_4394_List_Olexordp__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
( lexordp(A,R2,Xs,Ys)
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% List.lexordp_def
tff(fact_4395_listrel1p__def,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
( listrel1p(A,R2,Xs,Ys)
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% listrel1p_def
tff(fact_4396_lexord__take__index__conv,axiom,
! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y),lexord(A,R2))
<=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))
& ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
| ? [I2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
& ( take(A,I2,X) = take(A,I2,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I2)),aa(nat,A,nth(A,Y),I2)),R2) ) ) ) ).
% lexord_take_index_conv
tff(fact_4397_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),set2(B,Xs))) = aa(A,A,fold(B,A,comp(A,fun(A,A),B,inf_inf(A),F),Xs),top_top(A)) ) ).
% INF_set_fold
tff(fact_4398_Sup__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),set2(A,Xs)) = aa(A,A,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ).
% Sup_set_fold
tff(fact_4399_Inf__set__fold,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),set2(A,Xs)) = aa(A,A,fold(A,A,inf_inf(A),Xs),top_top(A)) ) ).
% Inf_set_fold
tff(fact_4400_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),set2(A,aa(list(A),list(A),cons(A,X),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),X) ) ).
% Sup_fin.set_eq_fold
tff(fact_4401_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Lcm(A,set2(A,Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).
% Lcm_set_eq_fold
tff(fact_4402_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),set2(A,Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).
% Lcm_fin.set_eq_fold
tff(fact_4403_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
=> ~ ! [I3: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys))
=> ( ( take(A,I3,Xs) = take(A,I3,Ys) )
=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys),I3)),R2) ) ) ) ) ).
% lex_take_index
tff(fact_4404_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),set2(B,Xs))) = aa(A,A,fold(B,A,comp(A,fun(A,A),B,sup_sup(A),F),Xs),bot_bot(A)) ) ).
% SUP_set_fold
tff(fact_4405_inv__image__partition,axiom,
! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(A,$o,P,X3) )
=> ( ! [Y2: A] :
( member(A,Y2,set2(A,Ys))
=> ~ aa(A,$o,P,Y2) )
=> ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),insert2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).
% inv_image_partition
tff(fact_4406_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),J: nat,X: A,Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
=> ( mset(product_prod(A,B),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J,Xs)),aa(list(A),list(A),cons(A,X),drop(A,J,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J,Ys)),aa(list(B),list(B),cons(B,Y),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),zip(A,B,Xs,Ys))) ) ) ) ).
% mset_zip_take_Cons_drop_twice
tff(fact_4407_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Aa2: set(A),X: A] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,Aa2)) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_4408_shuffles_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
( ( shuffles(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),insert2(list(A),Xa),bot_bot(set(list(A)))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ( ( ( Xa = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),insert2(list(A),X),bot_bot(set(list(A)))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ! [Y2: A,Ys3: list(A)] :
( ( Xa = aa(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))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,X3)),shuffles(A,Xs3,aa(list(A),list(A),cons(A,Y2),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,Y2)),shuffles(A,aa(list(A),list(A),cons(A,X3),Xs3),Ys3))) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_4409_shuffles_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
=> ( ! [Ys3: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3))
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys3) )
=> ( ! [Xs3: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A)))
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs3),nil(A)) )
=> ( ! [X3: A,Xs3: list(A),Y2: A,Ys3: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3)))
=> ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs3),aa(list(A),list(A),cons(A,Y2),Ys3))
=> ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X3),Xs3)),Ys3)
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X3),Xs3)),aa(list(A),list(A),cons(A,Y2),Ys3)) ) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ) ).
% shuffles.pinduct
tff(fact_4410_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
=> ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),insert2(list(A),Ys),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(1)
tff(fact_4411_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)))
=> ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),insert2(list(A),Xs),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(2)
tff(fact_4412_partition_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,$o)] : aa(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_4413_partition__P,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(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) )
=> ( ! [X2: A] :
( member(A,X2,set2(A,Yes2))
=> aa(A,$o,P,X2) )
& ! [X2: A] :
( member(A,X2,set2(A,No2))
=> ~ aa(A,$o,P,X2) ) ) ) ).
% partition_P
tff(fact_4414_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Xs)),aa(list(A),list(A),cons(A,Y),Ys)))
=> ( shuffles(A,aa(list(A),list(A),cons(A,X),Xs),aa(list(A),list(A),cons(A,Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,X)),shuffles(A,Xs,aa(list(A),list(A),cons(A,Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,Y)),shuffles(A,aa(list(A),list(A),cons(A,X),Xs),Ys))) ) ) ).
% shuffles.psimps(3)
tff(fact_4415_partition_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,$o),X: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),cons(A,X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_ob(fun(A,$o),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)),partition(A,P),Xs)) ).
% partition.simps(2)
tff(fact_4416_partition__set,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(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)),set2(A,Yes2)),set2(A,No2)) = set2(A,Xs) ) ) ).
% partition_set
tff(fact_4417_splice_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
=> ( ! [Ys3: list(A)] :
( aa(product_prod(list(A),list(A)),$o,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))
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys3) )
=> ( ! [X3: A,Xs3: list(A),Ys3: list(A)] :
( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs3)),Ys3))
=> ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Ys3),Xs3)
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X3),Xs3)),Ys3) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ).
% splice.pinduct
tff(fact_4418_set__rec,axiom,
! [A: $tType,Xs: list(A)] : set2(A,Xs) = rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_oc(A,fun(list(A),fun(set(A),set(A)))),Xs) ).
% set_rec
tff(fact_4419_map__tailrec__rev_Opelims,axiom,
! [B: $tType,A: $tType,X: fun(B,A),Xa: list(B),Xb: list(A),Y: list(A)] :
( ( map_tailrec_rev(B,A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(B,A),product_prod(list(B),list(A))),$o,accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A)),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),Xa),Xb)))
=> ( ( ( Xa = nil(B) )
=> ( ( Y = Xb )
=> ~ aa(product_prod(fun(B,A),product_prod(list(B),list(A))),$o,accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A)),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),nil(B)),Xb))) ) )
=> ~ ! [A4: B,As3: list(B)] :
( ( Xa = aa(list(B),list(B),cons(B,A4),As3) )
=> ( ( Y = map_tailrec_rev(B,A,X,As3,aa(list(A),list(A),cons(A,aa(B,A,X,A4)),Xb)) )
=> ~ aa(product_prod(fun(B,A),product_prod(list(B),list(A))),$o,accp(product_prod(fun(B,A),product_prod(list(B),list(A))),map_tailrec_rev_rel(B,A)),aa(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A))),aa(fun(B,A),fun(product_prod(list(B),list(A)),product_prod(fun(B,A),product_prod(list(B),list(A)))),product_Pair(fun(B,A),product_prod(list(B),list(A))),X),aa(list(A),product_prod(list(B),list(A)),aa(list(B),fun(list(A),product_prod(list(B),list(A))),product_Pair(list(B),list(A)),aa(list(B),list(B),cons(B,A4),As3)),Xb))) ) ) ) ) ) ).
% map_tailrec_rev.pelims
tff(fact_4420_set__remove1__eq,axiom,
! [A: $tType,Xs: list(A),X: A] :
( distinct(A,Xs)
=> ( set2(A,remove1(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ) ) ).
% set_remove1_eq
tff(fact_4421_distinct__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
<=> ( distinct(A,Xs)
& distinct(A,Ys)
& ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) ) ) ) ).
% distinct_append
tff(fact_4422_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( distinct(A,Xs)
=> ( distinct(A,Ys)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> distinct(A,Zs2) ) ) ) ) ).
% distinct_disjoint_shuffles
tff(fact_4423_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list(A),I: nat,J: nat] :
( distinct(A,Vs)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,take(A,I,Vs))),set2(A,drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).
% set_take_disj_set_drop_if_distinct
tff(fact_4424_splice_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( splice(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,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 )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( ( Y = aa(list(A),list(A),cons(A,X3),splice(A,Xa,Xs3)) )
=> ~ aa(product_prod(list(A),list(A)),$o,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),cons(A,X3),Xs3)),Xa)) ) ) ) ) ) ).
% splice.pelims
tff(fact_4425_set__update__distinct,axiom,
! [A: $tType,Xs: list(A),N: nat,X: A] :
( distinct(A,Xs)
=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
=> ( set2(A,list_update(A,Xs,N,X)) = aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),aa(set(A),set(A),insert2(A,aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).
% set_update_distinct
tff(fact_4426_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( aa(product_prod(list(A),list(A)),$o,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))
=> ( splice(A,nil(A),Ys) = Ys ) ) ).
% splice.psimps(1)
tff(fact_4427_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
( aa(product_prod(list(A),list(A)),$o,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),cons(A,X),Xs)),Ys))
=> ( splice(A,aa(list(A),list(A),cons(A,X),Xs),Ys) = aa(list(A),list(A),cons(A,X),splice(A,Ys,Xs)) ) ) ).
% splice.psimps(2)
tff(fact_4428_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I: nat,X: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I,X),list_update(B,Ys,I,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_update
tff(fact_4429_distinct__list__update,axiom,
! [A: $tType,Xs: list(A),A3: A,I: nat] :
( distinct(A,Xs)
=> ( ~ member(A,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),aa(set(A),set(A),insert2(A,aa(nat,A,nth(A,Xs),I)),bot_bot(set(A)))))
=> distinct(A,list_update(A,Xs,I,A3)) ) ) ).
% distinct_list_update
tff(fact_4430_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
<=> ? [Y3: A,N2: nat] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N2)),Y3),R2)
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
& ( Ys = list_update(A,Xs,N2,Y3) ) ) ) ).
% listrel1_iff_update
tff(fact_4431_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list(A),I4: fun(set(A),fun(B,$o)),Sigma_0: B,F: fun(B,fun(A,B))] :
( distinct(A,S)
=> ( aa(B,$o,aa(set(A),fun(B,$o),I4,set2(A,S)),Sigma_0)
=> ( ! [X3: A,It: set(A),Sigma: B] :
( member(A,X3,It)
=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),It),set2(A,S))
=> ( aa(B,$o,aa(set(A),fun(B,$o),I4,It),Sigma)
=> aa(B,$o,aa(set(A),fun(B,$o),I4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F,Sigma),X3)) ) ) )
=> aa(B,$o,aa(set(A),fun(B,$o),I4,bot_bot(set(A))),foldl(B,A,F,Sigma_0,S)) ) ) ) ).
% distinct_foldl_invar
tff(fact_4432_sum__list__update,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [K: nat,Xs: list(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
=> ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K,X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).
% sum_list_update
tff(fact_4433_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs2: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs2)) = map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_oe(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs2))) ).
% zip_left_commute
tff(fact_4434_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),cons(A,X),Xs),Ys) = case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_of(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs),Ys) ).
% zip_Cons1
tff(fact_4435_sum__list_ONil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).
% sum_list.Nil
tff(fact_4436_sum__list_OCons,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [X: A,Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),cons(A,X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ).
% sum_list.Cons
tff(fact_4437_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [Ns: list(A)] :
( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
<=> ! [X4: A] :
( member(A,X4,set2(A,Ns))
=> ( X4 = zero_zero(A) ) ) ) ) ).
% sum_list_eq_0_iff
tff(fact_4438_sum__list__append,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys)) ) ).
% sum_list_append
tff(fact_4439_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_og(B,A),Xs)) = zero_zero(A) ) ).
% sum_list_0
tff(fact_4440_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : map(product_prod(A,B),A,product_fst(A,B),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))),K),L)) = L ).
% map_fst_mk_snd
tff(fact_4441_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : map(product_prod(B,A),A,product_snd(B,A),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),L)) = L ).
% map_snd_mk_fst
tff(fact_4442_sum__list__map__remove1,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [X: A,Xs: list(A),F: fun(A,B)] :
( member(A,X,set2(A,Xs))
=> ( aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F,X)),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,remove1(A,X,Xs)))) ) ) ) ).
% sum_list_map_remove1
tff(fact_4443_sum__list__triv,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [R2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_cd(A,fun(B,A),R2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).
% sum_list_triv
tff(fact_4444_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& strict9044650504122735259up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ( Xs != nil(A) )
=> ( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ) ).
% sum_list_strict_mono
tff(fact_4445_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& ordere6658533253407199908up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X3)),aa(A,B,G,X3)) )
=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ).
% sum_list_mono
tff(fact_4446_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oh(fun(B,A),fun(fun(B,A),fun(B,A)),F),G),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,G,Xs))) ) ).
% sum_list_addf
tff(fact_4447_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),C2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_cj(fun(B,A),fun(A,fun(B,A)),F),C2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))),C2) ) ).
% sum_list_mult_const
tff(fact_4448_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ci(A,fun(fun(B,A),fun(B,A)),C2),F),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))) ) ).
% sum_list_const_mult
tff(fact_4449_sum__list__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs))) ) ).
% sum_list_abs
tff(fact_4450_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oi(fun(B,A),fun(fun(B,A),fun(B,A)),F),G),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,G,Xs))) ) ).
% sum_list_subtractf
tff(fact_4451_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,comp(A,A,B,uminus_uminus(A),F),Xs)) ) ).
% uminus_sum_list_map
tff(fact_4452_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : map(product_prod(B,C),A,F,zip(B,C,Xs,map(D,C,G,Ys))) = map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_oj(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G)),zip(B,D,Xs,Ys)) ).
% map_zip_map2
tff(fact_4453_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : map(product_prod(B,C),A,F,zip(B,C,map(D,B,G,Xs),Ys)) = map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_ok(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G)),zip(D,C,Xs,Ys)) ).
% map_zip_map
tff(fact_4454_sum_Odistinct__set__conv__list,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Xs: list(A),G: fun(A,B)] :
( distinct(A,Xs)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set2(A,Xs)) = aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs)) ) ) ) ).
% sum.distinct_set_conv_list
tff(fact_4455_sum__list__distinct__conv__sum__set,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Xs: list(A),F: fun(A,B)] :
( distinct(A,Xs)
=> ( aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),set2(A,Xs)) ) ) ) ).
% sum_list_distinct_conv_sum_set
tff(fact_4456_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,map(C,A,F,Xs),Ys) = map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_ol(fun(C,A),fun(C,fun(B,product_prod(A,B))),F)),zip(C,B,Xs,Ys)) ).
% zip_map1
tff(fact_4457_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,map(C,B,F,Ys)) = map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_om(fun(C,B),fun(A,fun(C,product_prod(A,B))),F)),zip(A,C,Xs,Ys)) ).
% zip_map2
tff(fact_4458_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_iq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G)),zip(C,D,Xs,Ys)) = zip(A,B,map(C,A,F,Xs),map(D,B,G,Ys)) ).
% map_prod_fun_zip
tff(fact_4459_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(C,A)),G: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_on(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F),G)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),Xs)) = foldl(A,C,F,A3,Xs) ).
% fst_foldl
tff(fact_4460_member__le__sum__list,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Xs: list(A)] :
( member(A,X,set2(A,Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).
% member_le_sum_list
tff(fact_4461_zip__same__conv__map,axiom,
! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = map(A,product_prod(A,A),aTP_Lamp_gz(A,product_prod(A,A)),Xs) ).
% zip_same_conv_map
tff(fact_4462_foldl__absorb1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Zs2: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),foldl(A,A,times_times(A),one_one(A),Zs2)) = foldl(A,A,times_times(A),X,Zs2) ) ).
% foldl_absorb1
tff(fact_4463_foldl__un__empty__eq,axiom,
! [A: $tType,I: set(A),Ww: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),I,Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),Ww)) ).
% foldl_un_empty_eq
tff(fact_4464_sum__list__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),zero_zero(A)) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A)) ) ) ).
% sum_list_nonpos
tff(fact_4465_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
=> ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
<=> ! [X4: A] :
( member(A,X4,set2(A,Xs))
=> ( X4 = zero_zero(A) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
tff(fact_4466_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).
% Groups_List.sum_list_nonneg
tff(fact_4467_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),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)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Ys)),product(A,B,Xs,Ys)) ).
% product.simps(2)
tff(fact_4468_zip__commute,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B)))),zip(B,A,Ys,Xs)) ).
% zip_commute
tff(fact_4469_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Xs: list(A)] :
( distinct(A,Xs)
=> ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_oo(A,A)),set2(A,Xs)) ) ) ) ).
% distinct_sum_list_conv_Sum
tff(fact_4470_foldl__set,axiom,
! [A: $tType,L: list(set(A))] : foldl(set(A),set(A),sup_sup(set(A)),bot_bot(set(A)),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_op(list(set(A)),fun(set(A),$o),L))) ).
% foldl_set
tff(fact_4471_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z2: B] :
( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2),set2(product_prod(A,B),Xs))
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
tff(fact_4472_elem__le__sum__list,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [K: nat,Ns: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns)) ) ) ).
% elem_le_sum_list
tff(fact_4473_Id__on__set,axiom,
! [A: $tType,Xs: list(A)] : id_on(A,set2(A,Xs)) = set2(product_prod(A,A),map(A,product_prod(A,A),aTP_Lamp_gz(A,product_prod(A,A)),Xs)) ).
% Id_on_set
tff(fact_4474_sum__list__sum__nth,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% sum_list_sum_nth
tff(fact_4475_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),cons(B,Y),Ys)) = case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_oq(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys),Xs) ).
% zip_Cons
tff(fact_4476_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1),set2(product_prod(A,B),Xs))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22),set2(product_prod(A,B),Xs))
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
tff(fact_4477_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : map(product_prod(B,A),A,product_snd(B,A),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_hd(A,fun(B,product_prod(B,A))),K),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_snd_mk_snd
tff(fact_4478_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : map(product_prod(A,B),A,product_fst(A,B),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_fst_mk_fst
tff(fact_4479_prod_Odistinct__set__conv__list,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Xs: list(A),G: fun(A,B)] :
( distinct(A,Xs)
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set2(A,Xs)) = aa(list(B),B,groups5270119922927024881d_list(B),map(A,B,G,Xs)) ) ) ) ).
% prod.distinct_set_conv_list
tff(fact_4480_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),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_4481_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_4482_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_4483_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] : zip(A,B,replicate(A,I,X),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).
% zip_replicate
tff(fact_4484_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( set2(A,replicate(A,N,X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ).
% set_replicate
tff(fact_4485_interv__sum__list__conv__sum__set__int,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(int,A),K: int,L: int] : aa(list(A),A,groups8242544230860333062m_list(A),map(int,A,F,upto(K,L))) = aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F),set2(int,upto(K,L))) ) ).
% interv_sum_list_conv_sum_set_int
tff(fact_4486_sum__set__upto__conv__sum__list__int,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(int,A),I: int,J: int] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F),set2(int,upto(I,J))) = aa(list(A),A,groups8242544230860333062m_list(A),map(int,A,F,upto(I,J))) ) ).
% sum_set_upto_conv_sum_list_int
tff(fact_4487_sum__list__replicate,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,C2: A] : aa(list(A),A,groups8242544230860333062m_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_4488_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( semiring_1(A)
& semiri3467727345109120633visors(A) )
=> ! [Xs: list(A)] :
( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
<=> member(A,zero_zero(A),set2(A,Xs)) ) ) ).
% prod_list_zero_iff
tff(fact_4489_sum__list__Suc,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,aTP_Lamp_or(fun(A,nat),fun(A,nat),F),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).
% sum_list_Suc
tff(fact_4490_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
set2(A,replicate(A,N,X)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ).
% set_replicate_conv_if
tff(fact_4491_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] : set2(A,replicate(A,aa(nat,nat,suc,N),X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).
% set_replicate_Suc
tff(fact_4492_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N,X),Ys) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),take(B,N,Ys)) ).
% zip_replicate1
tff(fact_4493_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list(A)] : map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))),K),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).
% map_zip1
tff(fact_4494_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list(B)] : map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).
% map_zip2
tff(fact_4495_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_he(B,fun(A,product_prod(A,B))),Y),take(A,N,Xs)) ).
% zip_replicate2
tff(fact_4496_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs2: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs2)) = map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_os(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs2)) ).
% zip_assoc
tff(fact_4497_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list(A),X5: set(A),F: fun(A,nat)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set2(A,Xs)),X5)
=> ( aa(set(A),$o,finite_finite(A),X5)
=> ( aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ot(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F)),X5) ) ) ) ).
% sum_list_map_eq_sum_count2
tff(fact_4498_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,set2(product_prod(A,C),Xys),set2(product_prod(C,B),Yzs)) = set2(product_prod(A,B),concat(product_prod(A,B),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_ov(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs),Xys))) ).
% set_relcomp
tff(fact_4499_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_ow(fun(A,nat),fun(list(A),fun(A,nat)),F),Xs)),set2(A,Xs)) ).
% sum_list_map_eq_sum_count
tff(fact_4500_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [F: fun(nat,A),Ns: list(nat)] :
( ! [X3: nat,Y2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y2)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,X3)),aa(nat,A,F,Y2)) )
=> ( sorted_wrt(nat,ord_less(nat),Ns)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(A),A,groups8242544230860333062m_list(A),map(nat,A,F,Ns))) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4501_length__concat,axiom,
! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),map(list(A),nat,size_size(list(A)),Xss)) ).
% length_concat
tff(fact_4502_product__concat__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ox(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).
% product_concat_map
tff(fact_4503_distinct__concat,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(list(A),Xs)
=> ( ! [Ys3: list(A)] :
( member(list(A),Ys3,set2(list(A),Xs))
=> distinct(A,Ys3) )
=> ( ! [Ys3: list(A),Zs: list(A)] :
( member(list(A),Ys3,set2(list(A),Xs))
=> ( member(list(A),Zs,set2(list(A),Xs))
=> ( ( Ys3 != Zs )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Ys3)),set2(A,Zs)) = bot_bot(set(A)) ) ) ) )
=> distinct(A,concat(A,Xs)) ) ) ) ).
% distinct_concat
tff(fact_4504_distinct__concat__iff,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(A,concat(A,Xs))
<=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
& ! [Ys4: list(A)] :
( member(list(A),Ys4,set2(list(A),Xs))
=> distinct(A,Ys4) )
& ! [Ys4: list(A),Zs3: list(A)] :
( ( member(list(A),Ys4,set2(list(A),Xs))
& member(list(A),Zs3,set2(list(A),Xs))
& ( Ys4 != Zs3 ) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Ys4)),set2(A,Zs3)) = bot_bot(set(A)) ) ) ) ) ).
% distinct_concat_iff
tff(fact_4505_product__code,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,set2(A,Xs),set2(B,Ys)) = set2(product_prod(A,B),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ox(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).
% product_code
tff(fact_4506_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
( sorted_wrt(A,X,Xa)
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) )
=> ~ ! [X3: A,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),Ys3) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),Ys3)))
=> ~ ( ! [Xa3: A] :
( member(A,Xa3,set2(A,Ys3))
=> aa(A,$o,aa(A,fun(A,$o),X,X3),Xa3) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
tff(fact_4507_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: $o] :
( ( sorted_wrt(A,X,Xa)
<=> (Y) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ( (Y)
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) ) )
=> ~ ! [X3: A,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),Ys3) )
=> ( ( (Y)
<=> ( ! [Xa2: A] :
( member(A,Xa2,set2(A,Ys3))
=> aa(A,$o,aa(A,fun(A,$o),X,X3),Xa2) )
& sorted_wrt(A,X,Ys3) ) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),Ys3))) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
tff(fact_4508_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
( ~ sorted_wrt(A,X,Xa)
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ~ ! [X3: A,Ys3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),Ys3) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),Ys3)))
=> ( ! [Xa4: A] :
( member(A,Xa4,set2(A,Ys3))
=> aa(A,$o,aa(A,fun(A,$o),X,X3),Xa4) )
& sorted_wrt(A,X,Ys3) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
tff(fact_4509_distinct__concat_H,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(list(A),filter2(list(A),aTP_Lamp_oy(list(A),$o),Xs))
=> ( ! [Ys3: list(A)] :
( member(list(A),Ys3,set2(list(A),Xs))
=> distinct(A,Ys3) )
=> ( ! [Ys3: list(A),Zs: list(A)] :
( member(list(A),Ys3,set2(list(A),Xs))
=> ( member(list(A),Zs,set2(list(A),Xs))
=> ( ( Ys3 != Zs )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Ys3)),set2(A,Zs)) = bot_bot(set(A)) ) ) ) )
=> distinct(A,concat(A,Xs)) ) ) ) ).
% distinct_concat'
tff(fact_4510_size__list__conv__sum__list,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : size_list(A,F,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).
% size_list_conv_sum_list
tff(fact_4511_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F),Fs)))
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
| ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,Fs)) ) ) ) ).
% in_measures(2)
tff(fact_4512_partition__rev_Opelims,axiom,
! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa),Xb)))
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ~ aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( ( Y = partition_rev(A,X,
$ite(aa(A,$o,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),cons(A,X3),No))),
Xs3) )
=> ~ aa(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),$o,accp(product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),cons(A,X3),Xs3)))) ) ) ) ) ) ) ).
% partition_rev.pelims
tff(fact_4513_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,nil(fun(A,nat)))) ).
% in_measures(1)
tff(fact_4514_partition__filter__conv,axiom,
! [A: $tType,F: fun(A,$o),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,F,Xs)),filter2(A,comp($o,$o,A,fNot,F),Xs)) ).
% partition_filter_conv
tff(fact_4515_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),X: A,Xs: list(A)] :
partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),aa(list(A),list(A),cons(A,X),Xs)) = partition_rev(A,P,
$ite(aa(A,$o,P,X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X),Yes2)),No2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),aa(list(A),list(A),cons(A,X),No2))),
Xs) ).
% partition_rev.simps(2)
tff(fact_4516_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) ).
% partition_rev.simps(1)
tff(fact_4517_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( monoid_add(A)
=> ! [F: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_oz(fun(B,A),fun(fun(B,$o),fun(B,A)),F),P),Xs)) ) ).
% sum_list_map_filter'
tff(fact_4518_sum__list__filter__le__nat,axiom,
! [A: $tType,F: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,filter2(A,P,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs))) ).
% sum_list_filter_le_nat
tff(fact_4519_sum__list__map__filter,axiom,
! [B: $tType,A: $tType] :
( monoid_add(B)
=> ! [Xs: list(A),P: fun(A,$o),F: fun(A,B)] :
( ! [X3: A] :
( member(A,X3,set2(A,Xs))
=> ( ~ aa(A,$o,P,X3)
=> ( aa(A,B,F,X3) = zero_zero(B) ) ) )
=> ( aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,filter2(A,P,Xs))) = aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs)) ) ) ) ).
% sum_list_map_filter
tff(fact_4520_set__minus__filter__out,axiom,
! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) = set2(A,filter2(A,aTP_Lamp_pa(A,fun(A,$o),Y),Xs)) ).
% set_minus_filter_out
tff(fact_4521_measures__less,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F),Fs))) ) ).
% measures_less
tff(fact_4522_measures__lesseq,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,Fs))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F),Fs))) ) ) ).
% measures_lesseq
tff(fact_4523_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_pb(list(A),fun(A,$o),Ys),Zs2) = Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
tff(fact_4524_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_pc(list(A),fun(A,$o),Ys),Zs2) = Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
tff(fact_4525_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_pb(list(A),fun(A,$o),Xs),Zs2) = Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
tff(fact_4526_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,Xs)),set2(A,Ys)) = bot_bot(set(A)) )
=> ( member(list(A),Zs2,shuffles(A,Xs,Ys))
=> ( filter2(A,aTP_Lamp_pc(list(A),fun(A,$o),Xs),Zs2) = Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
tff(fact_4527_partition__rev_Oelims,axiom,
! [A: $tType,X: fun(A,$o),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X,Xa,Xb) = Y )
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( Y != partition_rev(A,X,
$ite(aa(A,$o,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),cons(A,X3),No))),
Xs3) ) ) ) ) ) ).
% partition_rev.elims
tff(fact_4528_part__def,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),filter2(A,aa(B,fun(A,$o),aTP_Lamp_pd(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_pe(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_pf(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs))) ) ).
% part_def
tff(fact_4529_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( quicksort_by_rel(A,X,Xa,Xb) = Y )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( Y != aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_pg(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X3)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) ) ) ) ) ).
% quicksort_by_rel.elims
tff(fact_4530_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] : quicksort_by_rel(A,R,Sl2,aa(list(A),list(A),cons(A,X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_pg(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),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_4531_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A),X: A,Xs: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),cons(A,X),Xs))))
=> ( quicksort_by_rel(A,R,Sl2,aa(list(A),list(A),cons(A,X),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_pg(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R),Sl2),X)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),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_4532_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( quicksort_by_rel(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb)))
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A)))) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( Xb = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( ( Y = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_pg(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa),X3)),partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3)) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),aa(list(A),list(A),cons(A,X3),Xs3)))) ) ) ) ) ) ).
% quicksort_by_rel.pelims
tff(fact_4533_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R: fun(A,fun(A,$o)),Sl2: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),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))))
=> ( quicksort_by_rel(A,R,Sl2,nil(A)) = Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
tff(fact_4534_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)))] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22)))
=> ( ! [R11: fun(A,fun(A,$o)),Sl: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A))))
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R11),Sl),nil(A)) )
=> ( ! [R11: fun(A,fun(A,$o)),Sl: list(A),X3: A,Xs3: list(A)] :
( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R11),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),cons(A,X3),Xs3))))
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb3: list(A),Y4: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R11),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb3),Y4) = Xa3 )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R11),Sl),Y4) ) )
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb3: list(A),Y4: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),R11),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs3) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb3),Y4) = Xa3 )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R11),aa(list(A),list(A),cons(A,X3),quicksort_by_rel(A,R11,Sl,Y4))),Xb3) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,R11),Sl),aa(list(A),list(A),cons(A,X3),Xs3)) ) ) )
=> aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(A,fun(A,$o)),fun(list(A),fun(list(A),$o)),P,A0),A1),A22) ) ) ) ).
% quicksort_by_rel.pinduct
tff(fact_4535_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F,map(C,B,X,nil(C))) = bot_bot(set(A)) ).
% list_collect_set_map_simps(1)
tff(fact_4536_horner__sum__foldr,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_pi(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3),Xs,zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_4537_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( comm_semiring_0(B)
& comm_semiring_0(A) )
=> ! [Aa2: fun(A,fun(B,$o)),Ba: fun(C,fun(D,$o))] :
( aa(B,$o,aa(A,fun(B,$o),Aa2,zero_zero(A)),zero_zero(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),plus_plus(A)),plus_plus(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),times_times(A)),times_times(B))
=> aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,Ba,Aa2),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),Aa2,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,Ba),Aa2))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).
% horner_sum_transfer
tff(fact_4538_distinct__n__lists,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( distinct(A,Xs)
=> distinct(list(A),n_lists(A,N,Xs)) ) ).
% distinct_n_lists
tff(fact_4539_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,set(A))] : list_collect_set(B,A,F,nil(B)) = bot_bot(set(A)) ).
% list_collect_set_simps(1)
tff(fact_4540_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).
% sum_list.eq_foldr
tff(fact_4541_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = foldr(A,A,times_times(A),Xs,one_one(A)) ) ).
% prod_list.eq_foldr
tff(fact_4542_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).
% length_n_lists
tff(fact_4543_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& monoid_add(A) )
=> ! [Aa2: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),Aa2,zero_zero(A)),zero_zero(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),plus_plus(A)),plus_plus(B))
=> aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,Aa2),Aa2),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B)) ) ) ) ).
% sum_list_transfer
tff(fact_4544_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_mult(B)
& monoid_mult(A) )
=> ! [Aa2: fun(A,fun(B,$o))] :
( aa(B,$o,aa(A,fun(B,$o),Aa2,one_one(A)),one_one(B))
=> ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),Aa2,bNF_rel_fun(A,B,A,B,Aa2,Aa2)),times_times(A)),times_times(B))
=> aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,Aa2),Aa2),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B)) ) ) ) ).
% prod_list_transfer
tff(fact_4545_length__product__lists,axiom,
! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),map(list(A),nat,size_size(list(A)),Xss),one_one(nat)) ).
% length_product_lists
tff(fact_4546_partition__rev__filter__conv,axiom,
! [A: $tType,P: fun(A,$o),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,P,Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,comp($o,$o,A,fNot,P),Xs))),No2)) ).
% partition_rev_filter_conv
tff(fact_4547_lexordp__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp(A,Xs,Ys)
<=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A))))) ) ) ).
% lexordp_conv_lexord
tff(fact_4548_revg_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ~ ! [A4: A,As3: list(A)] :
( ( X = aa(list(A),list(A),cons(A,A4),As3) )
=> ( ( Y = revg(A,As3,aa(list(A),list(A),cons(A,A4),Xa)) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,A4),As3)),Xa)) ) ) ) ) ) ).
% revg.pelims
tff(fact_4549_sum__list__rev,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs)) = aa(list(A),A,groups8242544230860333062m_list(A),Xs) ) ).
% sum_list_rev
tff(fact_4550_prod__list_Orev,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),rev(A,Xs)) = aa(list(A),A,groups5270119922927024881d_list(A),Xs) ) ).
% prod_list.rev
tff(fact_4551_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs))) ) ).
% fold_plus_sum_list_rev
tff(fact_4552_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set2(B,Xs)) = aa(list(A),A,groups5270119922927024881d_list(A),map(B,A,G,remdups(B,Xs))) ) ).
% prod.set_conv_list
tff(fact_4553_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_4554_sum__code,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set2(B,Xs)) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,G,remdups(B,Xs))) ) ).
% sum_code
tff(fact_4555_sum__list__def,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).
% sum_list_def
tff(fact_4556_monoid__list_OF_Ocong,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] : groups_monoid_F(A,F,Z2) = groups_monoid_F(A,F,Z2) ).
% monoid_list.F.cong
tff(fact_4557_monoid__list_OCons,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,X: A,Xs: list(A)] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),aa(list(A),list(A),cons(A,X),Xs)) = aa(A,A,aa(A,fun(A,A),F,X),aa(list(A),A,groups_monoid_F(A,F,Z2),Xs)) ) ) ).
% monoid_list.Cons
tff(fact_4558_monoid__list_ONil,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),nil(A)) = Z2 ) ) ).
% monoid_list.Nil
tff(fact_4559_monoid__list_Oappend,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Xs: list(A),Ys: list(A)] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),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),F,aa(list(A),A,groups_monoid_F(A,F,Z2),Xs)),aa(list(A),A,groups_monoid_F(A,F,Z2),Ys)) ) ) ).
% monoid_list.append
tff(fact_4560_comm__monoid__list_Orev,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Xs: list(A)] :
( groups1828464146339083142d_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),rev(A,Xs)) = aa(list(A),A,groups_monoid_F(A,F,Z2),Xs) ) ) ).
% comm_monoid_list.rev
tff(fact_4561_monoid__list_Oeq__foldr,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Xs: list(A)] :
( groups_monoid_list(A,F,Z2)
=> ( aa(list(A),A,groups_monoid_F(A,F,Z2),Xs) = foldr(A,A,F,Xs,Z2) ) ) ).
% monoid_list.eq_foldr
tff(fact_4562_comm__monoid__list__set_Oset__conv__list,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A),Xs: list(B)] :
( groups4802862169904069756st_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set2(B,Xs)) = aa(list(A),A,groups_monoid_F(A,F,Z2),map(B,A,G,remdups(B,Xs))) ) ) ).
% comm_monoid_list_set.set_conv_list
tff(fact_4563_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Xs: list(B),G: fun(B,A)] :
( groups4802862169904069756st_set(A,F,Z2)
=> ( distinct(B,Xs)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set2(B,Xs)) = aa(list(A),A,groups_monoid_F(A,F,Z2),map(B,A,G,Xs)) ) ) ) ).
% comm_monoid_list_set.distinct_set_conv_list
tff(fact_4564_merge_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = Xa )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ( ! [V2: A,Va: list(A)] :
( ( X = aa(list(A),list(A),cons(A,V2),Va) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),list(A),cons(A,V2),Va) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,V2),Va)),nil(A))) ) ) )
=> ~ ! [X12: A,L12: list(A)] :
( ( X = aa(list(A),list(A),cons(A,X12),L12) )
=> ! [X23: A,L23: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X23),L23) )
=> ( ( Y = $ite(
aa(A,$o,aa(A,fun(A,$o),ord_less(A),X12),X23),
aa(list(A),list(A),cons(A,X12),merge(A,L12,aa(list(A),list(A),cons(A,X23),L23))),
$ite(X12 = X23,aa(list(A),list(A),cons(A,X12),merge(A,L12,L23)),aa(list(A),list(A),cons(A,X23),merge(A,aa(list(A),list(A),cons(A,X12),L12),L23))) ) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X12),L12)),aa(list(A),list(A),cons(A,X23),L23))) ) ) ) ) ) ) ) ) ).
% merge.pelims
tff(fact_4565_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A3: A] : enumerate(A,N,replicate(A,M2,A3)) = map(nat,product_prod(nat,A),aTP_Lamp_pj(A,fun(nat,product_prod(nat,A)),A3),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ).
% enumerate_replicate_eq
tff(fact_4566_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
=> ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M2,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_pk(nat,nat)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).
% sum_list_upt
tff(fact_4567_comm__monoid__mult__class_Oprod__def,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ( groups7121269368397514597t_prod(A,B) = groups_comm_monoid_F(B,A,times_times(B),one_one(B)) ) ) ).
% comm_monoid_mult_class.prod_def
tff(fact_4568_interv__sum__list__conv__sum__set__nat,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),M2: nat,N: nat] : aa(list(A),A,groups8242544230860333062m_list(A),map(nat,A,F,upt(M2,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set2(nat,upt(M2,N))) ) ).
% interv_sum_list_conv_sum_set_nat
tff(fact_4569_sum__set__upt__conv__sum__list__nat,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set2(nat,upt(M2,N))) = aa(list(A),A,groups8242544230860333062m_list(A),map(nat,A,F,upt(M2,N))) ) ).
% sum_set_upt_conv_sum_list_nat
tff(fact_4570_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: fun(nat,A),M2: nat] : enumerate(A,N,map(nat,A,F,upt(N,M2))) = map(nat,product_prod(nat,A),aTP_Lamp_pl(fun(nat,A),fun(nat,product_prod(nat,A)),F),upt(N,M2)) ).
% enumerate_map_upt
tff(fact_4571_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Ca: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ! [X3: set(B)] :
( member(set(B),X3,Ca)
=> aa(set(B),$o,finite_finite(B),X3) )
=> ( ! [X3: set(B)] :
( member(set(B),X3,Ca)
=> ! [Xa4: set(B)] :
( member(set(B),Xa4,Ca)
=> ( ( X3 != Xa4 )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X3),Xa4) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),Ca)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups_comm_monoid_F(A,set(B),F,Z2),groups_comm_monoid_F(A,B,F,Z2)),G),Ca) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
tff(fact_4572_comm__monoid__set_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,I4: set(B),Aa2: fun(B,set(C)),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),$o,finite_finite(B),I4)
=> ( ! [X3: B] :
( member(B,X3,I4)
=> aa(set(C),$o,finite_finite(C),aa(B,set(C),Aa2,X3)) )
=> ( ! [X3: B] :
( member(B,X3,I4)
=> ! [Xa4: B] :
( member(B,Xa4,I4)
=> ( ( X3 != Xa4 )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),Aa2,X3)),aa(B,set(C),Aa2,Xa4)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Aa2),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_pm(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F),Z2),Aa2),G)),I4) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
tff(fact_4573_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,S: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),$o,finite_finite(B),S)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_pn(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S) = $ite(member(B,A3,S),aa(A,A,aa(A,fun(A,A),F,aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))) ) ) ) ).
% comm_monoid_set.delta_remove
tff(fact_4574_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(B),Ba: set(B),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( aa(set(B),$o,finite_finite(B),Ba)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Aa2),Ba) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),Aa2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),Ba)) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
tff(fact_4575_comm__monoid__list__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups4802862169904069756st_set(A,F,Z2)
=> groups778175481326437816id_set(A,F,Z2) ) ).
% comm_monoid_list_set.axioms(2)
tff(fact_4576_comm__monoid__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> groups778175481326437816id_set(A,F,Z2) ) ).
% comm_monoid_set.intro
tff(fact_4577_comm__monoid__set_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups778175481326437816id_set(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% comm_monoid_set.axioms
tff(fact_4578_comm__monoid__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups778175481326437816id_set(A,F,Z2)
<=> comm_monoid(A,F,Z2) ) ).
% comm_monoid_set_def
tff(fact_4579_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_4580_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty
tff(fact_4581_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z2: A,P5: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z2),P5),bot_bot(set(B))) = Z2 ) ) ).
% comm_monoid_set.empty'
tff(fact_4582_comm__monoid__list__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups1828464146339083142d_list(A,F,Z2)
=> ( groups778175481326437816id_set(A,F,Z2)
=> groups4802862169904069756st_set(A,F,Z2) ) ) ).
% comm_monoid_list_set.intro
tff(fact_4583_comm__monoid__list__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( groups4802862169904069756st_set(A,F,Z2)
<=> ( groups1828464146339083142d_list(A,F,Z2)
& groups778175481326437816id_set(A,F,Z2) ) ) ).
% comm_monoid_list_set_def
tff(fact_4584_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(B),G: fun(B,A),X: B] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),insert2(B,X),Aa2)) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))) ) ) ) ).
% comm_monoid_set.insert_remove
tff(fact_4585_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(B),X: B,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( member(B,X,Aa2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),Aa2) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))) ) ) ) ) ).
% comm_monoid_set.remove
tff(fact_4586_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X,Xa) = Y )
=> ( aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X),Xa))
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),nil(A))) ) ) )
=> ( ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),cons(A,X3),nil(A)) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X3),Xs12)),Xs22) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),cons(A,X3),nil(A)))) ) ) )
=> ~ ! [Xs12: list(A),Xs22: list(A)] :
( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22) )
=> ! [X12: A,X23: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X12),aa(list(A),list(A),cons(A,X23),Xs3)) )
=> ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,X12),Xs12)),aa(list(A),list(A),cons(A,X23),Xs22)),Xs3) )
=> ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs22)),aa(list(A),list(A),cons(A,X12),aa(list(A),list(A),cons(A,X23),Xs3)))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
tff(fact_4587_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering(A,Less_eq,Less)
=> ( ordering_top_axioms(A,Less_eq,Top)
=> ordering_top(A,Less_eq,Less,Top) ) ) ).
% ordering_top.intro
tff(fact_4588_ordering__top__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
<=> ( ordering(A,Less_eq,Less)
& ordering_top_axioms(A,Less_eq,Top) ) ) ).
% ordering_top_def
tff(fact_4589_ordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
=> ordering(A,Less_eq,Less) ) ).
% ordering_dualI
tff(fact_4590_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 = B2 )
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
& aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3) ) ) ) ).
% ordering.eq_iff
tff(fact_4591_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3)
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
tff(fact_4592_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
tff(fact_4593_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
tff(fact_4594_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
tff(fact_4595_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 != B2 )
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A3),B2) ) ) ) ).
% ordering.not_eq_order_implies_strict
tff(fact_4596_ordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
| ( A4 = B3 ) ) )
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B3),A4) )
=> ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4)
=> ( ! [A4: A,B3: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B3),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),C3) ) )
=> ordering(A,Less_eq,Less) ) ) ) ) ).
% ordering_strictI
tff(fact_4597_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering(A,Less_eq,Less) ) ).
% ordering_top.axioms(1)
tff(fact_4598_order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.ordering_axioms
tff(fact_4599_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,aTP_Lamp_po(A,fun(A,$o)),aTP_Lamp_pp(A,fun(A,$o))) ) ).
% dual_order.ordering_axioms
tff(fact_4600_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),X: B,Aa2: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert2(B,X),Aa2)),S)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),insert2(B,X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F,Aa2)) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_4601_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y: option(B),D4: set(A)] :
restrict_map(A,B,fun_upd(A,option(B),M2,X,Y),D4) = $ite(member(A,X,D4),fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),X,Y),restrict_map(A,B,M2,D4)) ).
% restrict_fun_upd
tff(fact_4602_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set(A),M2: fun(A,option(B)),Y: option(B)] :
( member(A,X,D4)
=> ( fun_upd(A,option(B),restrict_map(A,B,M2,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),X,Y) ) ) ).
% fun_upd_restrict_conv
tff(fact_4603_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F,bot_bot(set(B))) = nil(B) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_4604_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),Aa2: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Aa2),S)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( ( sorted8670434370408473282of_set(A,B,Less_eq,F,Aa2) = nil(B) )
<=> ( Aa2 = bot_bot(set(B)) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4605_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),D4: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M2,D4),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),X,Y) ).
% fun_upd_restrict
tff(fact_4606_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F: fun(B,A),X: B,Aa2: set(B)] :
( folding_insort_key(A,B,Less_eq,Less,S,F)
=> ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert2(B,X),Aa2)),S)
=> ( aa(set(B),$o,finite_finite(B),Aa2)
=> ( sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),insert2(B,X),Aa2)) = insort_key(A,B,Less_eq,F,X,sorted8670434370408473282of_set(A,B,Less_eq,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Aa2),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_4607_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),D4: set(A),X: A] :
fun_upd(A,option(B),restrict_map(A,B,M2,D4),X,none(B)) = $ite(member(A,X,D4),restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))),restrict_map(A,B,M2,D4)) ).
% fun_upd_None_restrict
tff(fact_4608_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M2,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) ).
% restrict_upd_same
tff(fact_4609_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
( ! [X4: A,Y3: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
<=> ( V = none(product_prod(A,B)) ) ) ).
% not_Some_eq2
tff(fact_4610_restrict__map__to__empty,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),X2: A] : aa(A,option(B),restrict_map(A,B,M2,bot_bot(set(A))),X2) = none(B) ).
% restrict_map_to_empty
tff(fact_4611_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A] : fun_upd(A,option(B),M2,X,none(B)) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) ).
% map_upd_eq_restrict
tff(fact_4612_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = fun_upd(A,option(B),F,X,none(B)) ).
% restrict_complement_singleton_eq
tff(fact_4613_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_pq(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).
% rel_of_empty
tff(fact_4614_sup__Some,axiom,
! [A: $tType] :
( sup(A)
=> ! [X: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).
% sup_Some
tff(fact_4615_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Aa2: set(A),F: fun(A,B)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),Aa2))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_pr(fun(A,B),fun(A,option(B)),F)),Aa2)) ) ) ) ).
% Some_SUP
tff(fact_4616_singleton__None__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).
% singleton_None_Sup
tff(fact_4617_empty__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).
% empty_Sup
tff(fact_4618_bot__option__def,axiom,
! [A: $tType] :
( order(A)
=> ( bot_bot(option(A)) = none(A) ) ) ).
% bot_option_def
tff(fact_4619_top__option__def,axiom,
! [A: $tType] :
( order_top(A)
=> ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).
% top_option_def
tff(fact_4620_Some__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(A)] :
( ( Aa2 != bot_bot(set(A)) )
=> ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),Aa2)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),Aa2)) ) ) ) ).
% Some_Sup
tff(fact_4621_rel__of__def,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M2,P) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_ps(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M2),P))) ).
% rel_of_def
tff(fact_4622_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),K: A,V: B] : map_to_set(A,B,fun_upd(A,option(B),M2,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M2)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_pt(A,fun(product_prod(A,B),$o),K)))) ).
% map_to_set_upd
tff(fact_4623_extract__Some__iff,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs2: 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),Zs2))) )
<=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs2)) )
& aa(A,$o,P,Y)
& ~ ? [X4: A] :
( member(A,X4,set2(A,Ys))
& aa(A,$o,P,X4) ) ) ) ).
% extract_Some_iff
tff(fact_4624_extract__SomeE,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs2: 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),Zs2))) )
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs2)) )
& aa(A,$o,P,Y)
& ~ ? [X2: A] :
( member(A,X2,set2(A,Ys))
& aa(A,$o,P,X2) ) ) ) ).
% extract_SomeE
tff(fact_4625_map__to__set__empty,axiom,
! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_pq(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% map_to_set_empty
tff(fact_4626_map__to__set__empty__iff_I2_J,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B))] :
( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M2) )
<=> ! [X4: A] : aa(A,option(B),M2,X4) = none(B) ) ).
% map_to_set_empty_iff(2)
tff(fact_4627_map__to__set__empty__iff_I1_J,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B))] :
( ( map_to_set(A,B,M2) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] : aa(A,option(B),M2,X4) = none(B) ) ).
% map_to_set_empty_iff(1)
tff(fact_4628_Sup__option__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Aa2: set(option(A))] :
aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),Aa2) = $ite(
( ( Aa2 = bot_bot(set(option(A))) )
| ( Aa2 = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ),
none(A),
aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,Aa2))) ) ) ).
% Sup_option_def
tff(fact_4629_extract__Cons__code,axiom,
! [A: $tType,P: fun(A,$o),X: A,Xs: list(A)] :
extract(A,P,aa(list(A),list(A),cons(A,X),Xs)) = $ite(aa(A,$o,P,X),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))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_pv(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_4630_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Xs)))
=> ( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))),X,none(B)) ) ) ) ).
% map_of_distinct_upd4
tff(fact_4631_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),Xys))
=> ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_is_SomeI
tff(fact_4632_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
=> ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),Xys)) ) ) ).
% Some_eq_map_of_iff
tff(fact_4633_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
=> ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),set2(product_prod(A,B),Xys)) ) ) ).
% map_of_eq_Some_iff
tff(fact_4634_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list(product_prod(A,B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X),set2(product_prod(A,B),L))
=> ? [X3: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X3) ) ).
% weak_map_of_SomeI
tff(fact_4635_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y),set2(product_prod(B,A),Xs)) ) ).
% map_of_SomeD
tff(fact_4636_map__of__Cons__code_I2_J,axiom,
! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K: B] :
aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),cons(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V)),Ps)),K) = $ite(L = K,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K)) ).
% map_of_Cons_code(2)
tff(fact_4637_sup__option__def,axiom,
! [A: $tType] :
( sup(A)
=> ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_px(option(A),fun(option(A),fun(A,option(A))),X),Y),X) ) ).
% sup_option_def
tff(fact_4638_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ? [Ys3: list(product_prod(B,A)),Zs: list(product_prod(B,A))] :
( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys3),aa(list(product_prod(B,A)),list(product_prod(B,A)),cons(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),Zs)) )
& ( aa(B,option(A),map_of(B,A,Ys3),K) = none(A) ) ) ) ).
% map_of_Some_split
tff(fact_4639_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ks: list(A)] : map_of(A,B,map(A,product_prod(A,B),aTP_Lamp_kl(fun(A,B),fun(A,product_prod(A,B)),F),Ks)) = restrict_map(A,B,comp(B,option(B),A,some(B),F),set2(A,Ks)) ).
% map_of_map_restrict
tff(fact_4640_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A,P: fun(product_prod(B,A),$o)] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ( ~ aa(product_prod(B,A),$o,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V))
=> ( distinct(B,map(product_prod(B,A),B,product_fst(B,A),Xs))
=> ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P,Xs)),K) = none(A) ) ) ) ) ).
% map_of_Some_filter_not_in
tff(fact_4641_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),T2: list(product_prod(A,C)),K: A,X: C] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,option(C),map_of(A,C,T2),K) = aa(C,option(C),some(C),X) )
=> ( aa(B,option(C),map_of(B,C,map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_py(fun(A,B),fun(A,fun(C,product_prod(B,C))),F)),T2)),aa(A,B,F,K)) = aa(C,option(C),some(C),X) ) ) ) ).
% map_of_mapk_SomeI
tff(fact_4642_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Xs)))
=> ( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Ys)))
=> ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))),X) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_distinct_lookup
tff(fact_4643_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y7: B] :
( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Xs)))
=> ( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),Ys))),X,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd3
tff(fact_4644_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Xs)))
=> ( ~ member(A,X,set2(A,map(product_prod(A,B),A,product_fst(A,B),Ys)))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)),X,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd2
tff(fact_4645_these__empty,axiom,
! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).
% these_empty
tff(fact_4646_these__not__empty__eq,axiom,
! [A: $tType,Ba: set(option(A))] :
( ( these(A,Ba) != bot_bot(set(A)) )
<=> ( ( Ba != bot_bot(set(option(A))) )
& ( Ba != aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_not_empty_eq
tff(fact_4647_these__empty__eq,axiom,
! [A: $tType,Ba: set(option(A))] :
( ( these(A,Ba) = bot_bot(set(A)) )
<=> ( ( Ba = bot_bot(set(option(A))) )
| ( Ba = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_empty_eq
tff(fact_4648_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M2,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M2,K,none(B)))) ).
% graph_map_upd
tff(fact_4649_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list(A),Xa: list(A),Y: option($o)] :
( ( subset_eq_mset_impl(A,X,Xa) = Y )
=> ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa))
=> ( ( ( X = nil(A) )
=> ( ( Y = aa($o,option($o),some($o),Xa != nil(A)) )
=> ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa)) ) )
=> ~ ! [X3: A,Xs3: list(A)] :
( ( X = aa(list(A),list(A),cons(A,X3),Xs3) )
=> ( ( Y = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),aa(fun(list(A),fun(product_prod(A,list(A)),option($o))),fun(product_prod(list(A),product_prod(A,list(A))),option($o)),product_case_prod(list(A),product_prod(A,list(A)),option($o)),aTP_Lamp_qa(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs3)),extract(A,aa(A,fun(A,$o),fequal(A),X3),Xa)) )
=> ~ aa(product_prod(list(A),list(A)),$o,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),cons(A,X3),Xs3)),Xa)) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
tff(fact_4650_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B)),K: A,V: B] :
( inj_on(product_prod(A,B),A,product_fst(A,B),S)
=> ( ( aa(A,option(B),set_to_map(A,B,S),K) = aa(B,option(B),some(B),V) )
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),S) ) ) ).
% set_to_map_simp
tff(fact_4651_set__to__map__empty,axiom,
! [A: $tType,B: $tType,X2: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X2) = none(B) ).
% set_to_map_empty
tff(fact_4652_graph__empty,axiom,
! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_pq(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% graph_empty
tff(fact_4653_in__graphI,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A)),K: B,V: A] :
( ( aa(B,option(A),M2,K) = aa(A,option(A),some(A),V) )
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V),graph(B,A,M2)) ) ).
% in_graphI
tff(fact_4654_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: fun(A,option(B))] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,M2))
=> ( aa(A,option(B),M2,K) = aa(B,option(B),some(B),V) ) ) ).
% in_graphD
tff(fact_4655_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: fun(A,option(B)),Aa2: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M2,Aa2)))
=> member(A,K,Aa2) ) ).
% graph_restrictD(1)
tff(fact_4656_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S),X4) = none(B)
<=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(1)
tff(fact_4657_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: fun(A,option(B)),Aa2: set(A)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M2,Aa2)))
=> ( aa(A,option(B),M2,K) = aa(B,option(B),some(B),V) ) ) ).
% graph_restrictD(2)
tff(fact_4658_graph__def,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : graph(A,B,M2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_qb(fun(A,option(B)),fun(product_prod(A,B),$o),M2)) ).
% graph_def
tff(fact_4659_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
( ( aTP_Lamp_pq(A,option(B)) = set_to_map(A,B,S) )
<=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(2)
tff(fact_4660_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Aa2: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),Aa2) = the2(A,finite_fold(A,option(A),aTP_Lamp_qc(A,fun(option(A),option(A))),none(A),Aa2)) ) ).
% Sup_fin.eq_fold'
tff(fact_4661_the__dflt__None__nonempty,axiom,
! [A: $tType,S: set(A)] :
( ( S != bot_bot(set(A)) )
=> ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ).
% the_dflt_None_nonempty
tff(fact_4662_the__dflt__None__empty,axiom,
! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).
% the_dflt_None_empty
tff(fact_4663_dflt__None__set__def,axiom,
! [A: $tType,S: set(A)] :
dflt_None_set(A,S) = $ite(S = bot_bot(set(A)),none(set(A)),aa(set(A),option(set(A)),some(set(A)),S)) ).
% dflt_None_set_def
tff(fact_4664_the__dflt__None__set,axiom,
! [A: $tType,X: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X)) = X ).
% the_dflt_None_set
tff(fact_4665_set__to__map__def,axiom,
! [A: $tType,B: $tType,S: set(product_prod(B,A)),K: B] : aa(B,option(A),set_to_map(B,A,S),K) = eps_Opt(A,aa(B,fun(A,$o),aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),S),K)) ).
% set_to_map_def
tff(fact_4666_dom__fun__upd,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),X: A,Y: option(B)] :
dom(A,B,fun_upd(A,option(B),F,X,Y)) = $ite(Y = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))),aa(set(A),set(A),insert2(A,X),dom(A,B,F))) ).
% dom_fun_upd
tff(fact_4667_dom__eq__empty__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B))] :
( ( dom(A,B,F) = bot_bot(set(A)) )
<=> ! [X4: A] : aa(A,option(B),F,X4) = none(B) ) ).
% dom_eq_empty_conv
tff(fact_4668_dom__empty,axiom,
! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_pq(A,option(B))) = bot_bot(set(A)) ).
% dom_empty
tff(fact_4669_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] : graph(A,B,M2) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_qd(fun(A,option(B)),fun(A,product_prod(A,B)),M2)),dom(A,B,M2)) ).
% graph_eq_to_snd_dom
tff(fact_4670_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] :
( ( dom(A,B,F) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
<=> ? [V3: B] : F = fun_upd(A,option(B),aTP_Lamp_pq(A,option(B)),X,aa(B,option(B),some(B),V3)) ) ).
% dom_eq_singleton_conv
tff(fact_4671_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list(A),M2: fun(A,option(B))] :
( ( set2(A,Xs) = dom(A,B,M2) )
=> ( map_of(A,B,map(A,product_prod(A,B),aTP_Lamp_qd(fun(A,option(B)),fun(A,product_prod(A,B)),M2),Xs)) = M2 ) ) ).
% map_of_map_keys
tff(fact_4672_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: fun(B,option(A)),X: B,Y: A,Z2: A] :
( ( aa(B,option(A),M2,X) = aa(A,option(A),some(A),Y) )
=> ( inj_on(B,option(A),M2,dom(B,A,M2))
=> ( ~ member(A,Z2,ran(B,A,M2))
=> ( ran(B,A,fun_upd(B,option(A),M2,X,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M2)),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),aa(set(A),set(A),insert2(A,Z2),bot_bot(set(A)))) ) ) ) ) ).
% ran_map_upd_Some
tff(fact_4673_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_4674_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_om(fun(C,B),fun(A,fun(C,product_prod(A,B))),F)),Xs)) = comp(option(C),option(B),A,map_option(C,B,F),map_of(A,C,Xs)) ).
% map_of_map
tff(fact_4675_ran__empty,axiom,
! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_qe(B,option(A))) = bot_bot(set(A)) ).
% ran_empty
tff(fact_4676_ran__add,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F)),dom(A,B,G)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,F,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F)),ran(A,B,G)) ) ) ).
% ran_add
tff(fact_4677_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_4678_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_4679_map__add__left__comm,axiom,
! [B: $tType,A: $tType,Aa2: fun(A,option(B)),Ba: fun(A,option(B)),Ca: fun(A,option(B))] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,Aa2)),dom(A,B,Ba)) = bot_bot(set(A)) )
=> ( map_add(A,B,Aa2,map_add(A,B,Ba,Ca)) = map_add(A,B,Ba,map_add(A,B,Aa2,Ca)) ) ) ).
% map_add_left_comm
tff(fact_4680_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M2: fun(A,option(B)),M6: fun(A,option(B)),N: fun(A,option(B)),N5: fun(A,option(B))] :
( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M2),M6)
=> ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),N),N5)
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M6)),dom(A,B,N5)) = bot_bot(set(A)) )
=> aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M2,N)),map_add(A,B,M6,N5)) ) ) ) ) ).
% map_add_distinct_le
tff(fact_4681_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] : set_option(A,aa(A,option(A),some(A),X22)) = aa(set(A),set(A),insert2(A,X22),bot_bot(set(A))) ).
% option.simps(15)
tff(fact_4682_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B),Y: $o] :
( ( list_all_zip(A,B,X,Xa,Xb)
<=> (Y) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
=> ( ! [A4: A,As3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,A4),As3) )
=> ! [B3: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),cons(B,B3),Bs2) )
=> ( ( (Y)
<=> ( aa(B,$o,aa(A,fun(B,$o),X,A4),B3)
& list_all_zip(A,B,X,As3,Bs2) ) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As3)),aa(list(B),list(B),cons(B,B3),Bs2)))) ) ) )
=> ( ! [V2: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,V2),Va) )
=> ( ( Xb = nil(B) )
=> ( ~ (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,V2),Va)),nil(B)))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),cons(B,V2),Va) )
=> ( ~ (Y)
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),cons(B,V2),Va)))) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
tff(fact_4683_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X,Xa,Xb)
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) )
=> ~ ! [A4: A,As3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,A4),As3) )
=> ! [B3: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),cons(B,B3),Bs2) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As3)),aa(list(B),list(B),cons(B,B3),Bs2))))
=> ~ ( aa(B,$o,aa(A,fun(B,$o),X,A4),B3)
& list_all_zip(A,B,X,As3,Bs2) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
tff(fact_4684_set__empty__eq,axiom,
! [A: $tType,Xo: option(A)] :
( ( set_option(A,Xo) = bot_bot(set(A)) )
<=> ( Xo = none(A) ) ) ).
% set_empty_eq
tff(fact_4685_option_Osimps_I14_J,axiom,
! [A: $tType] : set_option(A,none(A)) = bot_bot(set(A)) ).
% option.simps(14)
tff(fact_4686_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X: fun(A,fun(B,$o)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X,Xa,Xb)
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb)))
=> ( ! [A4: A,As3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,A4),As3) )
=> ! [B3: B,Bs2: list(B)] :
( ( Xb = aa(list(B),list(B),cons(B,B3),Bs2) )
=> ( aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As3)),aa(list(B),list(B),cons(B,B3),Bs2))))
=> ( aa(B,$o,aa(A,fun(B,$o),X,A4),B3)
& list_all_zip(A,B,X,As3,Bs2) ) ) ) )
=> ( ! [V2: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,V2),Va) )
=> ( ( Xb = nil(B) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,V2),Va)),nil(B)))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V2: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),cons(B,V2),Va) )
=> ~ aa(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),cons(B,V2),Va)))) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
tff(fact_4687_nths__empty,axiom,
! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).
% nths_empty
tff(fact_4688_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
=> ordering_axioms(A,Less_eq,Less) ) ).
% ordering.axioms(2)
tff(fact_4689_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
! [A: $tType,H2: multiset(A)] : bot_bot(set(multiset(A))) != set_atMost(multiset(A),subseteq_mset(A),H2) ).
% subset_mset.not_empty_eq_Iic_eq_empty
tff(fact_4690_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
& ( A4 != B3 ) ) )
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A4)
=> ( A4 = B3 ) ) )
=> ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_axioms.intro
tff(fact_4691_ordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering_axioms(A,Less_eq,Less)
<=> ( ! [A5: A,B4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B4)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B4)
& ( A5 != B4 ) ) )
& ! [A5: A,B4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B4)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B4),A5)
=> ( A5 = B4 ) ) ) ) ) ).
% ordering_axioms_def
tff(fact_4692_ordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_def
tff(fact_4693_ordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
=> ( ordering_axioms(A,Less_eq,Less)
=> ordering(A,Less_eq,Less) ) ) ).
% ordering.intro
tff(fact_4694_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,aTP_Lamp_qf(A,fun(A,$o)),aTP_Lamp_qg(A,fun(A,$o))) ) ).
% dual_order.preordering_axioms
tff(fact_4695_partial__preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
<=> ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),A5)
& ! [A5: A,B4: A,C4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B4)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B4),C4)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),C4) ) ) ) ) ).
% partial_preordering_def
tff(fact_4696_preordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
| ( A4 = B3 ) ) )
=> ( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B3),A4) )
=> ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4)
=> ( ! [A4: A,B3: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B3),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less,A4),C3) ) )
=> preordering(A,Less_eq,Less) ) ) ) ) ).
% preordering_strictI
tff(fact_4697_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2) ) ) ).
% preordering.strict_implies_order
tff(fact_4698_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),A3) ) ) ) ).
% preordering.strict_iff_not
tff(fact_4699_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).
% preordering.strict_trans2
tff(fact_4700_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).
% preordering.strict_trans1
tff(fact_4701_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( partial_preordering(A,Less_eq)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),C2) ) ) ) ).
% partial_preordering.trans
tff(fact_4702_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o))] :
( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),A4)
=> ( ! [A4: A,B3: A,C3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),C3)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),C3) ) )
=> partial_preordering(A,Less_eq) ) ) ).
% partial_preordering.intro
tff(fact_4703_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,B2),C2)
=> aa(A,$o,aa(A,fun(A,$o),Less,A3),C2) ) ) ) ).
% preordering.strict_trans
tff(fact_4704_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A] :
( partial_preordering(A,Less_eq)
=> aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),A3) ) ).
% partial_preordering.refl
tff(fact_4705_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A] :
( preordering(A,Less_eq,Less)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,A3),A3) ) ).
% preordering.irrefl
tff(fact_4706_preordering_Oasym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( aa(A,$o,aa(A,fun(A,$o),Less,A3),B2)
=> ~ aa(A,$o,aa(A,fun(A,$o),Less,B2),A3) ) ) ).
% preordering.asym
tff(fact_4707_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% preordering.axioms(1)
tff(fact_4708_preordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less_eq),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Less))
=> preordering(A,Less_eq,Less) ) ).
% preordering_dualI
tff(fact_4709_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,ord_less_eq(A)) ) ).
% order.partial_preordering_axioms
tff(fact_4710_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( ordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% ordering.axioms(1)
tff(fact_4711_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,aTP_Lamp_qf(A,fun(A,$o))) ) ).
% dual_order.partial_preordering_axioms
tff(fact_4712_order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.preordering_axioms
tff(fact_4713_preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& preordering_axioms(A,Less_eq,Less) ) ) ).
% preordering_def
tff(fact_4714_preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( partial_preordering(A,Less_eq)
=> ( preordering_axioms(A,Less_eq,Less)
=> preordering(A,Less_eq,Less) ) ) ).
% preordering.intro
tff(fact_4715_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering(A,Less_eq,Less)
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering.axioms(2)
tff(fact_4716_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,$o)),Less_eq: fun(A,fun(A,$o))] :
( ! [A4: A,B3: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A4),B3)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A4),B3)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B3),A4) ) )
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering_axioms.intro
tff(fact_4717_preordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
( preordering_axioms(A,Less_eq,Less)
<=> ! [A5: A,B4: A] :
( aa(A,$o,aa(A,fun(A,$o),Less,A5),B4)
<=> ( aa(A,$o,aa(A,fun(A,$o),Less_eq,A5),B4)
& ~ aa(A,$o,aa(A,fun(A,$o),Less_eq,B4),A5) ) ) ) ).
% preordering_axioms_def
tff(fact_4718_curr__def,axiom,
! [A: $tType,C: $tType,B: $tType,Aa2: set(A),F: fun(product_prod(A,B),C),X2: A] :
bNF_Wellorder_curr(A,B,C,Aa2,F,X2) = $ite(member(A,X2,Aa2),aa(A,fun(B,C),aTP_Lamp_dk(fun(product_prod(A,B),C),fun(A,fun(B,C)),F),X2),undefined(fun(B,C))) ).
% curr_def
tff(fact_4719_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] : override_on(A,B,F,G,bot_bot(set(A))) = F ).
% override_on_emptyset
tff(fact_4720_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_el(int,fun(int,product_prod(int,int)))) ).
% Fract_def
tff(fact_4721_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
! [A: $tType,L: multiset(A)] : bot_bot(set(multiset(A))) != set_atLeast(multiset(A),subseteq_mset(A),L) ).
% subset_mset.not_empty_eq_Ici_eq_empty
tff(fact_4722_relH__def,axiom,
! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( relH(As,H2,H5)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
& ! [T6: typerep,X4: nat] :
( member(nat,X4,As)
=> ( ( refs(product_unit,H2,T6,X4) = refs(product_unit,H5,T6,X4) )
& ( arrays(product_unit,H2,T6,X4) = arrays(product_unit,H5,T6,X4) ) ) ) ) ) ).
% relH_def
tff(fact_4723_extract__def,axiom,
! [A: $tType,P: fun(A,$o),Xs: list(A)] : extract(A,P,Xs) = 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_qh(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs),dropWhile(A,comp($o,$o,A,fNot,P),Xs)) ).
% extract_def
tff(fact_4724_in__range_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
<=> (Y) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( ( (Y)
<=> ! [X4: nat] :
( member(nat,X4,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),lim(product_unit,H3)) ) )
=> ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)) ) ) ) ) ).
% in_range.pelims(1)
tff(fact_4725_in__range_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> ~ ! [X2: nat] :
( member(nat,X2,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),lim(product_unit,H3)) ) ) ) ) ) ).
% in_range.pelims(2)
tff(fact_4726_in__range_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> ! [X3: nat] :
( member(nat,X3,As3)
=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),lim(product_unit,H3)) ) ) ) ) ) ).
% in_range.pelims(3)
tff(fact_4727_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
<=> (Y) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( ( (Y)
<=> ( As3 = bot_bot(set(nat)) ) )
=> ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3)) ) ) ) ) ).
% one_assn_raw.pelims(1)
tff(fact_4728_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: 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),As3) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> ( As3 != bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
tff(fact_4729_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod(heap_ext(product_unit),set(nat))] :
( ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,one_assn_raw,X)
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X)
=> ~ ! [H3: heap_ext(product_unit),As3: set(nat)] :
( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3) )
=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As3))
=> ( As3 = bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
tff(fact_4730_relcomp__def,axiom,
! [A: $tType,C: $tType,B: $tType,X2: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X2,Xa3) = aa(fun(product_prod(A,C),$o),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,$o)),fun(product_prod(A,C),$o),product_case_prod(A,C,$o),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),X2),aTP_Lamp_qi(set(product_prod(B,C)),fun(B,fun(C,$o)),Xa3)))) ).
% relcomp_def
tff(fact_4731_times__int__def,axiom,
times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int_def
tff(fact_4732_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R2: fun(A,fun(A,$o)),X: A,Y: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,R2),X),Y)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),transitive_rtrancl(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).
% Enum.rtranclp_rtrancl_eq
tff(fact_4733_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R: fun(A,fun(C,$o))] : relcompp(A,C,B,R,bot_bot(fun(C,fun(B,$o)))) = bot_bot(fun(A,fun(B,$o))) ).
% relcompp_bot2
tff(fact_4734_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R: fun(C,fun(B,$o))] : relcompp(A,C,B,bot_bot(fun(A,fun(C,$o))),R) = bot_bot(fun(A,fun(B,$o))) ).
% relcompp_bot1
tff(fact_4735_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2))
=> ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
=> aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).
% rtranclp_induct2
tff(fact_4736_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Xa: A,Xb: B,Za: A,Zb: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
=> ~ ! [A4: A,B3: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
=> ~ aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)) ) ) ) ).
% converse_rtranclpE2
tff(fact_4737_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(A,B),fun(product_prod(A,B),$o)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,$o))] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
=> ( ! [A4: A,B3: B,Aa3: A,Ba2: B] :
( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),R2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2))
=> ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),transitive_rtranclp(product_prod(A,B),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa3),Ba2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))
=> ( aa(B,$o,aa(A,fun(B,$o),P,Aa3),Ba2)
=> aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) )
=> aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).
% converse_rtranclp_induct2
tff(fact_4738_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X2: A,Xa3: A] :
( aa(A,$o,aa(A,fun(A,$o),transitive_rtranclp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X2),Xa3)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa3),transitive_rtrancl(A,R2)) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_4739_relcompp__relcomp__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B)),X2: A,Xa3: B] :
( aa(B,$o,aa(A,fun(B,$o),relcompp(A,C,B,aTP_Lamp_qj(set(product_prod(A,C)),fun(A,fun(C,$o)),R2),aTP_Lamp_qk(set(product_prod(C,B)),fun(C,fun(B,$o)),S2)),X2),Xa3)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3),relcomp(A,C,B,R2,S2)) ) ).
% relcompp_relcomp_eq
tff(fact_4740_plus__int__def,axiom,
plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% plus_int_def
tff(fact_4741_minus__int__def,axiom,
minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% minus_int_def
tff(fact_4742_uminus__int__def,axiom,
uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat))))) ).
% uminus_int_def
tff(fact_4743_rtrancl__def,axiom,
! [A: $tType,X2: set(product_prod(A,A))] : transitive_rtrancl(A,X2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),transitive_rtranclp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),X2)))) ).
% rtrancl_def
tff(fact_4744_next_Osimps,axiom,
! [V: code_natural,W2: code_natural] :
aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),V),W2)) = $let(
v: code_natural,
v:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))))))))))))),
$let(
w2: code_natural,
w2:= minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))),
aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),v,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),w2),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),v),w2)) ) ) ).
% next.simps
tff(fact_4745_accp__acc__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A)),X2: A] :
( aa(A,$o,accp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X2)
<=> member(A,X2,acc(A,R2)) ) ).
% accp_acc_eq
tff(fact_4746_acc__def,axiom,
! [A: $tType,X2: set(product_prod(A,A))] : acc(A,X2) = aa(fun(A,$o),set(A),collect(A),accp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),X2))) ).
% acc_def
tff(fact_4747_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(code_natural,A),N4: code_natural,M3: code_natural] : X != 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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N4),M3)) ).
% Lazy_Sequence.iterate_upto.cases
tff(fact_4748_acc__induct__rule,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(A,A3,acc(A,R2))
=> ( ! [X3: A] :
( member(A,X3,acc(A,R2))
=> ( ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2)
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,A3) ) ) ).
% acc_induct_rule
tff(fact_4749_not__acc__down,axiom,
! [A: $tType,X: A,R: set(product_prod(A,A))] :
( ~ member(A,X,acc(A,R))
=> ~ ! [Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X),R)
=> member(A,Z3,acc(A,R)) ) ) ).
% not_acc_down
tff(fact_4750_acc__downward,axiom,
! [A: $tType,B2: A,R2: set(product_prod(A,A)),A3: A] :
( member(A,B2,acc(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R2)
=> member(A,A3,acc(A,R2)) ) ) ).
% acc_downward
tff(fact_4751_acc__induct,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A)),P: fun(A,$o)] :
( member(A,A3,acc(A,R2))
=> ( ! [X3: A] :
( member(A,X3,acc(A,R2))
=> ( ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R2)
=> aa(A,$o,P,Y4) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,A3) ) ) ).
% acc_induct
tff(fact_4752_acc_Ointros,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] :
( ! [Y2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X),R2)
=> member(A,Y2,acc(A,R2)) )
=> member(A,X,acc(A,R2)) ) ).
% acc.intros
tff(fact_4753_acc_Osimps,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
( member(A,A3,acc(A,R2))
<=> ? [X4: A] :
( ( A3 = X4 )
& ! [Xa2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2)
=> member(A,Xa2,acc(A,R2)) ) ) ) ).
% acc.simps
tff(fact_4754_acc_Ocases,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
( member(A,A3,acc(A,R2))
=> ! [Y4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),A3),R2)
=> member(A,Y4,acc(A,R2)) ) ) ).
% acc.cases
tff(fact_4755_log_Ocases,axiom,
! [X: product_prod(code_natural,code_natural)] :
~ ! [B3: code_natural,I3: 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),B3),I3) ).
% log.cases
tff(fact_4756_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( quickc658316121487927005ustive(B)
& cl_HOL_Oequal(A)
& quickc658316121487927005ustive(A) )
=> ! [X: product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(fun(A,B),option(product_prod($o,list(code_term)))),I3: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(fun(A,B),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(fun(A,B),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I3),D2)) ) ).
% exhaustive_fun'.cases
tff(fact_4757_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(code_natural,option(product_prod($o,list(code_term)))),D2: code_natural,I3: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(code_natural,option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F3),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),D2),I3)) ).
% exhaustive_natural'.cases
tff(fact_4758_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( quickc3360725361186068524ustive(B)
& cl_HOL_Oequal(A)
& quickc3360725361186068524ustive(A) )
=> ! [X: product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),I3: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I3),D2)) ) ).
% full_exhaustive_fun'.cases
tff(fact_4759_full__exhaustive__natural_H_Ocases,axiom,
! [X: product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D2: code_natural,I3: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_natural,code_natural)),F3),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),D2),I3)) ).
% full_exhaustive_natural'.cases
tff(fact_4760_acc__subset__induct,axiom,
! [A: $tType,D4: set(A),R: set(product_prod(A,A)),X: A,P: fun(A,$o)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),acc(A,R))
=> ( ! [X3: A,Z3: A] :
( member(A,X3,D4)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X3),R)
=> member(A,Z3,D4) ) )
=> ( member(A,X,D4)
=> ( ! [X3: A] :
( member(A,X3,D4)
=> ( ! [Z5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),X3),R)
=> aa(A,$o,P,Z5) )
=> aa(A,$o,P,X3) ) )
=> aa(A,$o,P,X) ) ) ) ) ).
% acc_subset_induct
tff(fact_4761_acc__downwards,axiom,
! [A: $tType,A3: A,R2: set(product_prod(A,A)),B2: A] :
( member(A,A3,acc(A,R2))
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),transitive_rtrancl(A,R2))
=> member(A,B2,acc(A,R2)) ) ) ).
% acc_downwards
tff(fact_4762_acc__downwards__aux,axiom,
! [A: $tType,B2: A,A3: A,R2: set(product_prod(A,A))] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),transitive_rtrancl(A,R2))
=> ( member(A,A3,acc(A,R2))
=> member(A,B2,acc(A,R2)) ) ) ).
% acc_downwards_aux
tff(fact_4763_split__seed__def,axiom,
! [S2: product_prod(code_natural,code_natural)] : split_seed(S2) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_qm(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),S2)),S2) ).
% split_seed_def
tff(fact_4764_list__decode_Opinduct,axiom,
! [A0: nat,P: fun(nat,$o)] :
( aa(nat,$o,accp(nat,nat_list_decode_rel),A0)
=> ( ( aa(nat,$o,accp(nat,nat_list_decode_rel),zero_zero(nat))
=> aa(nat,$o,P,zero_zero(nat)) )
=> ( ! [N4: nat] :
( aa(nat,$o,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N4))
=> ( ! [X2: nat,Y4: nat] :
( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y4) = nat_prod_decode(N4) )
=> aa(nat,$o,P,Y4) )
=> aa(nat,$o,P,aa(nat,nat,suc,N4)) ) )
=> aa(nat,$o,P,A0) ) ) ) ).
% list_decode.pinduct
tff(fact_4765_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( log(X,Xa) = Y )
=> ( aa(product_prod(code_natural,code_natural),$o,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))
=> ~ ( ( Y = $ite(
( aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less_eq(code_natural),X),one_one(code_natural))
| aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Xa),X) ),
one_one(code_natural),
aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,divide_divide(code_natural,Xa,X))) ) )
=> ~ aa(product_prod(code_natural,code_natural),$o,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_4766_Random_Orange__def,axiom,
! [K: code_natural] : range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_qo(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_qp(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_4767_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)))] :
( aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),A1),A22)))
=> ( ! [F3: fun(code_natural,A),N4: code_natural,M3: code_natural] :
( aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o,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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N4),M3)))
=> ( ! [X2: product_unit] :
( ~ aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),M3),N4)
=> aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,F3),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N4),one_one(code_natural))),M3) )
=> aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,F3),N4),M3) ) )
=> aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,$o)),P,A0),A1),A22) ) ) ).
% Predicate.iterate_upto.pinduct
tff(fact_4768_iter_Ocases,axiom,
! [A: $tType,X: product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))] :
~ ! [Random: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),Nrandom: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),aa(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),Random),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)),Nrandom),Seed)) ).
% iter.cases
tff(fact_4769_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X,product_Pair(B,C)) = X ).
% scomp_Pair
tff(fact_4770_iterate_Osimps,axiom,
! [A: $tType,B: $tType,K: code_natural,F: fun(B,fun(A,product_prod(B,A))),X: B] :
aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X) = $ite(K = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F))) ).
% iterate.simps
tff(fact_4771_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa))) ) ) ).
% iterate.elims
tff(fact_4772_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F) = aa(C,fun(A,B),F,X) ).
% Pair_scomp
tff(fact_4773_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb) = Y )
=> ( aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb)))
=> ~ ( ( Y = $ite(X = zero_zero(code_natural),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa))) )
=> ~ aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),$o,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb))) ) ) ) ).
% iterate.pelims
tff(fact_4774_iter_H_Ocases,axiom,
! [A: $tType] :
( quickcheck_random(A)
=> ! [X: product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))] :
~ ! [T3: itself(A),Nrandom: code_natural,Sz: code_natural,Seed: product_prod(code_natural,code_natural)] : X != aa(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),aa(itself(A),fun(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))),product_Pair(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),T3),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),aa(code_natural,fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),Nrandom),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)),Sz),Seed))) ) ).
% iter'.cases
tff(fact_4775_select__weight__def,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),Xs))),aTP_Lamp_qq(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_weight_def
tff(fact_4776_iterate_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A))] :
~ ! [K2: code_natural,F3: fun(A,fun(B,product_prod(A,B))),X3: A] : X != aa(product_prod(fun(A,fun(B,product_prod(A,B))),A),product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A)),aa(code_natural,fun(product_prod(fun(A,fun(B,product_prod(A,B))),A),product_prod(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A))),product_Pair(code_natural,product_prod(fun(A,fun(B,product_prod(A,B))),A)),K2),aa(A,product_prod(fun(A,fun(B,product_prod(A,B))),A),aa(fun(A,fun(B,product_prod(A,B))),fun(A,product_prod(fun(A,fun(B,product_prod(A,B))),A)),product_Pair(fun(A,fun(B,product_prod(A,B))),A),F3),X3)) ).
% iterate.cases
tff(fact_4777_select__weight__cons__zero,axiom,
! [A: $tType,X: A,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),cons(product_prod(code_natural,A),aa(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),zero_zero(code_natural)),X)),Xs)) = select_weight(A,Xs) ).
% select_weight_cons_zero
tff(fact_4778_select__weight__select,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( select_weight(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_4779_pick__same,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
=> ( pick(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_4780_cone__def,axiom,
bNF_Cardinal_cone = bNF_Ca6860139660246222851ard_of(product_unit,aa(set(product_unit),set(product_unit),insert2(product_unit,product_Unity),bot_bot(set(product_unit)))) ).
% cone_def
tff(fact_4781_UNIV__unit,axiom,
top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),insert2(product_unit,product_Unity),bot_bot(set(product_unit))) ).
% UNIV_unit
tff(fact_4782_bot__unit__def,axiom,
bot_bot(product_unit) = product_Unity ).
% bot_unit_def
tff(fact_4783_select__def,axiom,
! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(code_natural_of_nat(aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_qr(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_def
tff(fact_4784_Random__Pred_Ounion__def,axiom,
! [A: $tType,R12: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),R23: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),X2: product_prod(code_natural,code_natural)] : random_union(A,R12,R23,X2) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qt(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),R23)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R12,X2)) ).
% Random_Pred.union_def
tff(fact_4785_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( bit_or_not_num_neg(X,Xa) = Y )
=> ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
=> ( ( ( X = one2 )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit0,M3) )
=> ( ( Y = aa(num,num,bit1,M3) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,M3))) ) ) )
=> ( ( ( X = one2 )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit1,M3) )
=> ( ( Y = aa(num,num,bit1,M3) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa(num,num,bit0,N4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit0,one2) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N4)),one2)) ) ) )
=> ( ! [N4: num] :
( ( X = aa(num,num,bit0,N4) )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit0,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N4)),aa(num,num,bit0,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa(num,num,bit0,N4) )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit1,M3) )
=> ( ( Y = aa(num,num,bit0,bit_or_not_num_neg(N4,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N4)),aa(num,num,bit1,M3))) ) ) )
=> ( ! [N4: num] :
( ( X = aa(num,num,bit1,N4) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N4)),one2)) ) ) )
=> ( ! [N4: num] :
( ( X = aa(num,num,bit1,N4) )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit0,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N4)),aa(num,num,bit0,M3))) ) ) )
=> ~ ! [N4: num] :
( ( X = aa(num,num,bit1,N4) )
=> ! [M3: num] :
( ( Xa = aa(num,num,bit1,M3) )
=> ( ( Y = bitM(bit_or_not_num_neg(N4,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N4)),aa(num,num,bit1,M3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
tff(fact_4786_Random__Pred_Oempty__def,axiom,
! [A: $tType] : random_empty(A) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bot_bot(pred(A))) ).
% Random_Pred.empty_def
tff(fact_4787_Random__Pred_Obind__def,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),F: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),X2: product_prod(code_natural,code_natural)] : random_bind(B,A,R,F,X2) = aa(product_prod(pred(B),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(B),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(B),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qw(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),F)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),R,X2)) ).
% Random_Pred.bind_def
tff(fact_4788_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option(num)] :
( ( bit_un7362597486090784418nd_num(X,Xa) = Y )
=> ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = none(num) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N4))) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_qx(num,option(num)),bit_un7362597486090784418nd_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_4789_bottom__bind,axiom,
! [B: $tType,A: $tType,P: fun(B,pred(A))] : bind(B,A,bot_bot(pred(B)),P) = bot_bot(pred(A)) ).
% bottom_bind
tff(fact_4790_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option(num)] :
( ( bit_and_not_num(X,Xa) = Y )
=> ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N4))) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = none(num) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_qx(num,option(num)),bit_and_not_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_4791_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option(num)] :
( ( bit_un2480387367778600638or_num(X,Xa) = Y )
=> ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N4))) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M3,N4))) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M3)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M3,N4))) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
tff(fact_4792_singleton__sup,axiom,
! [A: $tType,Default: fun(product_unit,A),Aa2: pred(A),Ba: pred(A)] :
singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Aa2),Ba)) = $ite(
Aa2 = bot_bot(pred(A)),
singleton(A,Default,Ba),
$ite(
Ba = bot_bot(pred(A)),
singleton(A,Default,Aa2),
$ite(singleton(A,Default,Aa2) = singleton(A,Default,Ba),singleton(A,Default,Aa2),aa(product_unit,A,Default,product_Unity)) ) ) ).
% singleton_sup
tff(fact_4793_singleton__bot,axiom,
! [A: $tType,Default: fun(product_unit,A)] : singleton(A,Default,bot_bot(pred(A))) = aa(product_unit,A,Default,product_Unity) ).
% singleton_bot
tff(fact_4794_singleton__sup__aux,axiom,
! [A: $tType,Default: fun(product_unit,A),Aa2: pred(A),Ba: pred(A)] :
singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Aa2),Ba)) = $ite(
Aa2 = bot_bot(pred(A)),
singleton(A,Default,Ba),
$ite(Ba = bot_bot(pred(A)),singleton(A,Default,Aa2),singleton(A,Default,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),aa(A,pred(A),single(A),singleton(A,Default,Aa2))),aa(A,pred(A),single(A),singleton(A,Default,Ba))))) ) ).
% singleton_sup_aux
tff(fact_4795_take__bit__num__code,axiom,
! [N: nat,M2: num] : bit_take_bit_num(N,M2) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_rc(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M2)) ).
% take_bit_num_code
tff(fact_4796_or__num_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( bit_un6697907153464112080or_num(X,Xa) = Y )
=> ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa))
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,num,bit1,N4) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N4))) ) ) )
=> ( ( ( X = one2 )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,num,bit1,N4) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit1,M3) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,num,bit0,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit0,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit0,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M3)),aa(num,num,bit1,N4))) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit1,M3) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit0,N4) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit0,N4))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa(num,num,bit1,M3) )
=> ! [N4: num] :
( ( Xa = aa(num,num,bit1,N4) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M3,N4)) )
=> ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N4))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
tff(fact_4797_unit__pred__cases,axiom,
! [P: fun(pred(product_unit),$o),Q: pred(product_unit)] :
( aa(pred(product_unit),$o,P,bot_bot(pred(product_unit)))
=> ( aa(pred(product_unit),$o,P,aa(product_unit,pred(product_unit),single(product_unit),product_Unity))
=> aa(pred(product_unit),$o,P,Q) ) ) ).
% unit_pred_cases
tff(fact_4798_single__not__bot,axiom,
! [A: $tType,X: A] : aa(A,pred(A),single(A),X) != bot_bot(pred(A)) ).
% single_not_bot
tff(fact_4799_Random__Pred_ORandom__def,axiom,
! [A: $tType,G: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural)))] : random_Random(A,G) = product_scomp(product_prod(code_natural,code_natural),product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),G,comp(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_prod(A,fun(product_unit,code_term)),product_Pair(pred(A),product_prod(code_natural,code_natural)),comp(A,pred(A),product_prod(A,fun(product_unit,code_term)),single(A),product_fst(A,fun(product_unit,code_term))))) ).
% Random_Pred.Random_def
tff(fact_4800_Random__Pred_Osingle__def,axiom,
! [A: $tType,X: A] : random_single(A,X) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(A,pred(A),single(A),X)) ).
% Random_Pred.single_def
tff(fact_4801_pred__of__set__set__foldr__sup,axiom,
! [A: $tType,Xs: list(A)] : pred_of_set(A,set2(A,Xs)) = foldr(pred(A),pred(A),sup_sup(pred(A)),map(A,pred(A),single(A),Xs),bot_bot(pred(A))) ).
% pred_of_set_set_foldr_sup
tff(fact_4802_pred__of__set__fold__sup,axiom,
! [A: $tType,Aa2: set(A)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( pred_of_set(A,Aa2) = finite_fold(pred(A),pred(A),sup_sup(pred(A)),bot_bot(pred(A)),aa(set(A),set(pred(A)),image2(A,pred(A),single(A)),Aa2)) ) ) ).
% pred_of_set_fold_sup
tff(fact_4803_pred__of__set__set__fold__sup,axiom,
! [A: $tType,Xs: list(A)] : pred_of_set(A,set2(A,Xs)) = aa(pred(A),pred(A),fold(pred(A),pred(A),sup_sup(pred(A)),map(A,pred(A),single(A),Xs)),bot_bot(pred(A))) ).
% pred_of_set_set_fold_sup
tff(fact_4804_if__pred__eq,axiom,
! [B2: $o] :
if_pred((B2)) = $ite((B2),aa(product_unit,pred(product_unit),single(product_unit),product_Unity),bot_bot(pred(product_unit))) ).
% if_pred_eq
tff(fact_4805_contains__pred__def,axiom,
! [A: $tType,Aa2: set(A),X: A] :
predic7144156976422707464s_pred(A,Aa2,X) = $ite(member(A,X,Aa2),aa(product_unit,pred(product_unit),single(product_unit),product_Unity),bot_bot(pred(product_unit))) ).
% contains_pred_def
tff(fact_4806_Random__Pred_Onot__randompred__def,axiom,
! [P: fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),X2: product_prod(code_natural,code_natural)] : random6974930770145893639ompred(P,X2) = aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(fun(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_case_prod(pred(product_unit),product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_rd(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))))),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),P,X2)) ).
% Random_Pred.not_randompred_def
tff(fact_4807_eval__bot,axiom,
! [A: $tType] : eval(A,bot_bot(pred(A))) = bot_bot(fun(A,$o)) ).
% eval_bot
tff(fact_4808_not__bot,axiom,
! [A: $tType,Aa2: pred(A)] :
( ( Aa2 != bot_bot(pred(A)) )
=> ~ ! [X3: A] : ~ aa(A,$o,eval(A,Aa2),X3) ) ).
% not_bot
tff(fact_4809_botE,axiom,
! [A: $tType,X: A] : ~ aa(A,$o,eval(A,bot_bot(pred(A))),X) ).
% botE
tff(fact_4810_not__pred__eq,axiom,
! [P: pred(product_unit)] :
not_pred(P) = $ite(aa(product_unit,$o,eval(product_unit,P),product_Unity),bot_bot(pred(product_unit)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)) ).
% not_pred_eq
tff(fact_4811_bot__pred__def,axiom,
! [A: $tType] : bot_bot(pred(A)) = pred2(A,bot_bot(fun(A,$o))) ).
% bot_pred_def
tff(fact_4812_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F: fun(code_natural,A),N: code_natural,M2: code_natural] :
( aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N),M2)))
=> ( iterate_upto(A,F,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_re(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M2)) ) ) ).
% Predicate.iterate_upto.psimps
tff(fact_4813_bot__set__code,axiom,
! [A: $tType] : bot_bot(pred(A)) = seq2(A,aTP_Lamp_rf(product_unit,seq(A))) ).
% bot_set_code
tff(fact_4814_Predicate_Osingle__code,axiom,
! [A: $tType,X: A] : aa(A,pred(A),single(A),X) = seq2(A,aTP_Lamp_rg(A,fun(product_unit,seq(A)),X)) ).
% Predicate.single_code
tff(fact_4815_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
( ( iterate_upto(A,X,Xa,Xb) = Y )
=> ( aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o,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_re(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb)) )
=> ~ aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),$o,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_4816_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType,F: fun(code_natural,A),N: code_natural,M2: code_natural] : random_iterate_upto(A,F,N,M2) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),iterate_upto(A,F,N,M2)) ).
% Random_Pred.iterate_upto_def
tff(fact_4817_of__seq__code_I1_J,axiom,
! [A: $tType] : set_of_seq(A,empty(A)) = bot_bot(set(A)) ).
% of_seq_code(1)
tff(fact_4818_pred__of__seq_Osimps_I1_J,axiom,
! [A: $tType] : pred_of_seq(A,empty(A)) = bot_bot(pred(A)) ).
% pred_of_seq.simps(1)
tff(fact_4819_of__pred__code,axiom,
! [A: $tType,F: fun(product_unit,seq(A))] : set_of_pred(A,seq2(A,F)) = case_seq(set(A),A,bot_bot(set(A)),aTP_Lamp_rh(A,fun(pred(A),set(A))),aTP_Lamp_ri(pred(A),fun(seq(A),set(A))),aa(product_unit,seq(A),F,product_Unity)) ).
% of_pred_code
tff(fact_4820_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o),Aa2: set(A)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
=> ( comple1602240252501008431_chain(A,Ord,Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( member(A,X3,Aa2)
=> aa(A,$o,P,X3) )
=> aa(A,$o,P,aa(set(A),A,Lub,Aa2)) ) ) ) ) ).
% ccpo.admissibleD
tff(fact_4821_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: fun(A,fun(A,$o)),P: fun(A,$o),Lub: fun(set(A),A)] :
( ! [A7: set(A)] :
( comple1602240252501008431_chain(A,Ord,A7)
=> ( ( A7 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( member(A,X2,A7)
=> aa(A,$o,P,X2) )
=> aa(A,$o,P,aa(set(A),A,Lub,A7)) ) ) )
=> comple1908693960933563346ssible(A,Lub,Ord,P) ) ).
% ccpo.admissibleI
tff(fact_4822_ccpo_Oadmissible__def,axiom,
! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o)] :
( comple1908693960933563346ssible(A,Lub,Ord,P)
<=> ! [A8: set(A)] :
( comple1602240252501008431_chain(A,Ord,A8)
=> ( ( A8 != bot_bot(set(A)) )
=> ( ! [X4: A] :
( member(A,X4,A8)
=> aa(A,$o,P,X4) )
=> aa(A,$o,P,aa(set(A),A,Lub,A8)) ) ) ) ) ).
% ccpo.admissible_def
tff(fact_4823_fixp__induct,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,$o),F: fun(A,A)] :
( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> ( aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))))
=> ( ! [X3: A] :
( aa(A,$o,P,X3)
=> aa(A,$o,P,aa(A,A,F,X3)) )
=> aa(A,$o,P,comple115746919287870866o_fixp(A,F)) ) ) ) ) ) ).
% fixp_induct
tff(fact_4824_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),V: B] : map_mmupd(A,B,M2,bot_bot(set(A)),V) = M2 ).
% map_mmupd_empty
tff(fact_4825_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R4: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(B,A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R4)
=> ( bNF_Wellorder_iso(B,A,R2,R4,F)
=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),hilbert_inv_into(B,A,field2(B,R2),F,X)),hilbert_inv_into(B,A,field2(B,R2),F,Y)),R2) ) ) ).
% iso_backward
tff(fact_4826_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_4827_execute__change,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,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_change(A,F,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)),aa(A,A,F,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),ref_set(A,R2,aa(A,A,F,ref_get(A,H2,R2)),H2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))) ) ).
% execute_change
tff(fact_4828_successively_Opelims_I2_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
( successively(A,X,Xa)
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) )
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),cons(A,X3),nil(A)) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),nil(A)))) )
=> ~ ! [X3: A,Y2: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3)) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3))))
=> ~ ( aa(A,$o,aa(A,fun(A,$o),X,X3),Y2)
& successively(A,X,aa(list(A),list(A),cons(A,Y2),Xs3)) ) ) ) ) ) ) ) ).
% successively.pelims(2)
tff(fact_4829_comm__monoid__mset_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> comm_monoid_mset(A,F,Z2) ) ).
% comm_monoid_mset.intro
tff(fact_4830_comm__monoid__mset_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_mset(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% comm_monoid_mset.axioms
tff(fact_4831_comm__monoid__mset__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_mset(A,F,Z2)
<=> comm_monoid(A,F,Z2) ) ).
% comm_monoid_mset_def
tff(fact_4832_successively_Opelims_I3_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A)] :
( ~ successively(A,X,Xa)
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ~ ! [X3: A,Y2: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3)) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3))))
=> ( aa(A,$o,aa(A,fun(A,$o),X,X3),Y2)
& successively(A,X,aa(list(A),list(A),cons(A,Y2),Xs3)) ) ) ) ) ) ).
% successively.pelims(3)
tff(fact_4833_successively_Opelims_I1_J,axiom,
! [A: $tType,X: fun(A,fun(A,$o)),Xa: list(A),Y: $o] :
( ( successively(A,X,Xa)
<=> (Y) )
=> ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),Xa))
=> ( ( ( Xa = nil(A) )
=> ( (Y)
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),nil(A))) ) )
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),cons(A,X3),nil(A)) )
=> ( (Y)
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),nil(A)))) ) )
=> ~ ! [X3: A,Y2: A,Xs3: list(A)] :
( ( Xa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3)) )
=> ( ( (Y)
<=> ( aa(A,$o,aa(A,fun(A,$o),X,X3),Y2)
& successively(A,X,aa(list(A),list(A),cons(A,Y2),Xs3)) ) )
=> ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),X),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y2),Xs3)))) ) ) ) ) ) ) ).
% successively.pelims(1)
tff(fact_4834_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_4835_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ! [N4: nat,R3: A,H3: heap_ext(product_unit),N6: nat] : X != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N4),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H3),N6))))
=> ~ ! [N4: nat] : X != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N4),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).
% timeFrame.cases
tff(fact_4836_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( ! [X3: A,H4: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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)),X3),H4))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% Heap_cases
tff(fact_4837_execute__tap,axiom,
! [A: $tType,F: 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,F)),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,F,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_4838_execute__ureturn,axiom,
! [A: $tType,X: A] : heap_Time_execute(A,heap_Time_ureturn(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_rj(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% execute_ureturn
tff(fact_4839_execute__assert_I1_J,axiom,
! [A: $tType,P: fun(A,$o),X: A,H2: heap_ext(product_unit)] :
( aa(A,$o,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_4840_execute__return,axiom,
! [A: $tType,X: A] : heap_Time_execute(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_rk(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% execute_return
tff(fact_4841_execute__update,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),V: 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,V)),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,V,H2)),one_one(nat)))) ) ).
% execute_update
tff(fact_4842_ureturn__def,axiom,
! [A: $tType,X: A] : heap_Time_ureturn(A,X) = heap_Time_heap(A,aTP_Lamp_rj(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% ureturn_def
tff(fact_4843_return__def,axiom,
! [A: $tType,X: A] : heap_Time_return(A,X) = heap_Time_heap(A,aTP_Lamp_rk(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% return_def
tff(fact_4844_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),V: A] : ref_update(A,R2,V) = 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_rl(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R2),V)) ) ).
% Ref_Time.update_def
tff(fact_4845_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( ( heap_Time_timeFrame(A,X,Xa) = Y )
=> ( ! [R3: A,H3: heap_ext(product_unit),N6: nat] :
( ( Xa = 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),H3),N6))) )
=> ( Y != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N6)))) ) )
=> ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( Y != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).
% timeFrame.elims
tff(fact_4846_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( ( heap_Time_timeFrame(A,X,Xa) = Y )
=> ( aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),Xa))
=> ( ! [R3: A,H3: heap_ext(product_unit),N6: nat] :
( ( Xa = 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),H3),N6))) )
=> ( ( Y = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N6)))) )
=> ~ aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N6))))) ) )
=> ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ~ aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),none(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ) ) ) ) ) ).
% timeFrame.pelims
tff(fact_4847_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R2: A,H2: heap_ext(product_unit),N5: nat] : heap_Time_timeFrame(A,N,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),N5)))) = 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)),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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N5)))) ).
% timeFrame.simps(1)
tff(fact_4848_tap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F) = heap_Time_Heap2(A,aTP_Lamp_rm(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F)) ).
% tap_def
tff(fact_4849_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H5),N))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F,G)),H2) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,X)),H5)) ) ) ).
% execute_bind(1)
tff(fact_4850_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H9: heap_ext(product_unit),N5: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H5),N))) )
=> ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,X)),H5) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H9),N5))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F,G)),H2) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H9),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N5)))) ) ) ) ).
% execute_bind_eq_SomeI
tff(fact_4851_wait__def,axiom,
! [N: nat] : heap_Time_wait(N) = heap_Time_Heap2(product_unit,aTP_Lamp_rn(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).
% wait_def
tff(fact_4852_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H5),N))) )
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X),H5)
=> heap_Time_success(B,heap_Time_bind(A,B,F,G),H2) ) ) ).
% success_bind_executeI
tff(fact_4853_execute__nth_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H2: heap_ext(product_unit),A3: array(A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),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,I)),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)),I)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),one_one(nat)))) ) ) ) ).
% execute_nth(1)
tff(fact_4854_successE,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( heap_Time_success(A,F,H2)
=> ~ ! [R3: A,H4: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H4)) ) ).
% successE
tff(fact_4855_Array__Time_Onth__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat] : array_nth(A,A3,I) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_ro(array(A),fun(nat,fun(heap_ext(product_unit),$o)),A3),I),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rp(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),A3),I)) ) ).
% Array_Time.nth_def
tff(fact_4856_freeze__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_rq(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ).
% freeze_def
tff(fact_4857_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_4858_execute__swap_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H2: heap_ext(product_unit),A3: array(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),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_swap(A,I,X,A3)),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)),I)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I,X,H2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).
% execute_swap(1)
tff(fact_4859_swap__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X: A,A3: array(A)] : array_swap(A,I,X,A3) = heap_Time_guard(A,aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_rr(nat,fun(array(A),fun(heap_ext(product_unit),$o)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rs(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),I),X),A3)) ) ).
% swap_def
tff(fact_4860_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_4861_map__entry__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,F: fun(A,A),A3: array(A)] : array_map_entry(A,I,F,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_rr(nat,fun(array(A),fun(heap_ext(product_unit),$o)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rt(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I),F),A3)) ) ).
% map_entry_def
tff(fact_4862_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H2: heap_ext(product_unit),A3: array(A),F: fun(A,A)] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),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_map_entry(A,I,F,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,I,aa(A,A,F,aa(nat,A,nth(A,array_get(A,H2,A3)),I)),H2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).
% execute_map_entry(1)
tff(fact_4863_upd__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X: A,A3: array(A)] : array_upd(A,I,X,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_rr(nat,fun(array(A),fun(heap_ext(product_unit),$o)),I),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_ru(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I),X),A3)) ) ).
% upd_def
tff(fact_4864_execute__upd_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H2: heap_ext(product_unit),A3: array(A),X: A] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),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,I,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,I,X,H2)),one_one(nat)))) ) ) ) ).
% execute_upd(1)
tff(fact_4865_execute__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: 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,F)),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)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rv(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,map(nat,A,F,upt(zero_zero(nat),N)),H2))) ) ).
% execute_make
tff(fact_4866_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)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rw(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_4867_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)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rv(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,replicate(A,N,X),H2))) ) ).
% execute_new
tff(fact_4868_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_rx(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_4869_of__list__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_ry(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).
% of_list_def
tff(fact_4870_Array__Time_Omake__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A)] : array_make(A,N,F) = 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_rz(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),F)) ) ).
% Array_Time.make_def
tff(fact_4871_effect__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H5: heap_ext(product_unit),F: 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,map(nat,A,F,upt(zero_zero(nat),N)),H2) )
=> heap_Time_effect(array(A),array_make(A,N,F),H2,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_makeI
tff(fact_4872_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,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_4873_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_4874_effect__def,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N: nat] :
( heap_Time_effect(A,C2,H2,H5,R2,N)
<=> ( 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)),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),H5),N))) ) ) ).
% effect_def
tff(fact_4875_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),R2: A,H5: heap_ext(product_unit),N: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,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)),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),H5),N))) )
=> heap_Time_effect(A,C2,H2,H5,R2,N) ) ).
% effectI
tff(fact_4876_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] :
array_alloc(A,Xs,H2) = $let(
l: nat,
l:= lim(product_unit,H2),
$let(
r: array(A),
r:= array2(A,l),
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)),r),array_set(A,r,Xs,lim_update(product_unit,aTP_Lamp_sa(nat,fun(nat,nat),l),H2))) ) ) ) ).
% Array_Time.alloc_def
tff(fact_4877_Heap__lub__empty,axiom,
! [A: $tType] : heap_Time_Heap_lub(A,bot_bot(set(heap_Time_Heap(A)))) = heap_Time_Heap2(A,aTP_Lamp_sb(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% Heap_lub_empty
tff(fact_4878_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B)),F: fun(A,C),G: fun(B,D)] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R)
=> member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F,A3)),aa(B,D,G,B2)),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F,G)),R)) ) ).
% map_prod_imageI
tff(fact_4879_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,A3)),aa(D,B,G,B2)) ).
% map_prod_simp
tff(fact_4880_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F: fun(C,A),G: fun(D,B),R: set(product_prod(C,D))] :
( member(product_prod(A,B),C2,aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G)),R))
=> ~ ! [X3: C,Y2: D] :
( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X3)),aa(D,B,G,Y2)) )
=> ~ member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y2),R) ) ) ).
% prod_fun_imageE
tff(fact_4881_map__prod__def,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_sc(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F),G)) ).
% map_prod_def
tff(fact_4882_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [X: A,H2: heap_ext(product_unit)] :
ref_alloc(A,X,H2) = $let(
l: nat,
l:= lim(product_unit,H2),
$let(
r: ref(A),
r:= ref2(A,l),
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)),r),ref_set(A,r,X,lim_update(product_unit,aTP_Lamp_sa(nat,fun(nat,nat),l),H2))) ) ) ) ).
% Ref_Time.alloc_def
tff(fact_4883_bot__in__iterates,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] : member(A,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))),comple6359979572994053840erates(A,F)) ) ).
% bot_in_iterates
tff(fact_4884_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(B,C),A),A3: B,B2: C] : product_curry(B,C,A,F,A3,B2) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).
% curry_conv
tff(fact_4885_curryI,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A3: A,B2: B] :
( aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))
=> product_curry(A,B,$o,F,A3,B2) ) ).
% curryI
tff(fact_4886_curryE,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A3: A,B2: B] :
( product_curry(A,B,$o,F,A3,B2)
=> aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ).
% curryE
tff(fact_4887_curryD,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A3: A,B2: B] :
( product_curry(A,B,$o,F,A3,B2)
=> aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) ) ).
% curryD
tff(fact_4888_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa3: A,Xb3: B] : product_curry(A,B,C,X2,Xa3,Xb3) = aa(product_prod(A,B),C,X2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa3),Xb3)) ).
% curry_def
tff(fact_4889_effect__refI,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H5: heap_ext(product_unit),V: 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,V,H2) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(ref(A),ref_ref(A,V),H2,H5,R2,N) ) ) ) ).
% effect_refI
tff(fact_4890_ref__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A] : ref_ref(A,V) = heap_Time_heap(ref(A),aTP_Lamp_se(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),V)) ) ).
% ref_def
tff(fact_4891_execute__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: 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,V)),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)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_sd(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,V,H2))) ) ).
% execute_ref
tff(fact_4892_next__fresh,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H5: heap_ext(product_unit),X: A,H2: heap_ext(product_unit)] :
( ( 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,X,H2) )
=> ~ ref_present(A,H2,R2) ) ) ).
% next_fresh
tff(fact_4893_next__present,axiom,
! [A: $tType] :
( heap(A)
=> ! [R2: ref(A),H5: heap_ext(product_unit),X: A,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R2),H5) = ref_alloc(A,X,H2) )
=> ref_present(A,H5,R2) ) ) ).
% next_present
tff(fact_4894_eventually__prod__sequentially,axiom,
! [P: fun(product_prod(nat,nat),$o)] :
( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
<=> ? [N7: nat] :
! [M: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M)
=> ! [N2: nat] :
( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
=> aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N2),M)) ) ) ) ).
% eventually_prod_sequentially
tff(fact_4895_trivial__limit__sequentially,axiom,
at_top(nat) != bot_bot(filter(nat)) ).
% trivial_limit_sequentially
tff(fact_4896_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F: fun(nat,A)] : filtermap(nat,A,F,at_top(nat)) != bot_bot(filter(A)) ).
% filtermap_sequentually_ne_bot
tff(fact_4897_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ( at_top(A) != bot_bot(filter(A)) ) ) ).
% trivial_limit_at_top_linorder
tff(fact_4898_Nitpick_OEx1__unfold,axiom,
! [A: $tType,P: fun(A,$o)] :
( ? [X4: A] :
( aa(A,$o,P,X4)
& ! [Y3: A] :
( aa(A,$o,P,Y3)
=> ( Y3 = X4 ) ) )
<=> ? [X4: A] : aa(fun(A,$o),set(A),collect(A),P) = aa(set(A),set(A),insert2(A,X4),bot_bot(set(A))) ) ).
% Nitpick.Ex1_unfold
tff(fact_4899_adm__wf__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B))] :
( adm_wf(A,B,R,F4)
<=> ! [F6: fun(A,B),G5: fun(A,B),X4: A] :
( ! [Z4: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X4),R)
=> ( aa(A,B,F6,Z4) = aa(A,B,G5,Z4) ) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),F4,F6),X4) = aa(A,B,aa(fun(A,B),fun(A,B),F4,G5),X4) ) ) ) ).
% adm_wf_def
tff(fact_4900_eventually__frequentlyE,axiom,
! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
( eventually(A,P,F4)
=> ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
=> ( ( F4 != bot_bot(filter(A)) )
=> frequently(A,Q,F4) ) ) ) ).
% eventually_frequentlyE
tff(fact_4901_frequently__const,axiom,
! [A: $tType,F4: filter(A),P: $o] :
( ( F4 != bot_bot(filter(A)) )
=> ( frequently(A,aTP_Lamp_af($o,fun(A,$o),(P)),F4)
<=> (P) ) ) ).
% frequently_const
tff(fact_4902_frequently__const__iff,axiom,
! [A: $tType,P: $o,F4: filter(A)] :
( frequently(A,aTP_Lamp_af($o,fun(A,$o),(P)),F4)
<=> ( (P)
& ( F4 != bot_bot(filter(A)) ) ) ) ).
% frequently_const_iff
tff(fact_4903_eventually__frequently,axiom,
! [A: $tType,F4: filter(A),P: fun(A,$o)] :
( ( F4 != bot_bot(filter(A)) )
=> ( eventually(A,P,F4)
=> frequently(A,P,F4) ) ) ).
% eventually_frequently
tff(fact_4904_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),insert2(A,X),Aa2)) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).
% semilattice_neutr_set.insert_remove
tff(fact_4905_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( member(A,X,Aa2)
=> ( lattic5214292709420241887eutr_F(A,F,Z2,Aa2) = aa(A,A,aa(A,fun(A,A),F,X),lattic5214292709420241887eutr_F(A,F,Z2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Aa2),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).
% semilattice_neutr_set.remove
tff(fact_4906_transp__trans__eq,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( transp(A,aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
<=> trans(A,R2) ) ).
% transp_trans_eq
tff(fact_4907_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( lattic5214292709420241887eutr_F(A,F,Z2,bot_bot(set(A))) = Z2 ) ) ).
% semilattice_neutr_set.empty
tff(fact_4908_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Aa2: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( aa(set(A),$o,finite_finite(A),Aa2)
=> ( ( Aa2 != bot_bot(set(A)) )
=> ( ! [X3: A,Y2: A] : member(A,aa(A,A,aa(A,fun(A,A),F,X3),Y2),aa(set(A),set(A),insert2(A,X3),aa(set(A),set(A),insert2(A,Y2),bot_bot(set(A)))))
=> member(A,lattic5214292709420241887eutr_F(A,F,Z2,Aa2),Aa2) ) ) ) ) ).
% semilattice_neutr_set.closed
tff(fact_4909_semilattice__neutr__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
<=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_set_def
tff(fact_4910_semilattice__neutr__set_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_set.axioms
tff(fact_4911_semilattice__neutr__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> lattic5652469242046573047tr_set(A,F,Z2) ) ).
% semilattice_neutr_set.intro
tff(fact_4912_Fpow__not__empty,axiom,
! [A: $tType,Aa2: set(A)] : finite_Fpow(A,Aa2) != bot_bot(set(set(A))) ).
% Fpow_not_empty
tff(fact_4913_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A14: set(B)] :
( bNF_Ca8970107618336181345der_on(A,top_top(set(A)),R2)
=> ( bNF_Cardinal_cexp(sum_sum(B,product_unit),A,bNF_Cardinal_csum(B,product_unit,bNF_Ca6860139660246222851ard_of(B,A14),bNF_Cardinal_cone),R2) = bNF_Ca6860139660246222851ard_of(fun(A,sum_sum(B,product_unit)),bNF_Wellorder_Func(A,sum_sum(B,product_unit),top_top(set(A)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(set(sum_sum(B,product_unit)),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),sup_sup(set(sum_sum(B,product_unit))),aa(set(B),set(sum_sum(B,product_unit)),image2(B,sum_sum(B,product_unit),sum_Inl(B,product_unit)),A14)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),insert2(sum_sum(B,product_unit),aa(product_unit,sum_sum(B,product_unit),sum_Inr(product_unit,B),product_Unity)),bot_bot(set(sum_sum(B,product_unit))))))) ) ) ).
% card_order_csum_cone_cexp_def
tff(fact_4914_random__aux__set_Ocases,axiom,
! [X: product_prod(code_natural,code_natural)] :
( ! [J2: 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),zero_zero(code_natural)),J2)
=> ~ ! [I3: code_natural,J2: 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),code_Suc(I3)),J2) ) ).
% random_aux_set.cases
tff(fact_4915_prod_OPlus,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(C)
=> ! [Aa2: set(A),Ba: set(B),G: fun(sum_sum(A,B),C)] :
( aa(set(A),$o,finite_finite(A),Aa2)
=> ( aa(set(B),$o,finite_finite(B),Ba)
=> ( aa(set(sum_sum(A,B)),C,aa(fun(sum_sum(A,B),C),fun(set(sum_sum(A,B)),C),groups7121269368397514597t_prod(sum_sum(A,B),C),G),sum_Plus(A,B,Aa2,Ba)) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),comp(sum_sum(A,B),C,A,G,sum_Inl(A,B))),Aa2)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),comp(sum_sum(A,B),C,B,G,sum_Inr(B,A))),Ba)) ) ) ) ) ).
% prod.Plus
tff(fact_4916_empty__in__Fpow,axiom,
! [A: $tType,Aa2: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,Aa2)) ).
% empty_in_Fpow
tff(fact_4917_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(A,sum_sum(A,B),sum_Inl(A,B),X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).
% sum_set_simps(1)
tff(fact_4918_sum__set__simps_I4_J,axiom,
! [B: $tType,A: $tType,X: A] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(A,sum_sum(B,A),sum_Inr(A,B),X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).
% sum_set_simps(4)
tff(fact_4919_sum__set__simps_I3_J,axiom,
! [B: $tType,A: $tType,X: B] : aa(sum_sum(B,A),set(A),basic_setr(B,A),aa(B,sum_sum(B,A),sum_Inl(B,A),X)) = bot_bot(set(A)) ).
% sum_set_simps(3)
tff(fact_4920_sum__set__simps_I2_J,axiom,
! [B: $tType,A: $tType,X: B] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(B,sum_sum(A,B),sum_Inr(B,A),X)) = bot_bot(set(A)) ).
% sum_set_simps(2)
tff(fact_4921_Node__def,axiom,
! [B: $tType,A: $tType] : old_Node(A,B) = aa(fun(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o),set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),collect(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),aTP_Lamp_sg(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o)) ).
% Node_def
tff(fact_4922_sum__set__defs_I2_J,axiom,
! [A: $tType,B: $tType] : basic_setr(A,B) = sum_case_sum(A,set(B),B,aTP_Lamp_ik(A,set(B)),aTP_Lamp_sh(B,set(B))) ).
% sum_set_defs(2)
tff(fact_4923_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] : basic_setl(A,B) = sum_case_sum(A,set(A),B,aTP_Lamp_ff(A,set(A)),aTP_Lamp_fm(B,set(A))) ).
% sum_set_defs(1)
tff(fact_4924_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum(B,nat)] : member(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(fun(nat,sum_sum(A,nat)),fun(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aTP_Lamp_si(nat,sum_sum(A,nat))),A3),old_Node(A,B)) ).
% Node_K0_I
tff(fact_4925_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: sum_sum(A,nat)] : old_ndepth(A,B,old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_sj(nat,sum_sum(B,nat))),X))) = zero_zero(nat) ).
% ndepth_K0
tff(fact_4926_Atom__def,axiom,
! [B: $tType,A: $tType,X2: sum_sum(A,nat)] : old_Atom(A,B,X2) = aa(set(old_node(A,B)),set(old_node(A,B)),insert2(old_node(A,B),old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_sj(nat,sum_sum(B,nat))),X2))),bot_bot(set(old_node(A,B)))) ).
% Atom_def
tff(fact_4927_nth__item_Opinduct,axiom,
! [A0: nat,P: fun(nat,$o)] :
( aa(nat,$o,accp(nat,nth_item_rel),A0)
=> ( ( aa(nat,$o,accp(nat,nth_item_rel),zero_zero(nat))
=> aa(nat,$o,P,zero_zero(nat)) )
=> ( ! [N4: nat] :
( aa(nat,$o,accp(nat,nth_item_rel),aa(nat,nat,suc,N4))
=> ( ! [A9: nat,Aa4: nat] :
( ( nat_sum_decode(N4) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A9) )
=> ( ( nat_sum_decode(A9) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa4) )
=> aa(nat,$o,P,Aa4) ) )
=> ( ! [A9: nat,B9: nat] :
( ( nat_sum_decode(N4) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A9) )
=> ( ( nat_sum_decode(A9) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B9) )
=> aa(nat,$o,P,B9) ) )
=> ( ! [B9: nat,Ba3: nat,X2: nat,Y4: nat] :
( ( nat_sum_decode(N4) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B9) )
=> ( ( nat_sum_decode(B9) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba3) )
=> ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y4) = nat_prod_decode(Ba3) )
=> aa(nat,$o,P,X2) ) ) )
=> ( ! [B9: nat,Ba3: nat,X2: nat,Y4: nat] :
( ( nat_sum_decode(N4) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B9) )
=> ( ( nat_sum_decode(B9) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba3) )
=> ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y4) = nat_prod_decode(Ba3) )
=> aa(nat,$o,P,Y4) ) ) )
=> aa(nat,$o,P,aa(nat,nat,suc,N4)) ) ) ) ) )
=> aa(nat,$o,P,A0) ) ) ) ).
% nth_item.pinduct
tff(fact_4928_ntrunc__0,axiom,
! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,zero_zero(nat),M4) = bot_bot(set(old_node(A,B))) ).
% ntrunc_0
tff(fact_4929_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In1(A,B,M4)) = bot_bot(set(old_node(A,B))) ).
% ntrunc_one_In1
tff(fact_4930_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M4: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In0(A,B,M4)) = bot_bot(set(old_node(A,B))) ).
% ntrunc_one_In0
tff(fact_4931_dsum__In1I,axiom,
! [B: $tType,A: $tType,N3: set(old_node(A,B)),N8: set(old_node(A,B)),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N3),N8),S2)
=> member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,N3)),old_In1(A,B,N8)),old_dsum(A,B,R2,S2)) ) ).
% dsum_In1I
tff(fact_4932_is__empty__bot,axiom,
! [A: $tType] : is_empty(A,bot_bot(pred(A))) ).
% is_empty_bot
tff(fact_4933_dsum__In0I,axiom,
! [B: $tType,A: $tType,M4: set(old_node(A,B)),M7: set(old_node(A,B)),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M4),M7),R2)
=> member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,M4)),old_In0(A,B,M7)),old_dsum(A,B,R2,S2)) ) ).
% dsum_In0I
tff(fact_4934_dsum__def,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dsum(A,B,R2,S2) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),sup_sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_sk(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),R2))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_sl(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),S2))) ).
% dsum_def
tff(fact_4935_dsumE,axiom,
! [B: $tType,A: $tType,W2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),W2,old_dsum(A,B,R2,S2))
=> ( ! [X3: set(old_node(A,B)),X8: set(old_node(A,B))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X3),X8),R2)
=> ( W2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,X3)),old_In0(A,B,X8)) ) )
=> ~ ! [Y2: set(old_node(A,B)),Y6: set(old_node(A,B))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y2),Y6),S2)
=> ( W2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Y2)),old_In1(A,B,Y6)) ) ) ) ) ).
% dsumE
tff(fact_4936_Predicate_Ois__empty__def,axiom,
! [A: $tType,Aa2: pred(A)] :
( is_empty(A,Aa2)
<=> ( Aa2 = bot_bot(pred(A)) ) ) ).
% Predicate.is_empty_def
tff(fact_4937_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( linorder(A)
=> ( at_bot(A) != bot_bot(filter(A)) ) ) ).
% trivial_limit_at_bot_linorder
tff(fact_4938_bot__filter__def,axiom,
! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_sm(fun(A,$o),$o)) ).
% bot_filter_def
tff(fact_4939_subset_Osuc__Union__closed__empty,axiom,
! [A: $tType,Aa2: set(set(A))] : member(set(set(A)),bot_bot(set(set(A))),pred_s596693808085603175closed(set(A),Aa2,ord_less(set(A)))) ).
% subset.suc_Union_closed_empty
tff(fact_4940_pred__on_Osuc__Union__closed__empty,axiom,
! [A: $tType,Aa2: set(A),P: fun(A,fun(A,$o))] : member(set(A),bot_bot(set(A)),pred_s596693808085603175closed(A,Aa2,P)) ).
% pred_on.suc_Union_closed_empty
tff(fact_4941_dprod__def,axiom,
! [B: $tType,A: $tType,R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dprod(A,B,R2,S2) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_so(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),S2))),R2)) ).
% dprod_def
tff(fact_4942_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> order_mono(A,B,F) ) ) ).
% strict_mono_mono
tff(fact_4943_merge__list_Opinduct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A0: list(list(A)),A1: list(list(A)),P: fun(list(list(A)),fun(list(list(A)),$o))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1))
=> ( ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),nil(list(A))) )
=> ( ! [L3: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A)))))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A)))) )
=> ( ! [La: list(A),Acc22: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),nil(list(A))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),nil(list(A))) ) )
=> ( ! [La: list(A),Acc22: list(list(A)),L3: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A)))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L3),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)))
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A)))) ) )
=> ( ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),cons(list(A),L12),aa(list(list(A)),list(list(A)),cons(list(A),L23),Ls))))
=> ( aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,aa(list(list(A)),list(list(A)),cons(list(A),merge(A,L12,L23)),Acc22)),Ls)
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,Acc22),aa(list(list(A)),list(list(A)),cons(list(A),L12),aa(list(list(A)),list(list(A)),cons(list(A),L23),Ls))) ) )
=> aa(list(list(A)),$o,aa(list(list(A)),fun(list(list(A)),$o),P,A0),A1) ) ) ) ) ) ) ) ).
% merge_list.pinduct
tff(fact_4944_dprodI,axiom,
! [B: $tType,A: $tType,M4: set(old_node(A,B)),M7: set(old_node(A,B)),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),N3: set(old_node(A,B)),N8: set(old_node(A,B)),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M4),M7),R2)
=> ( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N3),N8),S2)
=> member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,M4,N3)),old_Scons(A,B,M7,N8)),old_dprod(A,B,R2,S2)) ) ) ).
% dprodI
tff(fact_4945_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_strict_mono(A,B,F)
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
<=> ( X = Y ) ) ) ) ).
% strict_mono_eq
tff(fact_4946_dprodE,axiom,
! [B: $tType,A: $tType,C2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R2: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S2: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),C2,old_dprod(A,B,R2,S2))
=> ~ ! [X3: set(old_node(A,B)),Y2: set(old_node(A,B)),X8: set(old_node(A,B))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X3),X8),R2)
=> ! [Y6: set(old_node(A,B))] :
( member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y2),Y6),S2)
=> ( C2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,X3,Y2)),old_Scons(A,B,X8,Y6)) ) ) ) ) ).
% dprodE
tff(fact_4947_merge__list_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: product_prod(list(list(A)),list(list(A)))] :
( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
=> ( ! [L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))))
=> ( ! [La: list(A),Acc22: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),nil(list(A)))
=> ( ! [La: list(A),Acc22: list(list(A)),L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))))
=> ~ ! [Acc22: list(list(A)),L12: list(A),L23: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),cons(list(A),L12),aa(list(list(A)),list(list(A)),cons(list(A),L23),Ls))) ) ) ) ) ) ).
% merge_list.cases
tff(fact_4948_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_strict_mono(A,B,F)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).
% strict_mono_less_eq
tff(fact_4949_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_strict_mono(A,B,F)
=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)) ) ) ) ).
% strict_monoD
tff(fact_4950_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X3: A,Y2: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y2)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,Y2)) )
=> order_strict_mono(A,B,F) ) ) ).
% strict_monoI
tff(fact_4951_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_strict_mono(A,B,F)
<=> ! [X4: A,Y3: A] :
( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X4)),aa(A,B,F,Y3)) ) ) ) ).
% strict_mono_def
tff(fact_4952_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_strict_mono(A,B,F)
=> ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y))
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).
% strict_mono_less
tff(fact_4953_merge__list_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X,Xa) = Y )
=> ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),Xa))
=> ( ( ( X = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = nil(A) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))) ) ) )
=> ( ( ( X = nil(list(A)) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))) )
=> ( ( Y = L3 )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))))) ) ) )
=> ( ! [La: list(A),Acc22: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),nil(list(A)))) ) ) )
=> ( ! [La: list(A),Acc22: list(list(A))] :
( ( X = aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22) )
=> ! [L3: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),cons(list(A),L3),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22))) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La),Acc22)),aa(list(list(A)),list(list(A)),cons(list(A),L3),nil(list(A))))) ) ) )
=> ~ ! [L12: list(A),L23: list(A),Ls: list(list(A))] :
( ( Xa = aa(list(list(A)),list(list(A)),cons(list(A),L12),aa(list(list(A)),list(list(A)),cons(list(A),L23),Ls)) )
=> ( ( Y = merge_list(A,aa(list(list(A)),list(list(A)),cons(list(A),merge(A,L12,L23)),X),Ls) )
=> ~ aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),aa(list(list(A)),list(list(A)),cons(list(A),L12),aa(list(list(A)),list(list(A)),cons(list(A),L23),Ls)))) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
tff(fact_4954_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc23: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),cons(list(A),L1),aa(list(list(A)),list(list(A)),cons(list(A),L22),Ls2))))
=> ( merge_list(A,Acc23,aa(list(list(A)),list(list(A)),cons(list(A),L1),aa(list(list(A)),list(list(A)),cons(list(A),L22),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),cons(list(A),merge(A,L1,L22)),Acc23),Ls2) ) ) ) ).
% merge_list.psimps(5)
tff(fact_4955_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc23: list(list(A)),L: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23)),aa(list(list(A)),list(list(A)),cons(list(A),L),nil(list(A)))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23),aa(list(list(A)),list(list(A)),cons(list(A),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),cons(list(A),L),aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23))) ) ) ) ).
% merge_list.psimps(4)
tff(fact_4956_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))
=> ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ) ).
% merge_list.psimps(1)
tff(fact_4957_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),L),nil(list(A)))))
=> ( merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),cons(list(A),L),nil(list(A)))) = L ) ) ) ).
% merge_list.psimps(2)
tff(fact_4958_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La2: list(A),Acc23: list(list(A))] :
( aa(product_prod(list(list(A)),list(list(A))),$o,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23)),nil(list(A))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),cons(list(A),La2),Acc23)) ) ) ) ).
% merge_list.psimps(3)
tff(fact_4959_uprod__def,axiom,
! [B: $tType,A: $tType,Aa2: set(set(old_node(A,B))),Ba: set(set(old_node(A,B)))] : old_uprod(A,B,Aa2,Ba) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_sq(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Ba)),Aa2)) ).
% uprod_def
tff(fact_4960_heap__step__admissible,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,P: fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),X: E] : comple1908693960933563346ssible(fun(A,option(product_prod(B,product_prod(C,D)))),partial_fun_lub(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_lub(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),partial_fun_ord(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_ord(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o),aTP_Lamp_sr(fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o)),P),X)) ).
% heap_step_admissible
tff(fact_4961_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).
% last_zip
tff(fact_4962_partial__function__definitions_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),U2: fun(C,fun(B,A)),F4: fun(C,C),Ca: fun(fun(B,A),C),F: C,P: fun(fun(B,A),$o)] :
( partia7178651479351089652itions(A,Leq,Lub)
=> ( ! [X3: B] : comple7038119648293358887notone(fun(B,A),A,partial_fun_ord(A,A,B,Leq),Leq,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_ss(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),U2),F4),Ca),X3))
=> ( ( F = aa(fun(B,A),C,Ca,comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_st(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),U2),F4),Ca))) )
=> ( ! [F3: fun(B,A)] : aa(C,fun(B,A),U2,aa(fun(B,A),C,Ca,F3)) = F3
=> ( comple1908693960933563346ssible(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),P)
=> ( aa(fun(B,A),$o,P,aTP_Lamp_su(fun(set(A),A),fun(B,A),Lub))
=> ( ! [F3: C] :
( aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,F3))
=> aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,aa(C,C,F4,F3))) )
=> aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,F)) ) ) ) ) ) ) ) ).
% partial_function_definitions.fixp_induct_uc
tff(fact_4963_in__rel__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X: A,Y: B] :
( fun_in_rel(A,B,R,X,Y)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R) ) ).
% in_rel_def
tff(fact_4964_ref_Oset,axiom,
! [A: $tType,X: nat] : set_ref(A,ref2(A,X)) = bot_bot(set(A)) ).
% ref.set
tff(fact_4965_array_Oset,axiom,
! [A: $tType,X: nat] : set_array(A,array2(A,X)) = bot_bot(set(A)) ).
% array.set
tff(fact_4966_wfrecI,axiom,
! [B: $tType,A: $tType,X: A,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B)),G: fun(A,B)] :
( ! [Z3: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X),R)
=> wfrec_rel(A,B,R,F4,Z3,aa(A,B,G,Z3)) )
=> wfrec_rel(A,B,R,F4,X,aa(A,B,aa(fun(A,B),fun(A,B),F4,G),X)) ) ).
% wfrecI
tff(fact_4967_wfrec__rel_Ocases,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B)),A1: A,A22: B] :
( wfrec_rel(A,B,R,F4,A1,A22)
=> ~ ! [G3: fun(A,B)] :
( ( A22 = aa(A,B,aa(fun(A,B),fun(A,B),F4,G3),A1) )
=> ~ ! [Z5: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),A1),R)
=> wfrec_rel(A,B,R,F4,Z5,aa(A,B,G3,Z5)) ) ) ) ).
% wfrec_rel.cases
tff(fact_4968_wfrec__rel_Osimps,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F4: fun(fun(A,B),fun(A,B)),A1: A,A22: B] :
( wfrec_rel(A,B,R,F4,A1,A22)
<=> ? [X4: A,G5: fun(A,B)] :
( ( A1 = X4 )
& ( A22 = aa(A,B,aa(fun(A,B),fun(A,B),F4,G5),X4) )
& ! [Xa2: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R)
=> wfrec_rel(A,B,R,F4,Xa2,aa(A,B,G5,Xa2)) ) ) ) ).
% wfrec_rel.simps
tff(fact_4969_comp__fun__idem__sup,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> finite_comp_fun_idem(A,A,sup_sup(A)) ) ).
% comp_fun_idem_sup
tff(fact_4970_ATP_Olambda__1,axiom,
! [Uu: product_prod(int,int)] :
aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_ec(product_prod(int,int),product_prod(int,int)),Uu) = $ite(aa(product_prod(int,int),int,product_fst(int,int),Uu) = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).
% ATP.lambda_1
tff(fact_4971_ATP_Olambda__2,axiom,
! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_gm(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_4972_ATP_Olambda__3,axiom,
! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_gq(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).
% ATP.lambda_3
tff(fact_4973_ATP_Olambda__4,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_ef(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ).
% ATP.lambda_4
tff(fact_4974_ATP_Olambda__5,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ns(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_5
tff(fact_4975_ATP_Olambda__6,axiom,
! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_ha(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).
% ATP.lambda_6
tff(fact_4976_ATP_Olambda__7,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_gz(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_7
tff(fact_4977_ATP_Olambda__8,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_cc(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_8
tff(fact_4978_ATP_Olambda__9,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( aa(A,$o,aTP_Lamp_ml(A,$o),Uu)
<=> dvd_dvd(A,Uu,one_one(A)) ) ) ).
% ATP.lambda_9
tff(fact_4979_ATP_Olambda__10,axiom,
! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_er(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).
% ATP.lambda_10
tff(fact_4980_ATP_Olambda__11,axiom,
! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_nt(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).
% ATP.lambda_11
tff(fact_4981_ATP_Olambda__12,axiom,
! [B: $tType,Uu: B] : aa(B,set(B),aTP_Lamp_sh(B,set(B)),Uu) = aa(set(B),set(B),insert2(B,Uu),bot_bot(set(B))) ).
% ATP.lambda_12
tff(fact_4982_ATP_Olambda__13,axiom,
! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_ff(A,set(A)),Uu) = aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))) ).
% ATP.lambda_13
tff(fact_4983_ATP_Olambda__14,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_qo(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_qn(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_14
tff(fact_4984_ATP_Olambda__15,axiom,
! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_gp(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).
% ATP.lambda_15
tff(fact_4985_ATP_Olambda__16,axiom,
! [A: $tType,Uu: list(A)] :
( aa(list(A),$o,aTP_Lamp_oy(list(A),$o),Uu)
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_16
tff(fact_4986_ATP_Olambda__17,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_oe(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_od(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_17
tff(fact_4987_ATP_Olambda__18,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_ni(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_nh(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).
% ATP.lambda_18
tff(fact_4988_ATP_Olambda__19,axiom,
! [A: $tType,Uu: multiset(A)] : aa(multiset(A),set(A),aTP_Lamp_lu(multiset(A),set(A)),Uu) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lt(multiset(A),fun(A,$o),Uu)) ).
% ATP.lambda_19
tff(fact_4989_ATP_Olambda__20,axiom,
! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_fv(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_fu(nat,fun(nat,$o),Uu)) ).
% ATP.lambda_20
tff(fact_4990_ATP_Olambda__21,axiom,
! [A: $tType,Uu: fun(A,nat)] :
( aa(fun(A,nat),$o,aTP_Lamp_mh(fun(A,nat),$o),Uu)
<=> aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mg(fun(A,nat),fun(A,$o),Uu))) ) ).
% ATP.lambda_21
tff(fact_4991_ATP_Olambda__22,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_qx(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).
% ATP.lambda_22
tff(fact_4992_ATP_Olambda__23,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_qy(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).
% ATP.lambda_23
tff(fact_4993_ATP_Olambda__24,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dm(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).
% ATP.lambda_24
tff(fact_4994_ATP_Olambda__25,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dl(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).
% ATP.lambda_25
tff(fact_4995_ATP_Olambda__26,axiom,
! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_dv(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).
% ATP.lambda_26
tff(fact_4996_ATP_Olambda__27,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aTP_Lamp_hu(product_prod(A,A),$o),Uu)
<=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).
% ATP.lambda_27
tff(fact_4997_ATP_Olambda__28,axiom,
! [A: $tType,B: $tType,Uu: product_prod(product_prod($o,A),product_prod($o,B))] :
( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_hs(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu)
<=> ? [X4: A,Y3: B] : Uu = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$true),X4)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) ) ).
% ATP.lambda_28
tff(fact_4998_ATP_Olambda__29,axiom,
! [B: $tType,A: $tType,Uu: product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))] :
( aa(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o,aTP_Lamp_sg(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),$o),Uu)
<=> ? [F6: fun(nat,sum_sum(A,nat)),X4: sum_sum(B,nat),K4: nat] :
( ( Uu = aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(fun(nat,sum_sum(A,nat)),fun(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),F6),X4) )
& ( aa(nat,sum_sum(A,nat),F6,K4) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ) ).
% ATP.lambda_29
tff(fact_4999_ATP_Olambda__30,axiom,
! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_sj(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).
% ATP.lambda_30
tff(fact_5000_ATP_Olambda__31,axiom,
! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_si(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).
% ATP.lambda_31
tff(fact_5001_ATP_Olambda__32,axiom,
! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_rb(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_qz(nat,fun(num,option(num)),Uua),aTP_Lamp_ra(nat,fun(num,option(num)),Uua),Uu) ).
% ATP.lambda_32
tff(fact_5002_ATP_Olambda__33,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] :
aa(nat,A,aTP_Lamp_bz(nat,fun(nat,A),Uu),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).
% ATP.lambda_33
tff(fact_5003_ATP_Olambda__34,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_ou(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),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_34
tff(fact_5004_ATP_Olambda__35,axiom,
! [Uu: pred(product_unit),Uua: product_prod(code_natural,code_natural)] :
aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_rd(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),Uu),Uua) = $ite(aa(product_unit,$o,eval(product_unit,Uu),product_Unity),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),bot_bot(pred(product_unit))),Uua),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)),Uua)) ).
% ATP.lambda_35
tff(fact_5005_ATP_Olambda__36,axiom,
! [Uu: int,Uua: int] :
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ea(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).
% ATP.lambda_36
tff(fact_5006_ATP_Olambda__37,axiom,
! [Uu: int,Uua: int] :
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_el(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uua = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Uu),Uua)) ).
% ATP.lambda_37
tff(fact_5007_ATP_Olambda__38,axiom,
! [A: $tType,Uu: set(fun(A,nat)),Uua: A] :
aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_mf(set(fun(A,nat)),fun(A,nat)),Uu),Uua) = $ite(Uu = bot_bot(set(fun(A,nat))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_lx(A,fun(fun(A,nat),nat),Uua)),Uu))) ).
% ATP.lambda_38
tff(fact_5008_ATP_Olambda__39,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] :
aa(nat,A,aTP_Lamp_ca(nat,fun(nat,A),Uu),Uua) = $ite(~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).
% ATP.lambda_39
tff(fact_5009_ATP_Olambda__40,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_kr(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),top_top(A)) ) ).
% ATP.lambda_40
tff(fact_5010_ATP_Olambda__41,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_kn(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,compow(fun(A,A),Uua,Uu),bot_bot(A)) ) ).
% ATP.lambda_41
tff(fact_5011_ATP_Olambda__42,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_rc(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_rb(num,fun(nat,option(num)),Uua),Uu) ).
% ATP.lambda_42
tff(fact_5012_ATP_Olambda__43,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_qz(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_qy(num,option(num)),bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_43
tff(fact_5013_ATP_Olambda__44,axiom,
! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_eg(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).
% ATP.lambda_44
tff(fact_5014_ATP_Olambda__45,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dc(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua))) ) ).
% ATP.lambda_45
tff(fact_5015_ATP_Olambda__46,axiom,
! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_eh(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).
% ATP.lambda_46
tff(fact_5016_ATP_Olambda__47,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_sl(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Uu)),old_In1(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_47
tff(fact_5017_ATP_Olambda__48,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_sk(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,Uu)),old_In0(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_48
tff(fact_5018_ATP_Olambda__49,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_de(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua))) ) ).
% ATP.lambda_49
tff(fact_5019_ATP_Olambda__50,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_kj(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),insert2(set(A),image(A,A,Uu,aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).
% ATP.lambda_50
tff(fact_5020_ATP_Olambda__51,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_co(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_51
tff(fact_5021_ATP_Olambda__52,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_52
tff(fact_5022_ATP_Olambda__53,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cu(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_53
tff(fact_5023_ATP_Olambda__54,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_rq(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_54
tff(fact_5024_ATP_Olambda__55,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
( aa(set(set(A)),$o,aTP_Lamp_gn(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
<=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
& ( Uua != bot_bot(set(set(A))) ) ) ) ).
% ATP.lambda_55
tff(fact_5025_ATP_Olambda__56,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_ke(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua)
<=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Uu),Uua)
& ! [A5: A,B4: A,C4: A] :
( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),Uua)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),C4),Uu) )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4),Uu) ) ) ) ).
% ATP.lambda_56
tff(fact_5026_ATP_Olambda__57,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_sp(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),insert2(set(old_node(A,B)),old_Scons(A,B,Uu,Uua)),bot_bot(set(set(old_node(A,B))))) ).
% ATP.lambda_57
tff(fact_5027_ATP_Olambda__58,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_fw(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).
% ATP.lambda_58
tff(fact_5028_ATP_Olambda__59,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_ie(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).
% ATP.lambda_59
tff(fact_5029_ATP_Olambda__60,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_mk(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
<=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),Uua),bot_bot(set(product_prod(A,B))))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).
% ATP.lambda_60
tff(fact_5030_ATP_Olambda__61,axiom,
! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_iy(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
<=> ( ( Uua = bot_bot(set(A)) )
| ? [A8: set(A),A5: A] :
( ( Uua = aa(set(A),set(A),insert2(A,A5),A8) )
& aa(set(A),$o,Uu,A8) ) ) ) ).
% ATP.lambda_61
tff(fact_5031_ATP_Olambda__62,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
( aa(fun(A,B),$o,aTP_Lamp_mu(set(B),fun(fun(A,B),$o),Uu),Uua)
<=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu) ) ).
% ATP.lambda_62
tff(fact_5032_ATP_Olambda__63,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_kk(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))),Uu) ).
% ATP.lambda_63
tff(fact_5033_ATP_Olambda__64,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_az(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,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),aa(num,num,bit0,one2))),Uua)))) ) ).
% ATP.lambda_64
tff(fact_5034_ATP_Olambda__65,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cg(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_65
tff(fact_5035_ATP_Olambda__66,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_kt(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ( aa(A,$o,Uu,Uua)
& ! [Y3: A] :
( aa(A,$o,Uu,Y3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3) ) ) ) ) ).
% ATP.lambda_66
tff(fact_5036_ATP_Olambda__67,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: fun(A,$o),Uua: A] :
( aa(A,$o,aTP_Lamp_kv(fun(A,$o),fun(A,$o),Uu),Uua)
<=> ( aa(A,$o,Uu,Uua)
& ! [Y3: A] :
( aa(A,$o,Uu,Y3)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).
% ATP.lambda_67
tff(fact_5037_ATP_Olambda__68,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_ji(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).
% ATP.lambda_68
tff(fact_5038_ATP_Olambda__69,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_km(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).
% ATP.lambda_69
tff(fact_5039_ATP_Olambda__70,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_fo(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),insert2(A,aa(B,A,Uu,Uua)),bot_bot(set(A))) ).
% ATP.lambda_70
tff(fact_5040_ATP_Olambda__71,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: A] :
( aa(A,$o,aTP_Lamp_bw(fun(A,B),fun(A,$o),Uu),Uua)
<=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).
% ATP.lambda_71
tff(fact_5041_ATP_Olambda__72,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ry(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rw(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ).
% ATP.lambda_72
tff(fact_5042_ATP_Olambda__73,axiom,
! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_be(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
& ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uua) ) ) ).
% ATP.lambda_73
tff(fact_5043_ATP_Olambda__74,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_ag(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,in_range,Uua)
& ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uua) ) ) ).
% ATP.lambda_74
tff(fact_5044_ATP_Olambda__75,axiom,
! [A: $tType,Uu: pred(A),Uua: seq(A)] : aa(seq(A),set(A),aa(pred(A),fun(seq(A),set(A)),aTP_Lamp_ri(pred(A),fun(seq(A),set(A))),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_of_pred(A,Uu)),set_of_seq(A,Uua)) ).
% ATP.lambda_75
tff(fact_5045_ATP_Olambda__76,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_cx(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_76
tff(fact_5046_ATP_Olambda__77,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_ox(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).
% ATP.lambda_77
tff(fact_5047_ATP_Olambda__78,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(set(product_prod(A,A))),aTP_Lamp_mj(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),Uu),Uua) = image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),insert2(set(product_prod(A,A)),Uu),bot_bot(set(set(product_prod(A,A)))))) ).
% ATP.lambda_78
tff(fact_5048_ATP_Olambda__79,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_df(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).
% ATP.lambda_79
tff(fact_5049_ATP_Olambda__80,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_dj(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).
% ATP.lambda_80
tff(fact_5050_ATP_Olambda__81,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bl(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),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_81
tff(fact_5051_ATP_Olambda__82,axiom,
! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_gr(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image2(set(A),set(A),insert2(A,Uu)),Uua)) ).
% ATP.lambda_82
tff(fact_5052_ATP_Olambda__83,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_rj(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),zero_zero(nat))) ).
% ATP.lambda_83
tff(fact_5053_ATP_Olambda__84,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_sd(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_84
tff(fact_5054_ATP_Olambda__85,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_rk(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_85
tff(fact_5055_ATP_Olambda__86,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_hm(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = image(B,A,Uu,aa(set(B),set(B),insert2(B,Uua),bot_bot(set(B)))) ).
% ATP.lambda_86
tff(fact_5056_ATP_Olambda__87,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_ja(fun(A,B),fun(B,set(A)),Uu),Uua) = vimage(A,B,Uu,aa(set(B),set(B),insert2(B,Uua),bot_bot(set(B)))) ).
% ATP.lambda_87
tff(fact_5057_ATP_Olambda__88,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_ku(set(A),fun(A,$o),Uu),Uua)
<=> ( Uu = aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))) ) ) ).
% ATP.lambda_88
tff(fact_5058_ATP_Olambda__89,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_pl(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_89
tff(fact_5059_ATP_Olambda__90,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_kl(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).
% ATP.lambda_90
tff(fact_5060_ATP_Olambda__91,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_qd(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),the2(B,aa(A,option(B),Uu,Uua))) ).
% ATP.lambda_91
tff(fact_5061_ATP_Olambda__92,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_at(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_92
tff(fact_5062_ATP_Olambda__93,axiom,
! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_op(list(set(A)),fun(set(A),$o),Uu),Uua)
<=> member(set(A),Uua,set2(set(A),Uu)) ) ).
% ATP.lambda_93
tff(fact_5063_ATP_Olambda__94,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( aa(A,$o,aTP_Lamp_pb(list(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,set2(A,Uu)) ) ).
% ATP.lambda_94
tff(fact_5064_ATP_Olambda__95,axiom,
! [A: $tType,Uu: A,Uua: pred(A)] : aa(pred(A),set(A),aa(A,fun(pred(A),set(A)),aTP_Lamp_rh(A,fun(pred(A),set(A))),Uu),Uua) = aa(set(A),set(A),insert2(A,Uu),set_of_pred(A,Uua)) ).
% ATP.lambda_95
tff(fact_5065_ATP_Olambda__96,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ba(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_96
tff(fact_5066_ATP_Olambda__97,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jg(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_97
tff(fact_5067_ATP_Olambda__98,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ax(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_98
tff(fact_5068_ATP_Olambda__99,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_qf(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_99
tff(fact_5069_ATP_Olambda__100,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_nf(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_100
tff(fact_5070_ATP_Olambda__101,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_po(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).
% ATP.lambda_101
tff(fact_5071_ATP_Olambda__102,axiom,
! [Uu: nat,Uua: nat] :
( aa(nat,$o,aTP_Lamp_ct(nat,fun(nat,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).
% ATP.lambda_102
tff(fact_5072_ATP_Olambda__103,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bb(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_103
tff(fact_5073_ATP_Olambda__104,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jh(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_104
tff(fact_5074_ATP_Olambda__105,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ay(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_105
tff(fact_5075_ATP_Olambda__106,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_qg(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_106
tff(fact_5076_ATP_Olambda__107,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ng(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_107
tff(fact_5077_ATP_Olambda__108,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pp(A,fun(A,$o)),Uu),Uua)
<=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).
% ATP.lambda_108
tff(fact_5078_ATP_Olambda__109,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_109
tff(fact_5079_ATP_Olambda__110,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_au(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).
% ATP.lambda_110
tff(fact_5080_ATP_Olambda__111,axiom,
! [A: $tType,Uu: A,Uua: multiset(A)] : aa(multiset(A),nat,aTP_Lamp_lv(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ).
% ATP.lambda_111
tff(fact_5081_ATP_Olambda__112,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ge(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).
% ATP.lambda_112
tff(fact_5082_ATP_Olambda__113,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_gk(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).
% ATP.lambda_113
tff(fact_5083_ATP_Olambda__114,axiom,
! [Uu: nat,Uua: nat] :
( aa(nat,$o,aTP_Lamp_fu(nat,fun(nat,$o),Uu),Uua)
<=> dvd_dvd(nat,Uua,Uu) ) ).
% ATP.lambda_114
tff(fact_5084_ATP_Olambda__115,axiom,
! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_es(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).
% ATP.lambda_115
tff(fact_5085_ATP_Olambda__116,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_he(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_116
tff(fact_5086_ATP_Olambda__117,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_pj(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_117
tff(fact_5087_ATP_Olambda__118,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_hd(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).
% ATP.lambda_118
tff(fact_5088_ATP_Olambda__119,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_ed(set(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,Uu) ) ) ).
% ATP.lambda_119
tff(fact_5089_ATP_Olambda__120,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu),Uua)
<=> member(A,Uua,Uu) ) ).
% ATP.lambda_120
tff(fact_5090_ATP_Olambda__121,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_ae(A,fun(A,$o),Uu),Uua)
<=> ( Uua = Uu ) ) ).
% ATP.lambda_121
tff(fact_5091_ATP_Olambda__122,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sa(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)) ).
% ATP.lambda_122
tff(fact_5092_ATP_Olambda__123,axiom,
! [A: $tType,Uu: A,Uua: product_unit] : aa(product_unit,seq(A),aTP_Lamp_rg(A,fun(product_unit,seq(A)),Uu),Uua) = insert(A,Uu,bot_bot(pred(A))) ).
% ATP.lambda_123
tff(fact_5093_ATP_Olambda__124,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_jc(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))) ).
% ATP.lambda_124
tff(fact_5094_ATP_Olambda__125,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_se(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_sd(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_125
tff(fact_5095_ATP_Olambda__126,axiom,
! [A: $tType,Uu: multiset(A),Uua: A] :
( aa(A,$o,aTP_Lamp_lt(multiset(A),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua)) ) ).
% ATP.lambda_126
tff(fact_5096_ATP_Olambda__127,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] :
( aa(A,$o,aTP_Lamp_mg(fun(A,nat),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).
% ATP.lambda_127
tff(fact_5097_ATP_Olambda__128,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: A] :
( aa(A,$o,aTP_Lamp_lw(set(multiset(A)),fun(A,$o),Uu),Uua)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_lv(A,fun(multiset(A),nat),Uua)),Uu))) ) ).
% ATP.lambda_128
tff(fact_5098_ATP_Olambda__129,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ap(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_129
tff(fact_5099_ATP_Olambda__130,axiom,
! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_mx(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).
% ATP.lambda_130
tff(fact_5100_ATP_Olambda__131,axiom,
! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_lx(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).
% ATP.lambda_131
tff(fact_5101_ATP_Olambda__132,axiom,
! [B: $tType,A: $tType,Uu: fun(set(A),A),Uua: B] : aa(B,A,aTP_Lamp_su(fun(set(A),A),fun(B,A),Uu),Uua) = aa(set(A),A,Uu,bot_bot(set(A))) ).
% ATP.lambda_132
tff(fact_5102_ATP_Olambda__133,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_qc(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).
% ATP.lambda_133
tff(fact_5103_ATP_Olambda__134,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ra(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_134
tff(fact_5104_ATP_Olambda__135,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option($o)),aTP_Lamp_qa(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Uu),Uua) = aa(fun(A,fun(list(A),option($o))),fun(product_prod(A,list(A)),option($o)),product_case_prod(A,list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_pz(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua)) ).
% ATP.lambda_135
tff(fact_5105_ATP_Olambda__136,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_pv(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_pu(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).
% ATP.lambda_136
tff(fact_5106_ATP_Olambda__137,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ep(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_eo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_137
tff(fact_5107_ATP_Olambda__138,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_en(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_em(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_138
tff(fact_5108_ATP_Olambda__139,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ej(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ei(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_139
tff(fact_5109_ATP_Olambda__140,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_rm(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_140
tff(fact_5110_ATP_Olambda__141,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_if(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_ie(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).
% ATP.lambda_141
tff(fact_5111_ATP_Olambda__142,axiom,
! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_sq(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_sp(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua)),Uu)) ).
% ATP.lambda_142
tff(fact_5112_ATP_Olambda__143,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: A] : aa(A,nat,aTP_Lamp_ly(set(multiset(A)),fun(A,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_lv(A,fun(multiset(A),nat),Uua)),Uu)) ).
% ATP.lambda_143
tff(fact_5113_ATP_Olambda__144,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_ov(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_ou(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua),Uu)) ).
% ATP.lambda_144
tff(fact_5114_ATP_Olambda__145,axiom,
! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_qn(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_145
tff(fact_5115_ATP_Olambda__146,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_aa(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_146
tff(fact_5116_ATP_Olambda__147,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bk(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_147
tff(fact_5117_ATP_Olambda__148,axiom,
! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_qr(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth(A,Uu),code_nat_of_natural(Uua))) ).
% ATP.lambda_148
tff(fact_5118_ATP_Olambda__149,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( aa(A,$o,aTP_Lamp_pc(list(A),fun(A,$o),Uu),Uua)
<=> ~ member(A,Uua,set2(A,Uu)) ) ).
% ATP.lambda_149
tff(fact_5119_ATP_Olambda__150,axiom,
! [A: $tType] :
( sup(A)
=> ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_pw(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).
% ATP.lambda_150
tff(fact_5120_ATP_Olambda__151,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_qp(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_151
tff(fact_5121_ATP_Olambda__152,axiom,
! [A: $tType,Uu: pred(A),Uua: pred(A)] : aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qs(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uu),Uua)) ).
% ATP.lambda_152
tff(fact_5122_ATP_Olambda__153,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_nh(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_153
tff(fact_5123_ATP_Olambda__154,axiom,
! [A: $tType,Uu: list(product_prod(code_natural,A)),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_qq(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),pick(A,Uu,Uua)) ).
% ATP.lambda_154
tff(fact_5124_ATP_Olambda__155,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_fx(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).
% ATP.lambda_155
tff(fact_5125_ATP_Olambda__156,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_ic(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_156
tff(fact_5126_ATP_Olambda__157,axiom,
! [A: $tType,Uu: A,Uua: A] :
( aa(A,$o,aTP_Lamp_pa(A,fun(A,$o),Uu),Uua)
<=> ( Uua != Uu ) ) ).
% ATP.lambda_157
tff(fact_5127_ATP_Olambda__158,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_iu(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A)))) ).
% ATP.lambda_158
tff(fact_5128_ATP_Olambda__159,axiom,
! [Uu: nat,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rn(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),Uu))) ).
% ATP.lambda_159
tff(fact_5129_ATP_Olambda__160,axiom,
! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_nd(fun(B,set(C)),fun(B,set(product_prod(C,C))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Uu,Uua)) ).
% ATP.lambda_160
tff(fact_5130_ATP_Olambda__161,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_ne(fun(A,set(B)),fun(A,set(product_prod(B,B))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_161
tff(fact_5131_ATP_Olambda__162,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ar(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_162
tff(fact_5132_ATP_Olambda__163,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_pr(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).
% ATP.lambda_163
tff(fact_5133_ATP_Olambda__164,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_ol(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_164
tff(fact_5134_ATP_Olambda__165,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_py(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).
% ATP.lambda_165
tff(fact_5135_ATP_Olambda__166,axiom,
! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_nj(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).
% ATP.lambda_166
tff(fact_5136_ATP_Olambda__167,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_nk(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_167
tff(fact_5137_ATP_Olambda__168,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_et(fun(A,set(B)),fun(A,nat),Uu),Uua) = finite_card(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_168
tff(fact_5138_ATP_Olambda__169,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_gl(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert2(B,aa(A,B,Uu,Uua)) ).
% ATP.lambda_169
tff(fact_5139_ATP_Olambda__170,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_or(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).
% ATP.lambda_170
tff(fact_5140_ATP_Olambda__171,axiom,
! [A: $tType,Uu: fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_hi(fun(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu,X4),Uua) ) ).
% ATP.lambda_171
tff(fact_5141_ATP_Olambda__172,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_jm(set(A),fun(A,$o),Uu),Uua)
<=> ! [X4: A] :
( member(A,X4,Uu)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X4) ) ) ) ).
% ATP.lambda_172
tff(fact_5142_ATP_Olambda__173,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( aa(A,$o,aTP_Lamp_jn(set(A),fun(A,$o),Uu),Uua)
<=> ! [X4: A] :
( member(A,X4,Uu)
=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Uua) ) ) ) ).
% ATP.lambda_173
tff(fact_5143_ATP_Olambda__174,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
( aa(multiset(A),$o,aTP_Lamp_lq(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,Uu)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Uua),X4) ) ) ).
% ATP.lambda_174
tff(fact_5144_ATP_Olambda__175,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
( aa(multiset(A),$o,aTP_Lamp_lp(set(multiset(A)),fun(multiset(A),$o),Uu),Uua)
<=> ! [X4: multiset(A)] :
( member(multiset(A),X4,Uu)
=> aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),Uua) ) ) ).
% ATP.lambda_175
tff(fact_5145_ATP_Olambda__176,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
( aa(set(A),$o,aTP_Lamp_jp(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
<=> ! [X4: A] :
( member(A,X4,Uua)
=> ! [Xa2: A] :
( member(A,Xa2,Uua)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu) ) ) ) ) ).
% ATP.lambda_176
tff(fact_5146_ATP_Olambda__177,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
( aa(A,$o,aTP_Lamp_mi(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
<=> ? [Y3: B] : member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y3),Uu) ) ).
% ATP.lambda_177
tff(fact_5147_ATP_Olambda__178,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_hl(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uu),Uua)
<=> ? [X4: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,aa(A,assn,Uu,X4)),Uua) ) ).
% ATP.lambda_178
tff(fact_5148_ATP_Olambda__179,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_pt(A,fun(product_prod(A,B),$o),Uu),Uua)
<=> ? [V4: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V4) ) ).
% ATP.lambda_179
tff(fact_5149_ATP_Olambda__180,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod($o,A),product_prod($o,B))] :
( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_hr(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu),Uua)
<=> ? [X4: A,Y3: B] :
( ( Uua = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$false),X4)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y3)) )
& member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),Uu) ) ) ).
% ATP.lambda_180
tff(fact_5150_ATP_Olambda__181,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aTP_Lamp_qb(fun(A,option(B)),fun(product_prod(A,B),$o),Uu),Uua)
<=> ? [A5: A,B4: B] :
( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
& ( aa(A,option(B),Uu,A5) = aa(B,option(B),some(B),B4) ) ) ) ).
% ATP.lambda_181
tff(fact_5151_ATP_Olambda__182,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_ju(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
<=> ? [X7: set(A),Y9: set(A)] :
( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X7),Y9) )
& ( X7 != bot_bot(set(A)) )
& ! [X4: A] :
( member(A,X4,Y9)
=> ? [Xa2: A] :
( member(A,Xa2,X7)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu) ) ) ) ) ).
% ATP.lambda_182
tff(fact_5152_ATP_Olambda__183,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_mw(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).
% ATP.lambda_183
tff(fact_5153_ATP_Olambda__184,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_na(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).
% ATP.lambda_184
tff(fact_5154_ATP_Olambda__185,axiom,
! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_mv(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_185
tff(fact_5155_ATP_Olambda__186,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_my(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_186
tff(fact_5156_ATP_Olambda__187,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_mz(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_187
tff(fact_5157_ATP_Olambda__188,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_oc(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = insert2(A,Uu) ).
% ATP.lambda_188
tff(fact_5158_ATP_Olambda__189,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_eb(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_189
tff(fact_5159_ATP_Olambda__190,axiom,
! [Uu: num,Uua: nat,Uub: nat] :
aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_dz(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_190
tff(fact_5160_ATP_Olambda__191,axiom,
! [Uu: num,Uua: int,Uub: int] :
aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dh(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_191
tff(fact_5161_ATP_Olambda__192,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_dg(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).
% ATP.lambda_192
tff(fact_5162_ATP_Olambda__193,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_bv(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).
% ATP.lambda_193
tff(fact_5163_ATP_Olambda__194,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bm(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).
% ATP.lambda_194
tff(fact_5164_ATP_Olambda__195,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bn(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).
% ATP.lambda_195
tff(fact_5165_ATP_Olambda__196,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_lz(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,zero_zero(A)) ) ).
% ATP.lambda_196
tff(fact_5166_ATP_Olambda__197,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_me(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uu = Uub,Uua,one_one(A)) ) ).
% ATP.lambda_197
tff(fact_5167_ATP_Olambda__198,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: B,Uua: A,Uub: B] :
aa(B,A,aa(A,fun(B,A),aTP_Lamp_ma(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,zero_zero(A)) ) ).
% ATP.lambda_198
tff(fact_5168_ATP_Olambda__199,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_md(B,fun(A,fun(B,A)),Uu),Uua),Uub) = $ite(Uub = Uu,Uua,one_one(A)) ) ).
% ATP.lambda_199
tff(fact_5169_ATP_Olambda__200,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_fs(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).
% ATP.lambda_200
tff(fact_5170_ATP_Olambda__201,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_gs(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).
% ATP.lambda_201
tff(fact_5171_ATP_Olambda__202,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_fr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).
% ATP.lambda_202
tff(fact_5172_ATP_Olambda__203,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_go(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),insert2(A,Uua),Uub),Uub) ).
% ATP.lambda_203
tff(fact_5173_ATP_Olambda__204,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_oz(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).
% ATP.lambda_204
tff(fact_5174_ATP_Olambda__205,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_bu(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).
% ATP.lambda_205
tff(fact_5175_ATP_Olambda__206,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_id(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_ic(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).
% ATP.lambda_206
tff(fact_5176_ATP_Olambda__207,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_gt(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_fx(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).
% ATP.lambda_207
tff(fact_5177_ATP_Olambda__208,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_np(fun(A,fun(A,$o)),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_208
tff(fact_5178_ATP_Olambda__209,axiom,
! [A: $tType] :
( sup(A)
=> ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_px(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_pw(A,fun(A,option(A)),Uub),Uua) ) ).
% ATP.lambda_209
tff(fact_5179_ATP_Olambda__210,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ph(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uua) ) ).
% ATP.lambda_210
tff(fact_5180_ATP_Olambda__211,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_fp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
$ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
Uub = one_one(code_integer)) ).
% ATP.lambda_211
tff(fact_5181_ATP_Olambda__212,axiom,
! [Uu: product_prod(code_natural,code_natural),Uua: code_natural,Uub: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_qm(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_ql(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uua),Uub)),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,code_natural),product_snd(code_natural,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,Uu))) ).
% ATP.lambda_212
tff(fact_5182_ATP_Olambda__213,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: pred(B),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qw(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aa(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aTP_Lamp_qv(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),Uu),Uua)),split_seed(Uub)) ).
% ATP.lambda_213
tff(fact_5183_ATP_Olambda__214,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dx(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_214
tff(fact_5184_ATP_Olambda__215,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ds(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_215
tff(fact_5185_ATP_Olambda__216,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dq(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_216
tff(fact_5186_ATP_Olambda__217,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_do(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dn(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_217
tff(fact_5187_ATP_Olambda__218,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_rp(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_218
tff(fact_5188_ATP_Olambda__219,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_ch(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_219
tff(fact_5189_ATP_Olambda__220,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_nr(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),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) )
& member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu)) ) ) ) ).
% ATP.lambda_220
tff(fact_5190_ATP_Olambda__221,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_nx(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),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),cons(A,Y3),Ys5)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu) ) ) ) ).
% ATP.lambda_221
tff(fact_5191_ATP_Olambda__222,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_js(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,field2(A,Uu))
& ! [X4: A] :
( member(A,X4,Uua)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu) ) ) ) ).
% ATP.lambda_222
tff(fact_5192_ATP_Olambda__223,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jt(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,field2(A,Uu))
& ! [X4: A] :
( member(A,X4,Uua)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub),Uu) ) ) ) ).
% ATP.lambda_223
tff(fact_5193_ATP_Olambda__224,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jr(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,field2(A,Uu))
& ! [X4: A] :
( member(A,X4,Uua)
=> ( ( Uub != X4 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu) ) ) ) ) ).
% ATP.lambda_224
tff(fact_5194_ATP_Olambda__225,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_jx(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,field2(A,Uu))
& ! [X4: A] :
( member(A,X4,Uua)
=> ( ( Uub != X4 )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub),Uu) ) ) ) ) ).
% ATP.lambda_225
tff(fact_5195_ATP_Olambda__226,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_jl(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).
% ATP.lambda_226
tff(fact_5196_ATP_Olambda__227,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_av(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_227
tff(fact_5197_ATP_Olambda__228,axiom,
! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_aw(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uu),Uub)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),$o),rep_assn,Uua),Uub) ) ) ).
% ATP.lambda_228
tff(fact_5198_ATP_Olambda__229,axiom,
! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jq(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uua,Uu)
& member(A,Uub,Uu) ) ) ).
% ATP.lambda_229
tff(fact_5199_ATP_Olambda__230,axiom,
! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ot(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uu,Uub)),aa(A,nat,Uua,Uub)) ).
% ATP.lambda_230
tff(fact_5200_ATP_Olambda__231,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_ow(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uua,Uub)),aa(A,nat,Uu,Uub)) ).
% ATP.lambda_231
tff(fact_5201_ATP_Olambda__232,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bt(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_232
tff(fact_5202_ATP_Olambda__233,axiom,
! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uu = Uub )
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_233
tff(fact_5203_ATP_Olambda__234,axiom,
! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub = Uu )
& aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_234
tff(fact_5204_ATP_Olambda__235,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_io(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).
% ATP.lambda_235
tff(fact_5205_ATP_Olambda__236,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_di(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).
% ATP.lambda_236
tff(fact_5206_ATP_Olambda__237,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_br(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
<=> ( member(A,Uub,Uu)
& ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).
% ATP.lambda_237
tff(fact_5207_ATP_Olambda__238,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ek(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
<=> ( member(B,Uub,Uua)
& ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_238
tff(fact_5208_ATP_Olambda__239,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_bj(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu)),Uua) ) ).
% ATP.lambda_239
tff(fact_5209_ATP_Olambda__240,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_bh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_240
tff(fact_5210_ATP_Olambda__241,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_bi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).
% ATP.lambda_241
tff(fact_5211_ATP_Olambda__242,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_bg(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_242
tff(fact_5212_ATP_Olambda__243,axiom,
! [B: $tType,C: $tType,Uu: set(product_prod(C,B)),Uua: C,Uub: B] :
( aa(B,$o,aa(C,fun(B,$o),aTP_Lamp_qk(set(product_prod(C,B)),fun(C,fun(B,$o)),Uu),Uua),Uub)
<=> member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uua),Uub),Uu) ) ).
% ATP.lambda_243
tff(fact_5213_ATP_Olambda__244,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
( aa(C,$o,aa(B,fun(C,$o),aTP_Lamp_qi(set(product_prod(B,C)),fun(B,fun(C,$o)),Uu),Uua),Uub)
<=> member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub),Uu) ) ).
% ATP.lambda_244
tff(fact_5214_ATP_Olambda__245,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_hj(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).
% ATP.lambda_245
tff(fact_5215_ATP_Olambda__246,axiom,
! [C: $tType,A: $tType,Uu: set(product_prod(A,C)),Uua: A,Uub: C] :
( aa(C,$o,aa(A,fun(C,$o),aTP_Lamp_qj(set(product_prod(A,C)),fun(A,fun(C,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uua),Uub),Uu) ) ).
% ATP.lambda_246
tff(fact_5216_ATP_Olambda__247,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_ak(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).
% ATP.lambda_247
tff(fact_5217_ATP_Olambda__248,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_hf(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ).
% ATP.lambda_248
tff(fact_5218_ATP_Olambda__249,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_jy(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua),Uu) ) ).
% ATP.lambda_249
tff(fact_5219_ATP_Olambda__250,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kd(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ).
% ATP.lambda_250
tff(fact_5220_ATP_Olambda__251,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_bq(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_251
tff(fact_5221_ATP_Olambda__252,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_eq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_252
tff(fact_5222_ATP_Olambda__253,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_cp(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_253
tff(fact_5223_ATP_Olambda__254,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oi(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_254
tff(fact_5224_ATP_Olambda__255,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_fl(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_255
tff(fact_5225_ATP_Olambda__256,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_ij(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_256
tff(fact_5226_ATP_Olambda__257,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ez(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu,Uub)),aa(A,assn,Uua,Uub)) ).
% ATP.lambda_257
tff(fact_5227_ATP_Olambda__258,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_it(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_258
tff(fact_5228_ATP_Olambda__259,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ip(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_259
tff(fact_5229_ATP_Olambda__260,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oh(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_260
tff(fact_5230_ATP_Olambda__261,axiom,
! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_lr(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).
% ATP.lambda_261
tff(fact_5231_ATP_Olambda__262,axiom,
! [B: $tType,A: $tType,Uu: fun(A,multiset(B)),Uua: fun(A,multiset(B)),Uub: A] : aa(A,multiset(B),aa(fun(A,multiset(B)),fun(A,multiset(B)),aTP_Lamp_ls(fun(A,multiset(B)),fun(fun(A,multiset(B)),fun(A,multiset(B))),Uu),Uua),Uub) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),aa(A,multiset(B),Uu,Uub)),aa(A,multiset(B),Uua,Uub)) ).
% ATP.lambda_262
tff(fact_5232_ATP_Olambda__263,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_iv(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_263
tff(fact_5233_ATP_Olambda__264,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_nl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).
% ATP.lambda_264
tff(fact_5234_ATP_Olambda__265,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ac(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
| aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).
% ATP.lambda_265
tff(fact_5235_ATP_Olambda__266,axiom,
! [Uu: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uua: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Uub: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o),aTP_Lamp_ad(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),$o)),Uu),Uua),Uub)
<=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uu,Uub)
& aa(product_prod(heap_ext(product_unit),set(nat)),$o,Uua,Uub) ) ) ).
% ATP.lambda_266
tff(fact_5236_ATP_Olambda__267,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_cf(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).
% ATP.lambda_267
tff(fact_5237_ATP_Olambda__268,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_kw(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,$o,Uua,Uub)
& ! [Y3: A] :
( aa(A,$o,Uua,Y3)
=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y3) ) ) ) ).
% ATP.lambda_268
tff(fact_5238_ATP_Olambda__269,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_ey(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_269
tff(fact_5239_ATP_Olambda__270,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_pd(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).
% ATP.lambda_270
tff(fact_5240_ATP_Olambda__271,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_cj(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_271
tff(fact_5241_ATP_Olambda__272,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_ew(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_272
tff(fact_5242_ATP_Olambda__273,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_gb(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).
% ATP.lambda_273
tff(fact_5243_ATP_Olambda__274,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_fe(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).
% ATP.lambda_274
tff(fact_5244_ATP_Olambda__275,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_gd(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_275
tff(fact_5245_ATP_Olambda__276,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_ex(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_276
tff(fact_5246_ATP_Olambda__277,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ga(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).
% ATP.lambda_277
tff(fact_5247_ATP_Olambda__278,axiom,
! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_fa(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_278
tff(fact_5248_ATP_Olambda__279,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_gi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).
% ATP.lambda_279
tff(fact_5249_ATP_Olambda__280,axiom,
! [A: $tType,B: $tType] :
( linord4140545234300271783up_add(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aq(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).
% ATP.lambda_280
tff(fact_5250_ATP_Olambda__281,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_li(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).
% ATP.lambda_281
tff(fact_5251_ATP_Olambda__282,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_pe(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_282
tff(fact_5252_ATP_Olambda__283,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ).
% ATP.lambda_283
tff(fact_5253_ATP_Olambda__284,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_rz(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rv(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,map(nat,A,Uua,upt(zero_zero(nat),Uu)),Uub)) ) ).
% ATP.lambda_284
tff(fact_5254_ATP_Olambda__285,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_rx(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rv(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,replicate(A,Uu,Uua),Uub)) ) ).
% ATP.lambda_285
tff(fact_5255_ATP_Olambda__286,axiom,
! [A: $tType,Uu: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qt(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_qs(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),Uu,Uub)) ).
% ATP.lambda_286
tff(fact_5256_ATP_Olambda__287,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kg(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub != Uua )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ) ).
% ATP.lambda_287
tff(fact_5257_ATP_Olambda__288,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ki(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ( ( Uub != Uua )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ) ).
% ATP.lambda_288
tff(fact_5258_ATP_Olambda__289,axiom,
! [B: $tType,A: $tType] :
( semiring_1(B)
=> ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ce(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_289
tff(fact_5259_ATP_Olambda__290,axiom,
! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
<=> ( ~ aa(A,$o,Uu,Uub)
| aa(A,$o,Uua,Uub) ) ) ).
% ATP.lambda_290
tff(fact_5260_ATP_Olambda__291,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_jw(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
<=> ( aa(set(A),$o,finite_finite(A),Uua)
& aa(set(A),$o,finite_finite(A),Uub)
& ( Uub != bot_bot(set(A)) )
& ! [X4: A] :
( member(A,X4,Uua)
=> ? [Xa2: A] :
( member(A,Xa2,Uub)
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu) ) ) ) ) ).
% ATP.lambda_291
tff(fact_5261_ATP_Olambda__292,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_jz(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),insert2(A,Uua),bot_bot(set(A)))) ).
% ATP.lambda_292
tff(fact_5262_ATP_Olambda__293,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dd(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_293
tff(fact_5263_ATP_Olambda__294,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_da(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_294
tff(fact_5264_ATP_Olambda__295,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_rv(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_295
tff(fact_5265_ATP_Olambda__296,axiom,
! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_kz(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))))) ) ).
% ATP.lambda_296
tff(fact_5266_ATP_Olambda__297,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_rw(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_297
tff(fact_5267_ATP_Olambda__298,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_dy(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),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).
% ATP.lambda_298
tff(fact_5268_ATP_Olambda__299,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
( aa(heap_ext(product_unit),$o,aa(array(A),fun(heap_ext(product_unit),$o),aTP_Lamp_rr(nat,fun(array(A),fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),array_length(A,Uub,Uua)) ) ) ).
% ATP.lambda_299
tff(fact_5269_ATP_Olambda__300,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
( aa(heap_ext(product_unit),$o,aa(nat,fun(heap_ext(product_unit),$o),aTP_Lamp_ro(array(A),fun(nat,fun(heap_ext(product_unit),$o)),Uu),Uua),Uub)
<=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),array_length(A,Uub,Uu)) ) ) ).
% ATP.lambda_300
tff(fact_5270_ATP_Olambda__301,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_cm(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_301
tff(fact_5271_ATP_Olambda__302,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_os(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_302
tff(fact_5272_ATP_Olambda__303,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_od(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_303
tff(fact_5273_ATP_Olambda__304,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fc(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_304
tff(fact_5274_ATP_Olambda__305,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_add(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fb(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_305
tff(fact_5275_ATP_Olambda__306,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_pf(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).
% ATP.lambda_306
tff(fact_5276_ATP_Olambda__307,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_ci(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_307
tff(fact_5277_ATP_Olambda__308,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_gc(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)),minus_minus(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_308
tff(fact_5278_ATP_Olambda__309,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_fd(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_309
tff(fact_5279_ATP_Olambda__310,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_gf(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_310
tff(fact_5280_ATP_Olambda__311,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_fz(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_311
tff(fact_5281_ATP_Olambda__312,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gj(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).
% ATP.lambda_312
tff(fact_5282_ATP_Olambda__313,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_ld(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).
% ATP.lambda_313
tff(fact_5283_ATP_Olambda__314,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_om(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_314
tff(fact_5284_ATP_Olambda__315,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_kc(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = image(B,A,Uu,aa(C,set(B),Uua,Uub)) ).
% ATP.lambda_315
tff(fact_5285_ATP_Olambda__316,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_fg(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),insert2(A,Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_316
tff(fact_5286_ATP_Olambda__317,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_ig(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).
% ATP.lambda_317
tff(fact_5287_ATP_Olambda__318,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(A),Uua: set(C),Uub: fun(A,B)] : aa(fun(A,B),set(fun(A,C)),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_mq(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,C,Uu,Uua) ).
% ATP.lambda_318
tff(fact_5288_ATP_Olambda__319,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_kf(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).
% ATP.lambda_319
tff(fact_5289_ATP_Olambda__320,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: A] : aa(A,set(sum_sum(B,C)),aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_nb(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).
% ATP.lambda_320
tff(fact_5290_ATP_Olambda__321,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_im(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert2(B,Uu),Uua) ).
% ATP.lambda_321
tff(fact_5291_ATP_Olambda__322,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_is(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).
% ATP.lambda_322
tff(fact_5292_ATP_Olambda__323,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_rl(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_323
tff(fact_5293_ATP_Olambda__324,axiom,
! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_nm(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
<=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).
% ATP.lambda_324
tff(fact_5294_ATP_Olambda__325,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_dk(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).
% ATP.lambda_325
tff(fact_5295_ATP_Olambda__326,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(multiset(A),B),Uua: fun(C,multiset(A)),Uub: C] : aa(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_lo(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(C,multiset(A),Uua,Uub)) ) ).
% ATP.lambda_326
tff(fact_5296_ATP_Olambda__327,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_jj(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_327
tff(fact_5297_ATP_Olambda__328,axiom,
! [C: $tType,B: $tType,A: $tType] :
( comm_monoid_mult(C)
=> ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_mo(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_328
tff(fact_5298_ATP_Olambda__329,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_cq(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_329
tff(fact_5299_ATP_Olambda__330,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_lf(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
<=> aa(B,$o,Uu,Uua) ) ).
% ATP.lambda_330
tff(fact_5300_ATP_Olambda__331,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_ia(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_hz(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).
% ATP.lambda_331
tff(fact_5301_ATP_Olambda__332,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: product_prod(code_natural,code_natural),Uub: B] : aa(B,pred(A),aa(product_prod(code_natural,code_natural),fun(B,pred(A)),aTP_Lamp_qu(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(B,pred(A))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),pred(A),product_fst(pred(A),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uu,Uub),Uua)) ).
% ATP.lambda_332
tff(fact_5302_ATP_Olambda__333,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),Uua: set(old_node(A,B)),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_so(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua),Uub) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_sn(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uua),Uub))),Uu)) ).
% ATP.lambda_333
tff(fact_5303_ATP_Olambda__334,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua: pred(B),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aa(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aTP_Lamp_qv(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(pred(B),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),Uu),Uua),Uub) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bind(B,A,Uua,aa(product_prod(code_natural,code_natural),fun(B,pred(A)),aTP_Lamp_qu(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(B,pred(A))),Uu),Uub))) ).
% ATP.lambda_334
tff(fact_5304_ATP_Olambda__335,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ih(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).
% ATP.lambda_335
tff(fact_5305_ATP_Olambda__336,axiom,
! [C: $tType,B: $tType,A: $tType] :
( condit1219197933456340205attice(C)
=> ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ii(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).
% ATP.lambda_336
tff(fact_5306_ATP_Olambda__337,axiom,
! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_nq(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ? [I2: nat] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I2)),aa(nat,B,nth(B,Uua),I2)) )
& aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).
% ATP.lambda_337
tff(fact_5307_ATP_Olambda__338,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
<=> ? [A5: A] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),aa(A,B,Uua,A5)) )
& member(A,A5,Uu) ) ) ).
% ATP.lambda_338
tff(fact_5308_ATP_Olambda__339,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_hq(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ? [A5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A5) )
& member(A,A5,Uu) ) ) ) ).
% ATP.lambda_339
tff(fact_5309_ATP_Olambda__340,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ho(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ? [A5: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A5) )
& member(A,A5,Uu) ) ) ) ).
% ATP.lambda_340
tff(fact_5310_ATP_Olambda__341,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_ll(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A5: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uua),A5) )
& member(multiset(A),A5,Uu) ) ) ).
% ATP.lambda_341
tff(fact_5311_ATP_Olambda__342,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_lm(set(multiset(A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A5: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uua),A5) )
& member(multiset(A),A5,Uu) ) ) ).
% ATP.lambda_342
tff(fact_5312_ATP_Olambda__343,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_kb(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
<=> ! [X4: product_prod(A,A)] :
( member(product_prod(A,A),X4,Uu)
=> aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ka(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)),X4) ) ) ).
% ATP.lambda_343
tff(fact_5313_ATP_Olambda__344,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_jv(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
<=> ? [X4: B] :
( member(B,X4,Uua)
& member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Uub),Uu) ) ) ).
% ATP.lambda_344
tff(fact_5314_ATP_Olambda__345,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,$o)),Uub: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_hh(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
<=> ? [A5: B,B4: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A5)),aa(B,A,Uu,B4)) )
& aa(B,$o,aa(B,fun(B,$o),Uua,A5),B4) ) ) ).
% ATP.lambda_345
tff(fact_5315_ATP_Olambda__346,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(B,A),fun(product_prod(A,A),$o),aTP_Lamp_jf(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
<=> ? [A15: B,A25: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A15)),aa(B,A,Uua,A25)) )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A15),A25),Uu) ) ) ).
% ATP.lambda_346
tff(fact_5316_ATP_Olambda__347,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_nz(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),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),cons(A,A5),V3)) )
| ? [U3: list(A),Aa5: A,B4: A,Va2: list(A),W3: list(A)] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa5),B4),Uu)
& ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U3),aa(list(A),list(A),cons(A,Aa5),Va2)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U3),aa(list(A),list(A),cons(A,B4),W3)) ) ) ) ) ).
% ATP.lambda_347
tff(fact_5317_ATP_Olambda__348,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_hp(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [A5: A,B4: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B4) )
& member(A,A5,Uu)
& member(A,B4,Uua) ) ) ) ).
% ATP.lambda_348
tff(fact_5318_ATP_Olambda__349,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_hn(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
<=> ? [A5: A,B4: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B4) )
& member(A,A5,Uu)
& member(A,B4,Uua) ) ) ) ).
% ATP.lambda_349
tff(fact_5319_ATP_Olambda__350,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_lk(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A5: multiset(A),B4: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A5),B4) )
& member(multiset(A),A5,Uu)
& member(multiset(A),B4,Uua) ) ) ).
% ATP.lambda_350
tff(fact_5320_ATP_Olambda__351,axiom,
! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(set(multiset(A)),fun(multiset(A),$o),aTP_Lamp_ln(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A5: multiset(A),B4: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A5),B4) )
& member(multiset(A),A5,Uu)
& member(multiset(A),B4,Uua) ) ) ).
% ATP.lambda_351
tff(fact_5321_ATP_Olambda__352,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),aTP_Lamp_lj(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),$o)),Uu),Uua),Uub)
<=> ? [A5: A,M03: multiset(A),K7: multiset(A)] :
( ( Uub = add_mset(A,A5,M03) )
& ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K7) )
& ! [B4: A] :
( member(A,B4,aa(multiset(A),set(A),set_mset(A),K7))
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5),Uu) ) ) ) ).
% ATP.lambda_352
tff(fact_5322_ATP_Olambda__353,axiom,
! [B: $tType,A: $tType,E: $tType,C: $tType,D: $tType,Uu: fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),Uua: E,Uub: fun(A,option(product_prod(B,product_prod(C,D))))] :
( aa(fun(A,option(product_prod(B,product_prod(C,D)))),$o,aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o),aTP_Lamp_sr(fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o)),Uu),Uua),Uub)
<=> ! [H: A,H6: C,R5: B,N2: D] :
( ( aa(A,option(product_prod(B,product_prod(C,D))),Uub,H) = aa(product_prod(B,product_prod(C,D)),option(product_prod(B,product_prod(C,D))),some(product_prod(B,product_prod(C,D))),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),R5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),H6),N2))) )
=> aa(D,$o,aa(B,fun(D,$o),aa(C,fun(B,fun(D,$o)),aa(A,fun(C,fun(B,fun(D,$o))),aa(E,fun(A,fun(C,fun(B,fun(D,$o)))),Uu,Uua),H),H6),R5),N2) ) ) ).
% ATP.lambda_353
tff(fact_5323_ATP_Olambda__354,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_oa(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
<=> ? [Us3: list(A),Z4: A,Z9: A,Vs3: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Z4),Vs3)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Z9),Uu)
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Z9),Vs3)) ) ) ) ).
% ATP.lambda_354
tff(fact_5324_ATP_Olambda__355,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] :
aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_gw(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).
% ATP.lambda_355
tff(fact_5325_ATP_Olambda__356,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_lb(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).
% ATP.lambda_356
tff(fact_5326_ATP_Olambda__357,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] :
aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_hg(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),insert2(B,Uub),Uuc),Uuc) ).
% ATP.lambda_357
tff(fact_5327_ATP_Olambda__358,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_bc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
aa(nat,A,Uua,Uuc),
$ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).
% ATP.lambda_358
tff(fact_5328_ATP_Olambda__359,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_bd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).
% ATP.lambda_359
tff(fact_5329_ATP_Olambda__360,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_ko(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_360
tff(fact_5330_ATP_Olambda__361,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_pn(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_361
tff(fact_5331_ATP_Olambda__362,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_by(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_362
tff(fact_5332_ATP_Olambda__363,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cn(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_363
tff(fact_5333_ATP_Olambda__364,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_bo(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).
% ATP.lambda_364
tff(fact_5334_ATP_Olambda__365,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_iz(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ).
% ATP.lambda_365
tff(fact_5335_ATP_Olambda__366,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_re(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Uu),Uua),Uub),Uuc) = $ite(aa(code_natural,$o,aa(code_natural,fun(code_natural,$o),ord_less(code_natural),Uub),Uua),empty(A),insert(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_366
tff(fact_5336_ATP_Olambda__367,axiom,
! [A: $tType,Uu: fun(A,$o),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_ob(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,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),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),cons(A,Uua),Uuc))) ).
% ATP.lambda_367
tff(fact_5337_ATP_Olambda__368,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_bx(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).
% ATP.lambda_368
tff(fact_5338_ATP_Olambda__369,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_hc(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_hb(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_369
tff(fact_5339_ATP_Olambda__370,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_gv(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_gu(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_370
tff(fact_5340_ATP_Olambda__371,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_nn(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_371
tff(fact_5341_ATP_Olambda__372,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_jo(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).
% ATP.lambda_372
tff(fact_5342_ATP_Olambda__373,axiom,
! [B: $tType,A: $tType,C: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_cl(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ck(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_373
tff(fact_5343_ATP_Olambda__374,axiom,
! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_ql(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uu),Uua),Uub),Uuc) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_Pair(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uu)),Uuc)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Uub),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uua))) ).
% ATP.lambda_374
tff(fact_5344_ATP_Olambda__375,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: set(A),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_ix(fun(A,fun(A,$o)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),product_Sigma(A,A,Uua,aTP_Lamp_in(set(A),fun(A,set(A)),Uua)))
& aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uuc) ) ) ).
% ATP.lambda_375
tff(fact_5345_ATP_Olambda__376,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ei(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).
% ATP.lambda_376
tff(fact_5346_ATP_Olambda__377,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B)),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_sn(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uu),Uua),Uub),Uuc) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,Uu,Uub)),old_Scons(A,B,Uua,Uuc))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_377
tff(fact_5347_ATP_Olambda__378,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rs(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_378
tff(fact_5348_ATP_Olambda__379,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(A),fun(A,fun(A,$o)),aTP_Lamp_jd(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uu)
& ~ member(A,Uub,Uua)
& ~ member(A,Uuc,Uua) ) ) ).
% ATP.lambda_379
tff(fact_5349_ATP_Olambda__380,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_ft(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
| ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua) ) ) ) ).
% ATP.lambda_380
tff(fact_5350_ATP_Olambda__381,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jk(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).
% ATP.lambda_381
tff(fact_5351_ATP_Olambda__382,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_ps(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
& aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).
% ATP.lambda_382
tff(fact_5352_ATP_Olambda__383,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_cw(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),binomial(Uub,Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).
% ATP.lambda_383
tff(fact_5353_ATP_Olambda__384,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_cv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),binomial(Uub,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).
% ATP.lambda_384
tff(fact_5354_ATP_Olambda__385,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ny(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
& ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),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),cons(A,Y3),Ys5)) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu) ) ) ) ).
% ATP.lambda_385
tff(fact_5355_ATP_Olambda__386,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_du(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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).
% ATP.lambda_386
tff(fact_5356_ATP_Olambda__387,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cr(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uu)
& ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_387
tff(fact_5357_ATP_Olambda__388,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_kp(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uua)
& ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).
% ATP.lambda_388
tff(fact_5358_ATP_Olambda__389,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_dt(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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).
% ATP.lambda_389
tff(fact_5359_ATP_Olambda__390,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_em(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).
% ATP.lambda_390
tff(fact_5360_ATP_Olambda__391,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_eo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).
% ATP.lambda_391
tff(fact_5361_ATP_Olambda__392,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_of(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)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).
% ATP.lambda_392
tff(fact_5362_ATP_Olambda__393,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_oq(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)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).
% ATP.lambda_393
tff(fact_5363_ATP_Olambda__394,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ks(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ( ( Uu = Uub )
& ( Uua = Uuc ) ) ) ).
% ATP.lambda_394
tff(fact_5364_ATP_Olambda__395,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_fk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).
% ATP.lambda_395
tff(fact_5365_ATP_Olambda__396,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bs(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
<=> ( member(A,Uuc,Uu)
& ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).
% ATP.lambda_396
tff(fact_5366_ATP_Olambda__397,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_no(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_397
tff(fact_5367_ATP_Olambda__398,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_pi(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_398
tff(fact_5368_ATP_Olambda__399,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( semiring_0(A)
& comm_monoid_add(B)
& times(B) )
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_mb(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_399
tff(fact_5369_ATP_Olambda__400,axiom,
! [B: $tType,A: $tType,C: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ck(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_400
tff(fact_5370_ATP_Olambda__401,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [Uu: fun(A,C),Uua: fun(B,C),Uub: A,Uuc: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_kx(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,Uub)),aa(B,C,Uua,Uuc)) ) ).
% ATP.lambda_401
tff(fact_5371_ATP_Olambda__402,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_gg(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_402
tff(fact_5372_ATP_Olambda__403,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_lg(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).
% ATP.lambda_403
tff(fact_5373_ATP_Olambda__404,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_iq(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_404
tff(fact_5374_ATP_Olambda__405,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_sc(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).
% ATP.lambda_405
tff(fact_5375_ATP_Olambda__406,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_1(C)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_cs(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cr(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_406
tff(fact_5376_ATP_Olambda__407,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(A,A),Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rt(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,aa(A,A,Uua,aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_407
tff(fact_5377_ATP_Olambda__408,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_ru(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_408
tff(fact_5378_ATP_Olambda__409,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_iw(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu,aa(set(A),set(A),insert2(A,Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_409
tff(fact_5379_ATP_Olambda__410,axiom,
! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option($o),aa(A,fun(list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_pz(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),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_410
tff(fact_5380_ATP_Olambda__411,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_ok(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_411
tff(fact_5381_ATP_Olambda__412,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_oj(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_412
tff(fact_5382_ATP_Olambda__413,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: fun(B,A)] : aa(fun(B,A),fun(B,A),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_st(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uuc))) ).
% ATP.lambda_413
tff(fact_5383_ATP_Olambda__414,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: fun(A,B),Uua: B,Uub: A,Uuc: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_nv(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Uu),Uua),Uub),Uuc) = aa(fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),list(A),product_prod(list(A),product_prod(list(A),list(A)))),aa(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),aa(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))),aa(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))))),aTP_Lamp_nu(fun(A,B),fun(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))))),Uu),Uua),Uub),Uuc)) ) ).
% ATP.lambda_414
tff(fact_5384_ATP_Olambda__415,axiom,
! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_hy(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_hx(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_415
tff(fact_5385_ATP_Olambda__416,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_hw(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_hv(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_416
tff(fact_5386_ATP_Olambda__417,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( semiring_0(A)
& comm_monoid_add(B)
& times(B) )
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: multiset(C),Uuc: B] : aa(B,A,aa(multiset(C),fun(B,A),aa(fun(C,A),fun(multiset(C),fun(B,A)),aTP_Lamp_mc(fun(B,A),fun(fun(C,A),fun(multiset(C),fun(B,A))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(A,image_mset(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_mb(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_417
tff(fact_5387_ATP_Olambda__418,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_lh(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_lg(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).
% ATP.lambda_418
tff(fact_5388_ATP_Olambda__419,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_gh(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gg(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_419
tff(fact_5389_ATP_Olambda__420,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dn(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_420
tff(fact_5390_ATP_Olambda__421,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_421
tff(fact_5391_ATP_Olambda__422,axiom,
! [A: $tType,Uu: fun(A,$o),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_qh(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,comp($o,$o,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_422
tff(fact_5392_ATP_Olambda__423,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_pu(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),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_423
tff(fact_5393_ATP_Olambda__424,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_424
tff(fact_5394_ATP_Olambda__425,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_dp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).
% ATP.lambda_425
tff(fact_5395_ATP_Olambda__426,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_la(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),image(B,B,Uu,aa(set(B),set(B),insert2(B,Uub),bot_bot(set(B)))))) ).
% ATP.lambda_426
tff(fact_5396_ATP_Olambda__427,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_hk(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
<=> ? [Y3: C] :
( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y3),Uu)
& member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc),Uua) ) ) ).
% ATP.lambda_427
tff(fact_5397_ATP_Olambda__428,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_ht(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu),Uua),Uub),Uuc)
<=> ? [A5: C] :
( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A5)),aa(C,B,Uub,A5)) )
& member(C,A5,Uu) ) ) ).
% ATP.lambda_428
tff(fact_5398_ATP_Olambda__429,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
( aa(product_prod(A,A),$o,aa(fun(A,B),fun(product_prod(A,A),$o),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o)),aTP_Lamp_kh(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),$o))),Uu),Uua),Uub),Uuc)
<=> ? [A15: A,A25: A] :
( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A15),A25) )
& member(A,A15,Uu)
& member(A,A25,Uu)
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A15)),aa(A,B,Uub,A25)),Uua) ) ) ).
% ATP.lambda_429
tff(fact_5399_ATP_Olambda__430,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_pm(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_430
tff(fact_5400_ATP_Olambda__431,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] :
aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_gu(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_431
tff(fact_5401_ATP_Olambda__432,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] :
aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_hb(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),insert2(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_432
tff(fact_5402_ATP_Olambda__433,axiom,
! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_pg(fun(A,fun(A,$o)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = quicksort_by_rel(A,Uu,aa(list(A),list(A),cons(A,Uub),quicksort_by_rel(A,Uu,Uua,Uud)),Uuc) ).
% ATP.lambda_433
tff(fact_5403_ATP_Olambda__434,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: B,Uud: fun(B,A)] : aa(fun(B,A),A,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_ss(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,A,aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uud))),Uuc) ).
% ATP.lambda_434
tff(fact_5404_ATP_Olambda__435,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_on(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).
% ATP.lambda_435
tff(fact_5405_ATP_Olambda__436,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_db(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,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)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_436
tff(fact_5406_ATP_Olambda__437,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_cy(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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_437
tff(fact_5407_ATP_Olambda__438,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_je(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_trancl(A,Uub))
| ( Uuc = Uu ) )
& ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_trancl(A,Uub))
| ( Uud = Uua ) ) ) ) ).
% ATP.lambda_438
tff(fact_5408_ATP_Olambda__439,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_cz(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)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_439
tff(fact_5409_ATP_Olambda__440,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_hz(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,Uua),aa(set(A),set(A),insert2(A,Uub),aa(set(A),set(A),insert2(A,Uuc),aa(set(A),set(A),insert2(A,Uud),bot_bot(set(A))))))),field2(A,Uu))
& ( ( ( Uua = Uuc )
& ( Uub = Uud ) )
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) )
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& ( Uua = Uuc )
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) ) ) ) ) ).
% ATP.lambda_440
tff(fact_5410_ATP_Olambda__441,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_jb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_rtrancl(A,Uub))
& member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_rtrancl(A,Uub)) ) ) ).
% ATP.lambda_441
tff(fact_5411_ATP_Olambda__442,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ka(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ( ( Uub = Uuc )
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),Uu) ) ) ).
% ATP.lambda_442
tff(fact_5412_ATP_Olambda__443,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_0(C)
=> ! [Uu: fun(A,C),Uua: fun(B,C),Uub: set(A),Uuc: set(B),Uud: C] :
( aa(C,$o,aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_ky(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),Uu),Uua),Uub),Uuc),Uud)
<=> ? [A5: A,B4: B] :
( ( Uud = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu,A5)),aa(B,C,Uua,B4)) )
& member(A,A5,Uub)
& member(B,B4,Uuc) ) ) ) ).
% ATP.lambda_443
tff(fact_5413_ATP_Olambda__444,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_hv(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu)
| ( ( Uub = Uud )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua) ) ) ) ).
% ATP.lambda_444
tff(fact_5414_ATP_Olambda__445,axiom,
! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_hx(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
<=> ( ( Uub = Uud )
& aa(A,$o,Uu,Uud)
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).
% ATP.lambda_445
tff(fact_5415_ATP_Olambda__446,axiom,
! [Uu: fun(a,b),Uua: b,Uub: a,Uuc: list(a),Uud: list(a),Uue: list(a)] :
aa(list(a),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))),aa(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))),aa(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))),aa(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))))),aTP_Lamp_nu(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
x: b,
x:= aa(a,b,Uu,Uub),
$ite(
aa(b,$o,aa(b,fun(b,$o),ord_less(b),x),Uua),
aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),cons(a,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)),Uud),Uue)),
$ite(aa(b,$o,aa(b,fun(b,$o),ord_less(b),Uua),x),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),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)),Uud),aa(list(a),list(a),cons(a,Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),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)),aa(list(a),list(a),cons(a,Uub),Uud)),Uue))) ) ) ).
% ATP.lambda_446
tff(fact_5416_ATP_Olambda__447,axiom,
! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ee($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
<=> (Uu) ) ).
% ATP.lambda_447
tff(fact_5417_ATP_Olambda__448,axiom,
! [A: $tType,B: $tType,Uu: multiset(B),Uua: A] : aa(A,multiset(B),aTP_Lamp_fq(multiset(B),fun(A,multiset(B)),Uu),Uua) = Uu ).
% ATP.lambda_448
tff(fact_5418_ATP_Olambda__449,axiom,
! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_eu(assn,fun(A,assn),Uu),Uua) = Uu ).
% ATP.lambda_449
tff(fact_5419_ATP_Olambda__450,axiom,
! [A: $tType,Uu: $o,Uua: A] :
( aa(A,$o,aTP_Lamp_af($o,fun(A,$o),(Uu)),Uua)
<=> (Uu) ) ).
% ATP.lambda_450
tff(fact_5420_ATP_Olambda__451,axiom,
! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_ir(set(D),fun(C,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_451
tff(fact_5421_ATP_Olambda__452,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_mp(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_452
tff(fact_5422_ATP_Olambda__453,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_mr(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_453
tff(fact_5423_ATP_Olambda__454,axiom,
! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_mt(set(B),fun(C,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_454
tff(fact_5424_ATP_Olambda__455,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ib(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_455
tff(fact_5425_ATP_Olambda__456,axiom,
! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_ms(set(A),fun(C,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_456
tff(fact_5426_ATP_Olambda__457,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_fh(set(A),fun(B,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_457
tff(fact_5427_ATP_Olambda__458,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_in(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_458
tff(fact_5428_ATP_Olambda__459,axiom,
! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_le(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).
% ATP.lambda_459
tff(fact_5429_ATP_Olambda__460,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ev(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_460
tff(fact_5430_ATP_Olambda__461,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_fj(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_461
tff(fact_5431_ATP_Olambda__462,axiom,
! [A: $tType,B: $tType] :
( linorder(B)
=> ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ai(B,fun(A,B),Uu),Uua) = Uu ) ).
% ATP.lambda_462
tff(fact_5432_ATP_Olambda__463,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_as(B,fun(A,B),Uu),Uua) = Uu ).
% ATP.lambda_463
tff(fact_5433_ATP_Olambda__464,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_fn(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_464
tff(fact_5434_ATP_Olambda__465,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_cd(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_465
tff(fact_5435_ATP_Olambda__466,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ah(A,fun(B,A),Uu),Uua) = Uu ).
% ATP.lambda_466
tff(fact_5436_ATP_Olambda__467,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_pk(nat,nat),Uu) = Uu ).
% ATP.lambda_467
tff(fact_5437_ATP_Olambda__468,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_oo(A,A),Uu) = Uu ) ).
% ATP.lambda_468
tff(fact_5438_ATP_Olambda__469,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_nw(A,A),Uu) = Uu ) ).
% ATP.lambda_469
tff(fact_5439_ATP_Olambda__470,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_am(A,A),Uu) = Uu ) ).
% ATP.lambda_470
tff(fact_5440_ATP_Olambda__471,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_mm(A,A),Uu) = Uu ) ).
% ATP.lambda_471
tff(fact_5441_ATP_Olambda__472,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_fy(B,A),Uu) = top_top(A) ) ).
% ATP.lambda_472
tff(fact_5442_ATP_Olambda__473,axiom,
! [A: $tType,Uu: A] : aa(A,set($o),aTP_Lamp_nc(A,set($o)),Uu) = top_top(set($o)) ).
% ATP.lambda_473
tff(fact_5443_ATP_Olambda__474,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_il(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_474
tff(fact_5444_ATP_Olambda__475,axiom,
! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_gx(C,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_475
tff(fact_5445_ATP_Olambda__476,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_fm(B,set(A)),Uu) = bot_bot(set(A)) ).
% ATP.lambda_476
tff(fact_5446_ATP_Olambda__477,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_fi(B,A),Uu) = bot_bot(A) ) ).
% ATP.lambda_477
tff(fact_5447_ATP_Olambda__478,axiom,
! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_gy(A,set(D)),Uu) = bot_bot(set(D)) ).
% ATP.lambda_478
tff(fact_5448_ATP_Olambda__479,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ik(A,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_479
tff(fact_5449_ATP_Olambda__480,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_og(B,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_480
tff(fact_5450_ATP_Olambda__481,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_aj(A,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_481
tff(fact_5451_ATP_Olambda__482,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_bp(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_482
tff(fact_5452_ATP_Olambda__483,axiom,
! [A: $tType,Uu: product_unit] : aa(product_unit,seq(A),aTP_Lamp_rf(product_unit,seq(A)),Uu) = empty(A) ).
% ATP.lambda_483
tff(fact_5453_ATP_Olambda__484,axiom,
! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_sb(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).
% ATP.lambda_484
tff(fact_5454_ATP_Olambda__485,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_qe(B,option(A)),Uu) = none(A) ).
% ATP.lambda_485
tff(fact_5455_ATP_Olambda__486,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_pq(A,option(B)),Uu) = none(B) ).
% ATP.lambda_486
tff(fact_5456_ATP_Olambda__487,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_lc(A,B),Uu) = undefined(B) ).
% ATP.lambda_487
tff(fact_5457_ATP_Olambda__488,axiom,
! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
( aa(product_prod(heap_ext(product_unit),set(nat)),$o,aTP_Lamp_ab(product_prod(heap_ext(product_unit),set(nat)),$o),Uu)
<=> $false ) ).
% ATP.lambda_488
tff(fact_5458_ATP_Olambda__489,axiom,
! [A: $tType,Uu: A] :
( aa(A,$o,aTP_Lamp_al(A,$o),Uu)
<=> $false ) ).
% ATP.lambda_489
tff(fact_5459_ATP_Olambda__490,axiom,
! [A: $tType,Uu: fun(A,$o)] :
( aa(fun(A,$o),$o,aTP_Lamp_sm(fun(A,$o),$o),Uu)
<=> $true ) ).
% ATP.lambda_490
% Type constructors (802)
tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(heap_Time_Heap(A16)) ) ).
tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(heap_Time_Heap(A16)) ) ).
tff(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
code_term_of(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
typerep2(code_natural) ).
tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
code_term_of(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
typerep2(code_integer) ).
tff(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of(code_term) ).
tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
typerep2(code_term) ).
tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(heap_ext(A16)) ) ).
tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(heap_ext(A16)) ) ).
tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
code_term_of(product_unit) ).
tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum(product_unit) ).
tff(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
typerep2(product_unit) ).
tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> code_term_of(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A16: $tType,A17: $tType] :
( ( enum(A16)
& enum(A17) )
=> enum(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> typerep2(product_prod(A16,A17)) ) ).
tff(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> typerep2(old_node(A16,A17)) ) ).
tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(multiset(A16)) ) ).
tff(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
code_term_of(typerep) ).
tff(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
typerep2(typerep) ).
tff(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
typerep2(assn) ).
tff(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_20,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(pred(A16)) ) ).
tff(tcon_Predicate_Opred___Typerep_Otyperep_21,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(pred(A16)) ) ).
tff(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_22,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(seq(A16)) ) ).
tff(tcon_Predicate_Oseq___Typerep_Otyperep_23,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(seq(A16)) ) ).
tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_24,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(option(A16)) ) ).
tff(tcon_Option_Ooption___Enum_Oenum_25,axiom,
! [A16: $tType] :
( enum(A16)
=> enum(option(A16)) ) ).
tff(tcon_Option_Ooption___Typerep_Otyperep_26,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(option(A16)) ) ).
tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_27,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(filter(A16)) ) ).
tff(tcon_Filter_Ofilter___Typerep_Otyperep_28,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(filter(A16)) ) ).
tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_29,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> code_term_of(sum_sum(A16,A17)) ) ).
tff(tcon_Sum__Type_Osum___Enum_Oenum_30,axiom,
! [A16: $tType,A17: $tType] :
( ( enum(A16)
& enum(A17) )
=> enum(sum_sum(A16,A17)) ) ).
tff(tcon_Sum__Type_Osum___Typerep_Otyperep_31,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> typerep2(sum_sum(A16,A17)) ) ).
tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_32,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(array(A16)) ) ).
tff(tcon_Heap_Oarray___Typerep_Otyperep_33,axiom,
! [A16: $tType] : typerep2(array(A16)) ).
tff(tcon_List_Olist___Code__Evaluation_Oterm__of_34,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(list(A16)) ) ).
tff(tcon_List_Olist___Typerep_Otyperep_35,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(list(A16)) ) ).
tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_36,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(ref(A16)) ) ).
tff(tcon_Heap_Oref___Typerep_Otyperep_37,axiom,
! [A16: $tType] : typerep2(ref(A16)) ).
tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_38,axiom,
code_term_of($o) ).
tff(tcon_HOL_Obool___Enum_Oenum_39,axiom,
enum($o) ).
tff(tcon_HOL_Obool___Typerep_Otyperep_40,axiom,
typerep2($o) ).
tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_41,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(set(A16)) ) ).
tff(tcon_Set_Oset___Enum_Oenum_42,axiom,
! [A16: $tType] :
( enum(A16)
=> enum(set(A16)) ) ).
tff(tcon_Set_Oset___Typerep_Otyperep_43,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(set(A16)) ) ).
tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_44,axiom,
code_term_of(rat) ).
tff(tcon_Rat_Orat___Typerep_Otyperep_45,axiom,
typerep2(rat) ).
tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_46,axiom,
code_term_of(num) ).
tff(tcon_Num_Onum___Typerep_Otyperep_47,axiom,
typerep2(num) ).
tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_48,axiom,
code_term_of(nat) ).
tff(tcon_Nat_Onat___Typerep_Otyperep_49,axiom,
typerep2(nat) ).
tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_50,axiom,
code_term_of(int) ).
tff(tcon_Int_Oint___Typerep_Otyperep_51,axiom,
typerep2(int) ).
tff(tcon_itself___Code__Evaluation_Oterm__of_52,axiom,
! [A16: $tType] :
( typerep2(A16)
=> code_term_of(itself(A16)) ) ).
tff(tcon_itself___Typerep_Otyperep_53,axiom,
! [A16: $tType] :
( typerep2(A16)
=> typerep2(itself(A16)) ) ).
tff(tcon_fun___Code__Evaluation_Oterm__of_54,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> code_term_of(fun(A16,A17)) ) ).
tff(tcon_fun___Enum_Oenum_55,axiom,
! [A16: $tType,A17: $tType] :
( ( enum(A16)
& enum(A17) )
=> enum(fun(A16,A17)) ) ).
tff(tcon_fun___Typerep_Otyperep_56,axiom,
! [A16: $tType,A17: $tType] :
( ( typerep2(A16)
& typerep2(A17) )
=> typerep2(fun(A16,A17)) ) ).
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A16: $tType,A17: $tType] :
( comple6319245703460814977attice(A17)
=> condit1219197933456340205attice(fun(A16,A17)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A16: $tType,A17: $tType] :
( comple592849572758109894attice(A17)
=> comple592849572758109894attice(fun(A16,A17)) ) ).
tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A16: $tType,A17: $tType] :
( ( cl_HOL_Oequal(A16)
& quickc3360725361186068524ustive(A16)
& quickc3360725361186068524ustive(A17) )
=> quickc3360725361186068524ustive(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A16: $tType,A17: $tType] :
( bounded_lattice(A17)
=> bounde4967611905675639751up_bot(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A16: $tType,A17: $tType] :
( bounded_lattice(A17)
=> bounde4346867609351753570nf_top(fun(A16,A17)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A16: $tType,A17: $tType] :
( comple6319245703460814977attice(A17)
=> comple6319245703460814977attice(fun(A16,A17)) ) ).
tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A16: $tType,A17: $tType] :
( ( cl_HOL_Oequal(A16)
& quickc658316121487927005ustive(A16)
& quickc658316121487927005ustive(A17) )
=> quickc658316121487927005ustive(fun(A16,A17)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A16: $tType,A17: $tType] :
( boolea8198339166811842893lgebra(A17)
=> boolea8198339166811842893lgebra(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A16: $tType,A17: $tType] :
( bounded_lattice(A17)
=> bounded_lattice_top(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A16: $tType,A17: $tType] :
( bounded_lattice(A17)
=> bounded_lattice_bot(fun(A16,A17)) ) ).
tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A16: $tType,A17: $tType] :
( comple6319245703460814977attice(A17)
=> comple9053668089753744459l_ccpo(fun(A16,A17)) ) ).
tff(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A16: $tType,A17: $tType] :
( ( code_term_of(A16)
& cl_HOL_Oequal(A16)
& quickcheck_random(A17) )
=> quickcheck_random(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A16: $tType,A17: $tType] :
( semilattice_sup(A17)
=> semilattice_sup(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A16: $tType,A17: $tType] :
( semilattice_inf(A17)
=> semilattice_inf(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A16: $tType,A17: $tType] :
( distrib_lattice(A17)
=> distrib_lattice(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A16: $tType,A17: $tType] :
( bounded_lattice(A17)
=> bounded_lattice(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Oorder__top,axiom,
! [A16: $tType,A17: $tType] :
( order_top(A17)
=> order_top(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Oorder__bot,axiom,
! [A16: $tType,A17: $tType] :
( order_bot(A17)
=> order_bot(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A16: $tType,A17: $tType] :
( preorder(A17)
=> preorder(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Olattice,axiom,
! [A16: $tType,A17: $tType] :
( lattice(A17)
=> lattice(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A16: $tType,A17: $tType] :
( order(A17)
=> order(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Otop,axiom,
! [A16: $tType,A17: $tType] :
( top(A17)
=> top(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A16: $tType,A17: $tType] :
( ord(A17)
=> ord(fun(A16,A17)) ) ).
tff(tcon_fun___Orderings_Obot,axiom,
! [A16: $tType,A17: $tType] :
( bot(A17)
=> bot(fun(A16,A17)) ) ).
tff(tcon_fun___Groups_Ouminus,axiom,
! [A16: $tType,A17: $tType] :
( uminus(A17)
=> uminus(fun(A16,A17)) ) ).
tff(tcon_fun___Lattices_Osup,axiom,
! [A16: $tType,A17: $tType] :
( semilattice_sup(A17)
=> sup(fun(A16,A17)) ) ).
tff(tcon_fun___Groups_Ominus,axiom,
! [A16: $tType,A17: $tType] :
( minus(A17)
=> minus(fun(A16,A17)) ) ).
tff(tcon_fun___HOL_Oequal,axiom,
! [A16: $tType,A17: $tType] :
( ( enum(A16)
& cl_HOL_Oequal(A17) )
=> cl_HOL_Oequal(fun(A16,A17)) ) ).
tff(tcon_itself___Quickcheck__Random_Orandom_57,axiom,
! [A16: $tType] :
( typerep2(A16)
=> quickcheck_random(itself(A16)) ) ).
tff(tcon_itself___HOL_Oequal_58,axiom,
! [A16: $tType] : cl_HOL_Oequal(itself(A16)) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder(int) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_59,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___Quickcheck__Exhaustive_Ofull__exhaustive_60,axiom,
quickc3360725361186068524ustive(int) ).
tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add(int) ).
tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict(int) ).
tff(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_61,axiom,
quickc658316121487927005ustive(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___Quickcheck__Random_Orandom_62,axiom,
quickcheck_random(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_63,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_64,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Lattices_Odistrib__lattice_65,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_66,axiom,
preorder(int) ).
tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top(int) ).
tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot(int) ).
tff(tcon_Int_Oint___Lattices_Olattice_67,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_68,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_69,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_70,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Lattices_Osup_71,axiom,
sup(int) ).
tff(tcon_Int_Oint___Groups_Otimes,axiom,
times(int) ).
tff(tcon_Int_Oint___Groups_Ominus_72,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_Int_Oint___HOL_Oequal_73,axiom,
cl_HOL_Oequal(int) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_74,axiom,
condit6923001295902523014norder(nat) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_75,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_76,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_77,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_78,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_79,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_80,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_81,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_82,axiom,
normal6328177297339901930cative(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_83,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_84,axiom,
semiri6575147826004484403cancel(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_85,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_86,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_87,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_88,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_89,axiom,
linord2810124833399127020strict(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_90,axiom,
quickc3360725361186068524ustive(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_91,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_92,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_93,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_94,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_95,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_96,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_97,axiom,
linord8928482502909563296strict(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_98,axiom,
quickc658316121487927005ustive(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_99,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_100,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_101,axiom,
semiri6843258321239162965malize(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_107,axiom,
ordere2520102378445227354miring(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_108,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_109,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring_110,axiom,
linordered_semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_111,axiom,
ordered_semiring_0(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_112,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Random_Orandom_113,axiom,
quickcheck_random(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_114,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_115,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_116,axiom,
distrib_lattice(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_117,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_118,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_119,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_120,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring_121,axiom,
ordered_semiring(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_122,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_123,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_124,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_125,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_126,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_127,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_128,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_129,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_130,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_131,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_132,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_133,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder__bot_134,axiom,
order_bot(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_135,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_136,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_137,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_138,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_139,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_140,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_141,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_142,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Orderings_Ono__top_143,axiom,
no_top(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_144,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_145,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_146,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Rings_Omult__zero_147,axiom,
mult_zero(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_148,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_149,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_150,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Orderings_Obot_151,axiom,
bot(nat) ).
tff(tcon_Nat_Onat___Lattices_Osup_152,axiom,
sup(nat) ).
tff(tcon_Nat_Onat___Groups_Otimes_153,axiom,
times(nat) ).
tff(tcon_Nat_Onat___Groups_Ominus_154,axiom,
minus(nat) ).
tff(tcon_Nat_Onat___Power_Opower_155,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_156,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_157,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_158,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_159,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_160,axiom,
heap(nat) ).
tff(tcon_Nat_Onat___HOL_Oequal_161,axiom,
cl_HOL_Oequal(nat) ).
tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_162,axiom,
quickc3360725361186068524ustive(num) ).
tff(tcon_Num_Onum___Quickcheck__Random_Orandom_163,axiom,
quickcheck_random(num) ).
tff(tcon_Num_Onum___Orderings_Opreorder_164,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_165,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_166,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_167,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Otimes_168,axiom,
times(num) ).
tff(tcon_Num_Onum___HOL_Oequal_169,axiom,
cl_HOL_Oequal(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_170,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_171,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_172,axiom,
semiri6575147826004484403cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_173,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_174,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_175,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_176,axiom,
linord2810124833399127020strict(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_177,axiom,
quickc3360725361186068524ustive(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_178,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_179,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_180,axiom,
linord715952674999750819strict(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_181,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_182,axiom,
linord181362715937106298miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_183,axiom,
linord8928482502909563296strict(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_184,axiom,
quickc658316121487927005ustive(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_185,axiom,
semiri3467727345109120633visors(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_186,axiom,
ordere6658533253407199908up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_187,axiom,
ordere166539214618696060dd_abs(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_188,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_189,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_190,axiom,
cancel2418104881723323429up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_191,axiom,
ring_15535105094025558882visors(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_192,axiom,
cancel1802427076303600483id_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_193,axiom,
linord4710134922213307826strict(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_194,axiom,
comm_s4317794764714335236cancel(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_195,axiom,
ordere2520102378445227354miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_196,axiom,
linord6961819062388156250ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_197,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_198,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring_199,axiom,
linordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_200,axiom,
ordered_semiring_0(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_201,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Random_Orandom_202,axiom,
quickcheck_random(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_203,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_204,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_205,axiom,
distrib_lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_206,axiom,
ab_semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_207,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_208,axiom,
ab_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring_209,axiom,
ordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_210,axiom,
ordered_ring_abs(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_211,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_212,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_213,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_214,axiom,
comm_semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_215,axiom,
comm_semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__mult_216,axiom,
semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom__divide_217,axiom,
semidom_divide(rat) ).
tff(tcon_Rat_Orat___Num_Osemiring__numeral_218,axiom,
semiring_numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__add_219,axiom,
semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__less__one_220,axiom,
zero_less_one(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring_221,axiom,
comm_semiring(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_222,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_223,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_224,axiom,
zero_neq_one(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring_225,axiom,
ordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_226,axiom,
idom_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_227,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_228,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__mult_229,axiom,
monoid_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_230,axiom,
comm_ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__add_231,axiom,
monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_232,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_233,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__top_234,axiom,
no_top(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__bot_235,axiom,
no_bot(rat) ).
tff(tcon_Rat_Orat___Lattices_Olattice_236,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_237,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Rings_Omult__zero_238,axiom,
mult_zero(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_239,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_240,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_241,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_242,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_243,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_244,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_245,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_246,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Lattices_Osup_247,axiom,
sup(rat) ).
tff(tcon_Rat_Orat___Groups_Otimes_248,axiom,
times(rat) ).
tff(tcon_Rat_Orat___Groups_Ominus_249,axiom,
minus(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Rat_Orat___Power_Opower_250,axiom,
power(rat) ).
tff(tcon_Rat_Orat___Num_Onumeral_251,axiom,
numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Ozero_252,axiom,
zero(rat) ).
tff(tcon_Rat_Orat___Rings_Oring_253,axiom,
ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom_254,axiom,
idom(rat) ).
tff(tcon_Rat_Orat___Groups_Oone_255,axiom,
one(rat) ).
tff(tcon_Rat_Orat___Rings_Odvd_256,axiom,
dvd(rat) ).
tff(tcon_Rat_Orat___HOL_Oequal_257,axiom,
cl_HOL_Oequal(rat) ).
tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_258,axiom,
! [A16: $tType] : condit1219197933456340205attice(set(A16)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_259,axiom,
! [A16: $tType] : comple592849572758109894attice(set(A16)) ).
tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_260,axiom,
! [A16: $tType] :
( quickc3360725361186068524ustive(A16)
=> quickc3360725361186068524ustive(set(A16)) ) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_261,axiom,
! [A16: $tType] : bounde4967611905675639751up_bot(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_262,axiom,
! [A16: $tType] : bounde4346867609351753570nf_top(set(A16)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_263,axiom,
! [A16: $tType] : comple6319245703460814977attice(set(A16)) ).
tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_264,axiom,
! [A16: $tType] :
( quickc658316121487927005ustive(A16)
=> quickc658316121487927005ustive(set(A16)) ) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_265,axiom,
! [A16: $tType] : boolea8198339166811842893lgebra(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_266,axiom,
! [A16: $tType] : bounded_lattice_top(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_267,axiom,
! [A16: $tType] : bounded_lattice_bot(set(A16)) ).
tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_268,axiom,
! [A16: $tType] : comple9053668089753744459l_ccpo(set(A16)) ).
tff(tcon_Set_Oset___Quickcheck__Random_Orandom_269,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(set(A16)) ) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_270,axiom,
! [A16: $tType] : semilattice_sup(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_271,axiom,
! [A16: $tType] : semilattice_inf(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Odistrib__lattice_272,axiom,
! [A16: $tType] : distrib_lattice(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_273,axiom,
! [A16: $tType] : bounded_lattice(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Oorder__top_274,axiom,
! [A16: $tType] : order_top(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Oorder__bot_275,axiom,
! [A16: $tType] : order_bot(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Opreorder_276,axiom,
! [A16: $tType] : preorder(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Olattice_277,axiom,
! [A16: $tType] : lattice(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Oorder_278,axiom,
! [A16: $tType] : order(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Otop_279,axiom,
! [A16: $tType] : top(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Oord_280,axiom,
! [A16: $tType] : ord(set(A16)) ).
tff(tcon_Set_Oset___Orderings_Obot_281,axiom,
! [A16: $tType] : bot(set(A16)) ).
tff(tcon_Set_Oset___Groups_Ouminus_282,axiom,
! [A16: $tType] : uminus(set(A16)) ).
tff(tcon_Set_Oset___Lattices_Osup_283,axiom,
! [A16: $tType] : sup(set(A16)) ).
tff(tcon_Set_Oset___Groups_Ominus_284,axiom,
! [A16: $tType] : minus(set(A16)) ).
tff(tcon_Set_Oset___HOL_Oequal_285,axiom,
! [A16: $tType] :
( cl_HOL_Oequal(A16)
=> cl_HOL_Oequal(set(A16)) ) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
condit1219197933456340205attice($o) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
comple592849572758109894attice($o) ).
tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_288,axiom,
quickc3360725361186068524ustive($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_289,axiom,
bounde4967611905675639751up_bot($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_290,axiom,
bounde4346867609351753570nf_top($o) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_291,axiom,
comple6319245703460814977attice($o) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_292,axiom,
boolea8198339166811842893lgebra($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_293,axiom,
bounded_lattice_top($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_294,axiom,
bounded_lattice_bot($o) ).
tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_295,axiom,
comple9053668089753744459l_ccpo($o) ).
tff(tcon_HOL_Obool___Quickcheck__Random_Orandom_296,axiom,
quickcheck_random($o) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_297,axiom,
semilattice_sup($o) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_298,axiom,
semilattice_inf($o) ).
tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_299,axiom,
distrib_lattice($o) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_300,axiom,
bounded_lattice($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder__top_301,axiom,
order_top($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder__bot_302,axiom,
order_bot($o) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_303,axiom,
preorder($o) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_304,axiom,
linorder($o) ).
tff(tcon_HOL_Obool___Lattices_Olattice_305,axiom,
lattice($o) ).
tff(tcon_HOL_Obool___Orderings_Oorder_306,axiom,
order($o) ).
tff(tcon_HOL_Obool___Orderings_Otop_307,axiom,
top($o) ).
tff(tcon_HOL_Obool___Orderings_Oord_308,axiom,
ord($o) ).
tff(tcon_HOL_Obool___Orderings_Obot_309,axiom,
bot($o) ).
tff(tcon_HOL_Obool___Groups_Ouminus_310,axiom,
uminus($o) ).
tff(tcon_HOL_Obool___Lattices_Osup_311,axiom,
sup($o) ).
tff(tcon_HOL_Obool___Groups_Ominus_312,axiom,
minus($o) ).
tff(tcon_HOL_Obool___Heap_Oheap_313,axiom,
heap($o) ).
tff(tcon_HOL_Obool___HOL_Oequal_314,axiom,
cl_HOL_Oequal($o) ).
tff(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_315,axiom,
! [A16: $tType] :
( typerep2(A16)
=> quickc3360725361186068524ustive(ref(A16)) ) ).
tff(tcon_Heap_Oref___Quickcheck__Random_Orandom_316,axiom,
! [A16: $tType] :
( typerep2(A16)
=> quickcheck_random(ref(A16)) ) ).
tff(tcon_Heap_Oref___Heap_Oheap_317,axiom,
! [A16: $tType] : heap(ref(A16)) ).
tff(tcon_Heap_Oref___HOL_Oequal_318,axiom,
! [A16: $tType] : cl_HOL_Oequal(ref(A16)) ).
tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_319,axiom,
! [A16: $tType] :
( quickc3360725361186068524ustive(A16)
=> quickc3360725361186068524ustive(list(A16)) ) ).
tff(tcon_List_Olist___Quickcheck__Random_Orandom_320,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(list(A16)) ) ).
tff(tcon_List_Olist___Heap_Oheap_321,axiom,
! [A16: $tType] :
( heap(A16)
=> heap(list(A16)) ) ).
tff(tcon_List_Olist___HOL_Oequal_322,axiom,
! [A16: $tType] : cl_HOL_Oequal(list(A16)) ).
tff(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_323,axiom,
! [A16: $tType] :
( typerep2(A16)
=> quickc3360725361186068524ustive(array(A16)) ) ).
tff(tcon_Heap_Oarray___Quickcheck__Random_Orandom_324,axiom,
! [A16: $tType] :
( typerep2(A16)
=> quickcheck_random(array(A16)) ) ).
tff(tcon_Heap_Oarray___Heap_Oheap_325,axiom,
! [A16: $tType] : heap(array(A16)) ).
tff(tcon_Heap_Oarray___HOL_Oequal_326,axiom,
! [A16: $tType] : cl_HOL_Oequal(array(A16)) ).
tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_327,axiom,
! [A16: $tType,A17: $tType] :
( ( quickc3360725361186068524ustive(A16)
& quickc3360725361186068524ustive(A17) )
=> quickc3360725361186068524ustive(sum_sum(A16,A17)) ) ).
tff(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_328,axiom,
! [A16: $tType,A17: $tType] :
( ( quickcheck_random(A16)
& quickcheck_random(A17) )
=> quickcheck_random(sum_sum(A16,A17)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_329,axiom,
! [A16: $tType,A17: $tType] :
( ( heap(A16)
& heap(A17) )
=> heap(sum_sum(A16,A17)) ) ).
tff(tcon_Sum__Type_Osum___HOL_Oequal_330,axiom,
! [A16: $tType,A17: $tType] : cl_HOL_Oequal(sum_sum(A16,A17)) ).
tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_331,axiom,
! [A16: $tType] : condit1219197933456340205attice(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_332,axiom,
! [A16: $tType] : bounde4967611905675639751up_bot(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_333,axiom,
! [A16: $tType] : bounde4346867609351753570nf_top(filter(A16)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_334,axiom,
! [A16: $tType] : comple6319245703460814977attice(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_335,axiom,
! [A16: $tType] : bounded_lattice_top(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_336,axiom,
! [A16: $tType] : bounded_lattice_bot(filter(A16)) ).
tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_337,axiom,
! [A16: $tType] : comple9053668089753744459l_ccpo(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_338,axiom,
! [A16: $tType] : semilattice_sup(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_339,axiom,
! [A16: $tType] : semilattice_inf(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_340,axiom,
! [A16: $tType] : distrib_lattice(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_341,axiom,
! [A16: $tType] : bounded_lattice(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__top_342,axiom,
! [A16: $tType] : order_top(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_343,axiom,
! [A16: $tType] : order_bot(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Opreorder_344,axiom,
! [A16: $tType] : preorder(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Olattice_345,axiom,
! [A16: $tType] : lattice(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder_346,axiom,
! [A16: $tType] : order(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Otop_347,axiom,
! [A16: $tType] : top(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Oord_348,axiom,
! [A16: $tType] : ord(filter(A16)) ).
tff(tcon_Filter_Ofilter___Orderings_Obot_349,axiom,
! [A16: $tType] : bot(filter(A16)) ).
tff(tcon_Filter_Ofilter___Lattices_Osup_350,axiom,
! [A16: $tType] : sup(filter(A16)) ).
tff(tcon_Filter_Ofilter___HOL_Oequal_351,axiom,
! [A16: $tType] :
( cl_HOL_Oequal(A16)
=> cl_HOL_Oequal(filter(A16)) ) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_352,axiom,
! [A16: $tType] :
( comple5582772986160207858norder(A16)
=> condit6923001295902523014norder(option(A16)) ) ).
tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_353,axiom,
! [A16: $tType] :
( comple6319245703460814977attice(A16)
=> condit1219197933456340205attice(option(A16)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_354,axiom,
! [A16: $tType] :
( comple592849572758109894attice(A16)
=> comple592849572758109894attice(option(A16)) ) ).
tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_355,axiom,
! [A16: $tType] :
( quickc3360725361186068524ustive(A16)
=> quickc3360725361186068524ustive(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_356,axiom,
! [A16: $tType] :
( lattice(A16)
=> bounde4967611905675639751up_bot(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_357,axiom,
! [A16: $tType] :
( bounded_lattice_top(A16)
=> bounde4346867609351753570nf_top(option(A16)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A16: $tType] :
( comple5582772986160207858norder(A16)
=> comple5582772986160207858norder(option(A16)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_358,axiom,
! [A16: $tType] :
( comple6319245703460814977attice(A16)
=> comple6319245703460814977attice(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_359,axiom,
! [A16: $tType] :
( bounded_lattice_top(A16)
=> bounded_lattice_top(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_360,axiom,
! [A16: $tType] :
( lattice(A16)
=> bounded_lattice_bot(option(A16)) ) ).
tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_361,axiom,
! [A16: $tType] :
( comple6319245703460814977attice(A16)
=> comple9053668089753744459l_ccpo(option(A16)) ) ).
tff(tcon_Option_Ooption___Quickcheck__Random_Orandom_362,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_363,axiom,
! [A16: $tType] :
( semilattice_sup(A16)
=> semilattice_sup(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_364,axiom,
! [A16: $tType] :
( semilattice_inf(A16)
=> semilattice_inf(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_365,axiom,
! [A16: $tType] :
( distrib_lattice(A16)
=> distrib_lattice(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice_366,axiom,
! [A16: $tType] :
( bounded_lattice_top(A16)
=> bounded_lattice(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Owellorder_367,axiom,
! [A16: $tType] :
( wellorder(A16)
=> wellorder(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__top_368,axiom,
! [A16: $tType] :
( order_top(A16)
=> order_top(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__bot_369,axiom,
! [A16: $tType] :
( order(A16)
=> order_bot(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Opreorder_370,axiom,
! [A16: $tType] :
( preorder(A16)
=> preorder(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Olinorder_371,axiom,
! [A16: $tType] :
( linorder(A16)
=> linorder(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Olattice_372,axiom,
! [A16: $tType] :
( lattice(A16)
=> lattice(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder_373,axiom,
! [A16: $tType] :
( order(A16)
=> order(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Otop_374,axiom,
! [A16: $tType] :
( order_top(A16)
=> top(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Oord_375,axiom,
! [A16: $tType] :
( preorder(A16)
=> ord(option(A16)) ) ).
tff(tcon_Option_Ooption___Orderings_Obot_376,axiom,
! [A16: $tType] :
( order(A16)
=> bot(option(A16)) ) ).
tff(tcon_Option_Ooption___Lattices_Osup_377,axiom,
! [A16: $tType] :
( sup(A16)
=> sup(option(A16)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_378,axiom,
! [A16: $tType] :
( heap(A16)
=> heap(option(A16)) ) ).
tff(tcon_Option_Ooption___HOL_Oequal_379,axiom,
! [A16: $tType] : cl_HOL_Oequal(option(A16)) ).
tff(tcon_Predicate_Oseq___HOL_Oequal_380,axiom,
! [A16: $tType] : cl_HOL_Oequal(seq(A16)) ).
tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_381,axiom,
! [A16: $tType] : condit1219197933456340205attice(pred(A16)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_382,axiom,
! [A16: $tType] : comple592849572758109894attice(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_383,axiom,
! [A16: $tType] : bounde4967611905675639751up_bot(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_384,axiom,
! [A16: $tType] : bounde4346867609351753570nf_top(pred(A16)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_385,axiom,
! [A16: $tType] : comple6319245703460814977attice(pred(A16)) ).
tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_386,axiom,
! [A16: $tType] : boolea8198339166811842893lgebra(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_387,axiom,
! [A16: $tType] : bounded_lattice_top(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_388,axiom,
! [A16: $tType] : bounded_lattice_bot(pred(A16)) ).
tff(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_389,axiom,
! [A16: $tType] : comple9053668089753744459l_ccpo(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_390,axiom,
! [A16: $tType] : semilattice_sup(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_391,axiom,
! [A16: $tType] : semilattice_inf(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Odistrib__lattice_392,axiom,
! [A16: $tType] : distrib_lattice(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_393,axiom,
! [A16: $tType] : bounded_lattice(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder__top_394,axiom,
! [A16: $tType] : order_top(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder__bot_395,axiom,
! [A16: $tType] : order_bot(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Opreorder_396,axiom,
! [A16: $tType] : preorder(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Olattice_397,axiom,
! [A16: $tType] : lattice(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder_398,axiom,
! [A16: $tType] : order(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Otop_399,axiom,
! [A16: $tType] : top(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Oord_400,axiom,
! [A16: $tType] : ord(pred(A16)) ).
tff(tcon_Predicate_Opred___Orderings_Obot_401,axiom,
! [A16: $tType] : bot(pred(A16)) ).
tff(tcon_Predicate_Opred___Groups_Ouminus_402,axiom,
! [A16: $tType] : uminus(pred(A16)) ).
tff(tcon_Predicate_Opred___Lattices_Osup_403,axiom,
! [A16: $tType] : sup(pred(A16)) ).
tff(tcon_Predicate_Opred___Groups_Ominus_404,axiom,
! [A16: $tType] : minus(pred(A16)) ).
tff(tcon_Predicate_Opred___HOL_Oequal_405,axiom,
! [A16: $tType] : cl_HOL_Oequal(pred(A16)) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_406,axiom,
bounde4967611905675639751up_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_407,axiom,
bounde4346867609351753570nf_top(assn) ).
tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_408,axiom,
boolea8198339166811842893lgebra(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_409,axiom,
bounded_lattice_top(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_410,axiom,
bounded_lattice_bot(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_411,axiom,
semilattice_sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_412,axiom,
semilattice_inf(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_413,axiom,
distrib_lattice(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_414,axiom,
bounded_lattice(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_415,axiom,
ab_semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_416,axiom,
comm_monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_417,axiom,
semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__top_418,axiom,
order_top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_419,axiom,
order_bot(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Opreorder_420,axiom,
preorder(assn) ).
tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_421,axiom,
monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Olattice_422,axiom,
lattice(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder_423,axiom,
order(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Otop_424,axiom,
top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oord_425,axiom,
ord(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Obot_426,axiom,
bot(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ouminus_427,axiom,
uminus(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osup_428,axiom,
sup(assn) ).
tff(tcon_Assertions_Oassn___Groups_Otimes_429,axiom,
times(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ominus_430,axiom,
minus(assn) ).
tff(tcon_Assertions_Oassn___Power_Opower_431,axiom,
power(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oone_432,axiom,
one(assn) ).
tff(tcon_Assertions_Oassn___Rings_Odvd_433,axiom,
dvd(assn) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_434,axiom,
heap(typerep) ).
tff(tcon_Typerep_Otyperep___HOL_Oequal_435,axiom,
cl_HOL_Oequal(typerep) ).
tff(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_436,axiom,
! [A16: $tType] :
( quickc3360725361186068524ustive(A16)
=> quickc3360725361186068524ustive(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_437,axiom,
! [A16: $tType] :
( preorder(A16)
=> ordere6658533253407199908up_add(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_438,axiom,
! [A16: $tType] : cancel2418104881723323429up_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_439,axiom,
! [A16: $tType] : cancel1802427076303600483id_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_440,axiom,
! [A16: $tType] : cancel_semigroup_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_441,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_442,axiom,
! [A16: $tType] : comm_monoid_diff(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_443,axiom,
! [A16: $tType] : ab_semigroup_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_444,axiom,
! [A16: $tType] : comm_monoid_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_445,axiom,
! [A16: $tType] : semigroup_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Orderings_Opreorder_446,axiom,
! [A16: $tType] :
( preorder(A16)
=> preorder(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_447,axiom,
! [A16: $tType] : monoid_add(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Orderings_Oorder_448,axiom,
! [A16: $tType] :
( preorder(A16)
=> order(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Orderings_Oord_449,axiom,
! [A16: $tType] :
( preorder(A16)
=> ord(multiset(A16)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ominus_450,axiom,
! [A16: $tType] : minus(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___Groups_Ozero_451,axiom,
! [A16: $tType] : zero(multiset(A16)) ).
tff(tcon_Multiset_Omultiset___HOL_Oequal_452,axiom,
! [A16: $tType] :
( cl_HOL_Oequal(A16)
=> cl_HOL_Oequal(multiset(A16)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_453,axiom,
! [A16: $tType,A17: $tType] :
( ( quickc3360725361186068524ustive(A16)
& quickc3360725361186068524ustive(A17) )
=> quickc3360725361186068524ustive(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_454,axiom,
! [A16: $tType,A17: $tType] :
( ( quickc658316121487927005ustive(A16)
& quickc658316121487927005ustive(A17) )
=> quickc658316121487927005ustive(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_455,axiom,
! [A16: $tType,A17: $tType] :
( ( quickcheck_random(A16)
& quickcheck_random(A17) )
=> quickcheck_random(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_456,axiom,
! [A16: $tType,A17: $tType] :
( ( heap(A16)
& heap(A17) )
=> heap(product_prod(A16,A17)) ) ).
tff(tcon_Product__Type_Oprod___HOL_Oequal_457,axiom,
! [A16: $tType,A17: $tType] : cl_HOL_Oequal(product_prod(A16,A17)) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
condit6923001295902523014norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_459,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_460,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_461,axiom,
quickc3360725361186068524ustive(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_462,axiom,
bounde4967611905675639751up_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_463,axiom,
bounde4346867609351753570nf_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_464,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_465,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_466,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_467,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_468,axiom,
bounded_lattice_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_469,axiom,
comple9053668089753744459l_ccpo(product_unit) ).
tff(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_470,axiom,
quickcheck_random(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_471,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_472,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_473,axiom,
distrib_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_474,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_475,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_476,axiom,
order_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_477,axiom,
order_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_478,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_479,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_480,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_481,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Otop_482,axiom,
top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_483,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Obot_484,axiom,
bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_485,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osup_486,axiom,
sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ominus_487,axiom,
minus(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_488,axiom,
heap(product_unit) ).
tff(tcon_Product__Type_Ounit___HOL_Oequal_489,axiom,
cl_HOL_Oequal(product_unit) ).
tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_490,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(heap_ext(A16)) ) ).
tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_491,axiom,
! [A16: $tType] : cl_HOL_Oequal(heap_ext(A16)) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_492,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_493,axiom,
semiri1453513574482234551roduct(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_494,axiom,
euclid5411537665997757685th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_495,axiom,
euclid8789492081693882211th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_496,axiom,
ordere1937475149494474687imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_497,axiom,
euclid3128863361964157862miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_498,axiom,
euclid4440199948858584721cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_499,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_500,axiom,
euclid8851590272496341667cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_501,axiom,
semiri6575147826004484403cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_502,axiom,
strict9044650504122735259up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_503,axiom,
ordere580206878836729694up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_504,axiom,
ordere2412721322843649153imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_505,axiom,
bit_se359711467146920520ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_506,axiom,
linord2810124833399127020strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_507,axiom,
quickc3360725361186068524ustive(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_508,axiom,
strict7427464778891057005id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_509,axiom,
ordere8940638589300402666id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_510,axiom,
euclid3725896446679973847miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_511,axiom,
linord715952674999750819strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_512,axiom,
linord4140545234300271783up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_513,axiom,
bit_ri3973907225187159222ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_514,axiom,
linord181362715937106298miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_515,axiom,
linord8928482502909563296strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_516,axiom,
quickc658316121487927005ustive(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_517,axiom,
semiri3467727345109120633visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_518,axiom,
ordere6658533253407199908up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_519,axiom,
ordere166539214618696060dd_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_520,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_521,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_522,axiom,
cancel2418104881723323429up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_523,axiom,
ring_15535105094025558882visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_524,axiom,
cancel1802427076303600483id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_525,axiom,
linord4710134922213307826strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_526,axiom,
comm_s4317794764714335236cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_527,axiom,
bit_semiring_bits(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_528,axiom,
ordere2520102378445227354miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_529,axiom,
linord6961819062388156250ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_530,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_531,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_532,axiom,
linordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_533,axiom,
ordered_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_534,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_535,axiom,
quickcheck_random(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_536,axiom,
ab_semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_537,axiom,
algebraic_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_538,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_539,axiom,
ab_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_540,axiom,
ordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_541,axiom,
ordered_ring_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_542,axiom,
semiring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_543,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_544,axiom,
semiring_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_545,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_546,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_547,axiom,
comm_semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_548,axiom,
comm_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_549,axiom,
semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_550,axiom,
semidom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_551,axiom,
semidom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_552,axiom,
semiring_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_553,axiom,
semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_554,axiom,
zero_less_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_555,axiom,
comm_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_556,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_557,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_558,axiom,
zero_neq_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_559,axiom,
ordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_560,axiom,
idom_abs_sgn(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_561,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_562,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_563,axiom,
monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_564,axiom,
comm_ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_565,axiom,
monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_566,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_567,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_568,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_569,axiom,
mult_zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_570,axiom,
comm_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_571,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_572,axiom,
neg_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_573,axiom,
ring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_574,axiom,
semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_575,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_576,axiom,
uminus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_577,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_578,axiom,
times(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_579,axiom,
minus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Power_Opower_580,axiom,
power(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_581,axiom,
numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_582,axiom,
zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring_583,axiom,
ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_584,axiom,
idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oone_585,axiom,
one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_586,axiom,
dvd(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_587,axiom,
cl_HOL_Oequal(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_588,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_589,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_590,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_591,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_592,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_593,axiom,
semiri6575147826004484403cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_594,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_595,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_596,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_597,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_598,axiom,
linord2810124833399127020strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_599,axiom,
quickc3360725361186068524ustive(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_600,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_601,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_602,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_603,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_604,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_605,axiom,
linord8928482502909563296strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_606,axiom,
quickc658316121487927005ustive(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_607,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_608,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_609,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_610,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_611,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_612,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_613,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_614,axiom,
ordere2520102378445227354miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_615,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_616,axiom,
linordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_617,axiom,
ordered_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_618,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_619,axiom,
quickcheck_random(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_620,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_621,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_622,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_623,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_624,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_625,axiom,
ordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_626,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_627,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_628,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_629,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_630,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_631,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_632,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_633,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_634,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_635,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_636,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_637,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_638,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_639,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_640,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_641,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_642,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_643,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_644,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_645,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_646,axiom,
mult_zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_647,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_648,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_649,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Otimes_650,axiom,
times(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ominus_651,axiom,
minus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_652,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_653,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_654,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_655,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_656,axiom,
dvd(code_natural) ).
tff(tcon_Code__Numeral_Onatural___HOL_Oequal_657,axiom,
cl_HOL_Oequal(code_natural) ).
tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_658,axiom,
! [A16: $tType] :
( quickcheck_random(A16)
=> quickcheck_random(heap_Time_Heap(A16)) ) ).
tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_659,axiom,
! [A16: $tType] : cl_HOL_Oequal(heap_Time_Heap(A16)) ).
% Helper facts (5)
tff(help_fNot_2_1_U,axiom,
! [P: $o] :
( (P)
| aa($o,$o,fNot,(P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: $o] :
( ~ aa($o,$o,fNot,(P))
| ~ (P) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( X != Y )
| aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y)
| ( X = Y ) ) ).
tff(help_fChoice_1_1_T,axiom,
! [A: $tType,P: fun(A,$o)] :
( aa(A,$o,P,fChoice(A,P))
= ( ? [X7: A] : aa(A,$o,P,X7) ) ) ).
% Free types (1)
tff(tfree_0,hypothesis,
linorder(b) ).
% Conjectures (1)
tff(conj_0,conjecture,
entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),a2),b2)),c),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),a2),c)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),b2),c))) ).
%------------------------------------------------------------------------------